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0, and then discretizing time, into TO , Tl , . . . , where or
TO :=
0,
Tn + ! :=
inf{t > Tn :
I X (t) - X ( Tn ) 1 >
t5X } ,
n ::::
0,
and deal with the resulting random walk (�n) ' where This approximation scheme is accurate, reasonably fast, and very flexible: it is capable of handling a wide variety of problems, with moving as well fixed barriers. For the theory, and detailed comparison with other available methods, see Rogers and Stapleton (1998) ; another approach is due to Ait Sahlia and Lai (1998b). Techniques useful here include continuity corrections for approximations to normality, Edgeworth expansions, and Richardson ex trapolation. as
4 . 8 . 2 Lookback Options
Lookback - or hindsight - options, which we discuss in more detail in §6.3.4 in continuous time, are options that convey the right to 'buy at the low,
4.8 Further Contingent Claim Valuation in Discrete Time
145
sell at the high' - in other words, to eliminate the regret that an investor operating in real time on current, partial knowledge would feel looking back in time with complete knowledge. Again, most of the theory is for continuous time (see e.g. Zhang (1997) , Chapter 12), but a discrete-time framework may be preferred - or needed, if the only prices available are those sampled at certain discrete time-points. Care is obviously needed here, as discretization of time will miss the extremes of the peaks and troughs giving the highs and lows in continuous time. Discrete lookback options have been studied from several viewpoints; see e.g. Heynen and Kat (1995) , Kat (1995) and Levy and Mantion (1997). An interesting approach using duality theory for random walks has been given by AitSahlia and Lai (1998a). 4.8.3 A Three-period Example
Assume we have two basic securities: a risk-free bond and a risky stock. The one-year risk-free interest rate (continuously compounded) is r = 0.06 and the volatility of the stock is 20%. We price calls and puts in a three-period Cox-Ross-Rubinstein model. The up and down movements of the stock price are given by 1 + u = eUvzs. = 1. 1224 and 1 + d = (1 + u ) - l
= e-uvzs.
=
0.8910,
with a = 0.2 and ..::1 = 1/3. We obtain risk-neutral probabilities by (4. 10) p*
=
eT4 - d
u-d
=
0.5584.
We assume that the price of the stock at time t = 0 is 8(0) = 100. To price a European call option with maturity one year (N 3) and strike K = 10) we can either use the valuation formula (4.13) or work our way backwards through the tree. Prices of the stock and the call are given in Figure 4.2 below. One can implement the simple evaluation formulae for the CRR- and the BS-models and compare the values. Figure 4.3 is for 8 = 100, K = 90, r = 0.06, a = 0.2, T = 1 . =
To price a European put, with price process denoted by p(t) , and an Amer ican put, P(t) , (maturity N = 3, strike 100), we can for the European put either use the put-call parity ( 1 . 1 ) , the risk-neutral pricing formula, or work backwards through the tree. For the prices of the American put we use the technique outlined in §4.8. 1 . Prices of the two puts are given in Figure 4.4. We indicate the early exercise times of the American put in bold type. Recall that the discrete-time rule is to exercise if the intrinsic value K - 8(t) is larger than the value of the corresponding European put.
146
4 . Mathematical Finance in Discrete Time
t ) ( the 'future' ) and
lP(A I X (t) , B) = lP(A I X (t) ) .
That is, if you know where you are ( at time t ) , how you got there doesn't matter so far as predicting the future is concerned - equivalently, past and future are conditionally independent given the present. The same definition applied to Markov processes in discrete time. X is said to be strong Markov if the above holds with the fixed time t replaced by a stopping time ( a random variable ) . This is a real restriction of the Markov property in the continuous-time case ( though not in discrete time ) . Perhaps the simplest example of a Markov process that is not strong Markov is given by T
5 . 2 Classes of Processes
1 59
X(t) : = 0 (t ::; r ) , t - r (t � r ) ,
where is an exponentially distributed random variable. Then X is Markov (from the lack of memory property of the exponential distribution) , but not strong Markov (the Markov property fails at the stopping time r) . One must expect the strong Markov property to fail in cases, as here, when 'all the action is at random times'. Another example of a Markov but not strong Markov process is a left-continuous Poisson process - obtained by taking a Poisson process (see below) and modifying its paths to be left-continuous rather than right-continuous. For background and further properties, see e.g. Cox and Miller (1972 ) , Chapter 5, Ethier and Kurtz (1986) , Chapter 4, and Karatzas and Shreve (1991) , §2.6. r
5.2.4 Diffusions
A diffusion is a path-continuous strong Markov process such that for each time t and state x the following limits exist: I-£(t, x ) a 2 (t, x )
: = limh./.O : = limh./.O
-hl lE [(X(t + h ) - X(t) ) I X (t) = -hl lE [(X(t + h ) - X(t)) 2 I X(t)
xl ,
=
x] .
Then 1-£ ( t, x ) is called the drift, a 2 (t, x ) the diffusion coefficient. The term 'diffusion' derives from physical situations involving Brownian motion (§5.3 below) . The mathematics of heat diffusing through a conducting medium (which goes back to Fourier in the early 19th century) is intimately linked with Brownian motion (the mathematics of which is 20th century). The theory of diffusions can be split according to dimension. For one dimensional diffusions, there are a number of ways of treating the theory; see for instance the classic treatments of Breiman (1992) and Doob ( 1953 ) . For higher-dimensional diffusions, there is basically one way: via the stochastic differential equation methodology (or its reformulation in terms of a martin gale problem) . This shows the best way to treat the one-dimensional case: the best method is the one that generalises. It also shows that Markov pro cesses and martingales, as well as being the two general classes of stochastic process with which one can get anywhere mathematically, are also intimately linked technically. We will encounter diffusions largely as solutions of stochas tic differential equations in §5.6; for further background see Grimmett and Stirzaker (2001) , Chapter 13, Revuz and Yor (1991), Chapter 7, and Stroock and Varadhan (1979) .
1 60
5. Stochastic Processes in Continuous Time
5 . 3 Brownian Motion
Brownian motion originates in work of the botanist Robert Brown in 1828. It was introduced into finance by Louis Bachelier in 1900, and developed in physics by Albert Einstein in 1905; see §5.3.4. for background and reference. Trle fact that Brownian motion exists is quite deep, and was first proved by Norbert Wiener (1894-1964) in 1923. In honour of this, Brownian motion is also known as the Wiener process, and the probability measure generating it - the measure IP. on e[O, 1] (one can extend to e[o, 00 ) ) by IP* ( A ) = IP( W. E A)
=
IP({t --+ Wt (w ) } E A)
for all Borel sets A E e[O, 1] - is called Wiener measure. 5 . 3 . 1 Definition and Existence Definition 5.3. 1 . A stochastic process X
= (X(t)h�o is a standard (one dimensional) Brownian motion, BM or BM(IR) , on some probability space (il, :F, IP) , if (i) X (O) = 0 a.s. , (ii) X has independent increments: X (t+u) - X (t) is independent ofa(X(s) : s ::; t) for u � 0, (iii) X has stationary increments: the law of X (t + u) - X(t) depends only on u, (iv) X has Gaussian increments: X (t + u) - X (t) is normally distributed with mean 0 and variance u, X (t + u) - X (t) N(O, u) , (v) X has continuous paths: X(t) is a continuous function of t, i. e. t --+ X (t, w ) is continuous in t for all W E il. rv
The path continuity in (v) can be relaxed by assuming it only a.s.; we can then get continuity by excluding a suitable null-set from our probability space. We shall henceforth denote standard Brownian motion BM(IR) by W (W(t)) (W for Wiener), though B = (B(t)) (B for Brown) is also common. Standard Brownian motion BM(IRd ) in d dimensions is defined by W(t) (W1 (t) , . . . , Wd (t) ) , where Wb . . . , Wd are independent standard Brownian motions in one dimension (independent copies of BM(IR) ) . We turn next to Wiener's theorem, on existence of Brownian motion. The proof, in which we follow Steele (2001), Ch. 3, is a streamlined version of the classical due to Levy in his book of 1948 and Cieselski in 1961 (see below for references) . =
:=
Theorem 5 . 3 . 1 (Wiener) . Brownian motion exists.
5.3 Brownian Motion
161
Covariance. Before addressing existence, we first find the covariance func tion. For s ::::: t, Wt = Ws + (Wt - Ws ) , so as JE(Wt ) = 0,
The last term is JE(Ws )JE(Wt - Ws ) by independent increments, and this is zero, so Cov(W. , Wt) = JE(W; ) =
(s ::::: t) : Cov(Ws , Wt)
s
=
min(s, t) .
A Gaussian process (one whose finite-dimensional distributions are Gaus sian) is specified by its mean function and its covariance function, so among centered (zero-mean) Gaussian processes, the covariance function min(s, t) serves as the signature of Brownian motion. For a ::::: tl < . . . < tn , the joint law of X(td , X(t 2 ) ' . . . ' X (tn ) can be obtained from that of X(td, X (t 2 ) X(h ) , . . . , X (tn) - X (tn - d · These are jointly Gaussian, hence so are X (td , . , X(tn) : the finite-dimensional distributions are multivariate normal. Re call §5.2.2 that the multivariate normal law in n dimensions, Nn (p" E) is specified by the mean vector p, and the covariance matrix E (non-negative definite) . So to check the finite-dimensional distributions of BM stationary independent increments with Wt N(O, t) it suffices to show that they are multivariate normal with mean zero and covariance Cov(Ws , Wt ) = min(s, t) as above. Finite-dimensional Distributions.
.
.
-
-
rv
for t E [0, 1] ) . This gives 00. First, take L 2 [0, 1] , and any complete orthonormal system (cons) (¢n ) on it. Now L 2 is a Hilbert space, under the inner product Construction of HM.
It suffices to construct
BM
t E [0, n] by dilation, and t E [0, 00) by letting n
--+
1 (f , g)
=
J
f (x) g (x) dx
o
so norm
I l f ll
(or
J
fg ) ,
: = (J f2 ) 1/2 ) . By Parseval's identity, 1
J o
00
fg
= "L (f , ¢n ) (g , ¢n) n=O
(where convergence of the series on the right is in L 2 , or in mean square: Il f - L � (f, ¢k ) ¢k ll --+ a as n --+ 00) . Now take, for s, t E [0, 1 ] ' f ( x ) = l [o, s] (x) ,
g (x )
= l [o, t] (x) .
1 62
5 . Stochastic Processes in Continuous Time
Parseval's identity becomes min(s, t) Now take
s
00
=
t
nL=O J cPn(x)dx J cPn(x)dx. 0
0
(Zn) independent and identically distributed N(O, 1 ) , and write t Wt L Zn J cPn(x)dx. n=O 00
=
0
This is a sum of independent random variables. Kolmogorov's theorem on random series ('three-series theorem', Shiryaev (1996) , IV, Theorem 3) says that it converges a.s. if the sum of the variances converges. This is t by above. So the series above converges a.s., and by excluding the exceptional null set from our probability space (as we may), everywhere.
§2,
L:=o (J� cPn(x)dx)2,
=
The Haar System. Define
on [0, ! ) , else. Write o ( t 1 , and for � 1 , express in dyadic form as = + k for a unique j 0, 1 , . . . and k 0, 1 , . . . - 1 . Using this notation for j, k throughout, write - k) (t) (so has support [k/2j , (k + 1 / ) . So if ( f. ) have the same j, 0, while if have different js, one can check that is on half its support, on the other half, so = 0. Also is on [k/2 , (k + 1 ) /2 ] , so 1. Combining:
H )
==
=
n
n
, 2j Hn 2j/2 H(2j t ) 2j ) j _2(jj t+h)/2J H� J HmHn c5mn,
n
=
2j
n,
:=
Hn HmHn 2(j t+h)/2 H� 2j ==
m, n
m
m, n
=
n
HmHn J HmHn
=
and form an orthonormal system, called the Haar system. For complete ness: the indicator of any dyadic interval [k/ (k + 1 /2 ] is in the linear span of the (difference two consecutive and scale) . Linear combinations of such indicators are dense in £2 [0, 1] . Combining: the Haar system is a complete orthonormal system in £2 [0, 1] .
(Hn) Hn
Hns
2j,
) j
(Hn)
The Schauder System. We obtain the Schauder system by integrating the Haar system. Consider the triangular function (or 'tent function')
5 . 3 Brownian Motion
2t
L\(t)
�
1:
(1
�
163
t) on [�, 1] , else.
Llo (t) t, Lll(t) := Ll(t), and define the nth Schauder function Lln Lln(t) : = Ll(2jt - k) ( n 2j + k 1). Note that Lln has support [k/2j, (k + 1)/2j] ( so is 'localized' on this dyadic
Write by
:=
;:::
=
interval, which is small for n, j large). We see that t
Jo H(u)du
and similarly
where
lo
=
=
� Ll(t),
t
Jo Hn(u)du = lnLln(t),
1 and for n ;:::
1,
In �2 Tj/2 (n 2j + k 1). The Schauder system ( Ll n) is again a cons on L2 [0, 1] . =
Theorem 5 . 3 . 2 . For (Zn ) O' independent N(O,
as above,
converges uniformly on Brownian motion.
: = nL=O lnZn Lln(t)
[0, 1] ' a. s . The process
Lemma 5 . 3 . 1 . For Zn independent N(O, 1 ) ,
for some random variable
So for any
1) random variables,
In, Lln
00
Wt
Proof.
;:::
=
X
For x > 1 ,
a
> 1,
C < 00 a. s.
W
( Wt
t E
[0, 1 ] ) is
164
5. Stochastic Processes in Continuous Time
JP ( J Zn J > y'2a log n)
Since E n- a
< 00
for a > 1 ,
:::;
y'2/ 7l' exp{ -a log n}
= y'2/ the Borel-Cantelli lemma gives
JP ( J Zn J > y'2a log n
So
for infinitely many n)
C : = sup � n > 2 y'log n
< 00
=
7l'
n-a .
0.
a.s. o
Proof of Theorem 5.3.2.
1 . Convergence.
Choose J and M � 2 J ; then 00
L
n =M
In J Zn J Lln (t) :::;
00
C L In y'log nLln (t) . M
The right is majorized by 00 2i _ 1
�
C L L Tj/ 2 y'j + 1 Ll 2i +k (t) J k =O (perhaps including some extra terms at the beginning, using 2j + k < 2j + 1 , log n :::; (j + 1) log 2, and Lln ( . ) � 0, so the series is absolutely conver gent). In the inner sum, only one term is non-zero (t can belong to only one dyadic interval [k/2j , (k + 1)/2j ) ) , and each Lln (t) E [0, 1] . So n =
00
C L "21 T j / 2 v'.7+1 \:It E [0, 1] ' j=J and this tends to ° J -+ 00 , so M -+ 00 . So the series E In Zn Lln (t) is absolutely and uniformly convergent, a.s. Since continuity is preserved under uniform convergence and each Lln (t) (so each partial sum) is continuous, Wt is continuous in t. LHS :::; as
2. Covariance.
as
By absolute convergence and Fubini's theorem,
So the covariance is
5 . 3 Brownian Motion
2: n
165
t
s
J ¢ J ¢n
=
m
0
0
min ( s, t) ,
by the Parseval calculation above. Take h , . . . , tm E [0, 1] ; we have to show that (W(t1 ) , . . . , W(t n ) ) is multivariate normal, with mean vector 0 and covari ance matrix ( min ( ti, tj ) ) . The multivariate characteristic function is 3. Joint Distributions.
which by independence of the Zn is
Since each Zn is N(O, 1), the right-hand side is
The sum in the exponent on the right is tj
tk
2: l� ?= 2: Uj Uk .:::l n (tj )Lln (tk ) = ?= 2: Uj Uk 2: J Hn (u)du J Hn (u)du, 00
m
m
m
n =O 3=1 k =1
m
3 =1 k =1
giving
m
00
n=O 0
0
m
2: 2: Uj Uk min(tj , tk) ,
j =1 k = 1 by the Parseval calculation, as (Hn ) are a cons. Combining,
This says that (W(t1 ) , . . . , W(t n ) ) is multinormal with mean 0 and covari ance function min ( tj , t k) required. This completes the construction of BM. o
as
5. Stochastic Processes in Continuous Time
166
Wavelets. The Haar system ( and the Schauder system obtained by integration from it, are examples of wavelet systems. The original function, H or is a mother wavelet, and the 'daughter wavelets' are obtained from it by dilation and translation. The expansion of the theorem is the wavelet expansion of BM with respect to the Schauder system For any f E C[O, l] , we can form its wavelet expansion
Hn),
(.1n)
.1,
(.1n).
)
00
f ( t = n=O cn.1n(t ), with wavelet coefficients Cn. Here Cn are given by k + � ) - "21 [f ( 2ki ) + f (k----v-+ 1)] . cn = f ( � This is the form that gives the .1n ( ) term its correct triangular influence, localized on the dyadic interval [k/2i , (k + 1)/2i ] . Thus for f EM, Cn lnZn, with In, Zn as above. The wavelet construction of BM above is, in modern language, the classical 'broken-line' construction of BM due to Levy in his L
.
=
book of 1948 - the Levy representation of EM using the Schauder system, and extended to general cons by Cieselski in 1961; see McKean (1969) , §1.2 for a textbook account. The earliest expansion of BAI 'Fourier-Wiener ex pansion' - used the trigonometric cons ( Paley and Zygmund 1930-32, Paley, Wiener and Zygmund 1932) ; see Kahane (1985), Preface and §16.3. -
We shall see that Brownian motion is a fractal, and wavelets are a useful tool for the analysis of fractals more generally. For background, see e.g. Holschneider (1995) , §4.4. Note.
For further background, see any measure-theoretic text on stochastic pro cesses. A treatment starting directly from our main reference of measure theoretic results ( Williams (1991)) is Rogers and Williams (1994) , Chapter 1 . The classic is Doob (1953), VIII.2. Excellent modern texts include Karatzas and Shreve (1991) and Revuz and Yor ( 1991) ( see particularly Karatzas and Shreve (1991) , §2.2-4 for construction ) . From the mathematical point of view, Brownian motion owes much of its importance to belonging to all the important classes of stochastic processes: it is ( strong ) Markov, a ( continuous ) martingale, Gaussian, a diffusion, a Levy process etc. From an applied point of view, as its diverse origins Brown's work in botany, Bachelier's in economics, Einstein's in statistical mechanics etc. - suggest, Brownian motion has a universal character, and is ubiquitous both in theory and in applied modeling. The universal nature of Brownian motion as a stochastic process is simply the dynamic counterpart - where we work with evolution in time - of the universal nature of its static counterpart, the normal ( or Gaussian ) distribution - in probability,
5.3 Brownian Motion
167
statistics, science, economics etc. Both arise from the same source, the central limit theorem. This says that when we average large numbers of independent and comparable objects, we obtain the normal distribution (see §2.8) in a static context, or Brownian motion in a dynamic context (see §5. 1 1 for the machinery - weak convergence - needed to handle such limiting results for stochastic processes; cf. §2.6 for its static counterpart) . What the central limit theorem really says is that, when what we observe is the result of a very large number of individually very small influences, the normal distribution or Brownian motion will inevitably and automatically emerge. This explains the central role of the normal distribution in statistics - basically, this is why statistics works. It also explains the central role of Brownian motion as the basic model of random fluctuations, or random noise as one often says. As the word noise suggests, this usage comes from electrical engineering and the early days of radio (see e.g. Wax (1954)) . When we come to studying the dynamics of stochastic processes by means of stochastic differential equations (§5.8 below), we will usually find a 'driving noise' term. The most basic driving noise process is Brownian motion; its role is to represent the 'random buffeting' of the object under study by a myriad of influences which we have no hope of studying in detail - and indeed, no need to. By using the central limit theorem, we make the very complexity of the situation work on our side: Brownian motion is a comparatively simple and tractable process to work with - vastly simpler than the underlying random buffeting whose effect it approximates and represents. The precise circumstances in which one obtains the normal or Gaussian distribution, or Brownian motion, have been much studied (this was the predominant theme in Levy's life's work, for instance) . One needs means and variances to exist (which is why the mean p, and the variance 0' 2 are needed to parametrize the normal or Gaussian family) . One also needs either independence, or something not too far removed from it, such as suitable martingale dependence (for martingale central limit theory, see the excellent book Hall and Heyde (1980)) or Markov dependence (see Ethier and Kurtz (1986) ) . 5.3.2 Quadratic Variation of Brownian Motion
Recall that a N(p" 0' 2 ) distributed random variable � has moment-generating function JvI (t) .IE (exp{tO ) exp P,t + (j 2 t 2 ;=
=
{ � }.
We take p, 0 below; we can recover the general case by adding p, back on. So, for � N(0, (j 2 ) distributed, =
168
5 . Stochastic Processes in Continuous Time
{�0'2e } 1 �0'2t2 ;! ( �0'2 t2 ) 2 (t6) = 1 � 0'2 t 2 ! 0' 4t4 ( t6 ) . 4! 2! As the Taylor coefficients of the moment-generating function are the mo ments (hence the name moment-generating function! ) , JE (e) Var (O = 0'motion 2 ,JE(t;,on4) =JR3, 0'4this, sogivesVar (e) = JE (t;,4 ) - [JE (eW 20'4. For W Brownian JE (W(t)) 0, Var (W(t)) JE« W(t)2) = t, Var (W(t)2) 2t2. In particular, for t > 0 small, this shows that the variance of W(t ) 2 is neg ligible compared with its expected value. Thus, the randomness in W(t ) 2 is negligible compared to its mean for t small. This suggests that if we take a fine enough partition P of [0, tJ a finite set of points 0 = to tl . t n t with grid mesh I I P I I max I ti - ti- Il small enough - then writing LlW(ti ) W(ti ) - W(ti - d and Llti ti - ti-I , n �(LlW(ti))2 i=1 will closely resemble n n n �JE i=1 - ti d t. i=1 « LlW(ti ))2) � i=1 Llti = �(ti This is in fact true: n n � i=1 (LlW(ti))2 -+ � i=1 Llti = t in probability (max I ti ti - l 0) . This limit is called the quadratic variation of W over [0 , tJ : Start with the formal definitions. A partition 1l'n of [0, tJ is a finite set of points tni such that 0 = tno t n = tj the mesh of the tn partition is l 1l'n l maxi( tni - tn , ( i - I » ) , the maximal subinterval length. We consider nested sequences (1l'n ) of partitions (each refines its predecessors by adding further partition points) , with l 1l'n l O. Call (writing ti for tni for M(t) = exp +
+
=
+
+
+0
+
0
=
=
=
=
=
:=
=
-
0, write
with W BM. Then We is Gaussian, with mean 0, variance c - 2 x c2t = t and covariance
min(s, t) = Cov( W(s), W(t)) . Also We has continuous paths, W does. So We has all the properties of Brownian motion. So, We is Brownian motion. It is said to be derived from W by Brownian scaling with scale-factor c > O. Since (W( ut) : t 2 0) ( v'uW(t) : t 2 0) in law, Vu > 0, W is called self-similar with index 1/2 ( Bingham, Goldie, and Teugels (1987), §8.5) . Brownian motion is thus a fractal. A piece of Brownian path, looked at under a microscope, still looks Brownian, however much we 'zoom in and magnify'. Of course, the contrast with a function f with some smoothness is stark: a differentiable function begins to look straight under repeated zoom ing and magnification, because it has a tangent. =
as
=
1 72
5. Stochastic Processes in Continuous Time
Time-Inversion.
Write
Xt
:=
Then X has mean 0 and covariance Cov(Xs , Xt )
=
tW( l /t) .
s t.Cov(B( l / s ) , B( l /t))
=
s t. min( l / s , l /t)
min(t, s ) = min(s, t) . Since X has continuous paths also, as above, X is Brownian motion. We say that X is obtained from W by time-inversion. This property is useful in transforming properties of BM 'in the large' (t � 00 ) to properties 'in the small ' , or local properties (t � 0) . For example, one can translate the law of the iterated logarithm (LIL) from global to local form. Using time-inversion, we see that - as the zero-set of Brownian motion Z := {t 2: 0 Wt = O} is unbounded (contains infinitely many points increas ing to infinity) , it must also contain infinitely many points decreasing to zero. That is, any zero of Brownian motion (e.g., time t 0, as we are choosing to start our BM at the origin) produces an 'echo' - an infinite sequence of zeros at positive times decreasing to zero. How can we hope to graph such a function? (We can't!) How on earth does it manage to escape from zero, when hitting zero at one time, say, forces zero to be hit infinitely many times in any time-interval [ + ] ( > O)? The answer to these questions in volves excursion theory, one of Ito's great contributions to probability theory (1 9 70) . When BM is at zero, it is as likely to leave to the right as to the left, by symmetry - but it will leave, immediately, with probability one. These 'excursions away from zero' - above and below - happen according to a Pois son random measure governing the excursions - the excursion measure - on path-space. As there are infinitely many excursions in finite time-intervals, the excursion measure has infinite mass - it is a-finite but not finite. For details of the form of the Brownian excursion, background, proofs etc., we refer to Rogers and Williams (2000), or Bertoin (1996) , IV. Note however that, far from being pathological as one might at first imagine, the behaviour described above is what one expects of a normal, well-behaved process: the technical term is ' {O} is regular for 0' (Bertoin ( 1996) , IV) , and 'regular' is used to describe good, not bad, behaviour. Since Brownian motion has continuous paths, its zero-set Z is closed. Since each zero is, by above, a limit-point of zeros, Z is a perfect set. The zero-set is also uncountable ('big', in one sense) , but Lebesgue-null - has Lebesgue measure zero ('small', in another sense) . The machinery for measuring the size of small sets such as Z is that of Hausdorff measures. The Hausdorff measure properties of Z have been studied in great detail. The zero-set Z has a fractal structure, which it inherits from that of W under Brownian scaling. The natural machinery for studying the fine detail of the structure of fractals is, as above, that of Hausdorff measures. =
:
=
u
u, u
t
t
5.3 Brownian Motion
1 73
Parameters of Brownian Motion - Estimation and Hypothesis Test ing. If we form J.Lt + aWt - or replace N(O, t ) by N ( J.Lt , at) in the defini
tion of Brownian increments - we obtain a Levy process that has contin uous paths and Gaussian increments, called Brownian motion with drift J.L and diffusion coefficient a, BM(J.L, a ) , rather than standard Brownian mo tion BM = BM(O, 1 ) as above. By above, the quadratic variation of a seg ment of BM(J.L, a ) path on the time-interval [0, t] is a 2 t, a.s. So, if we can observe a Brownian path completely over any time-interval however short, then in principle we can determine the diffusion coefficient a with probability one. In particular, we can distinguish between two different a s - a1 and a2 , say - with certainty. In technical language: the Wiener measures IP.l and IP. 2 representing these two Brownian motions with different as on function space are mutually singular. By contrast, if the two as are the same, the two measures are mutually absolutely continuous. We can then test a hypothesis Ho : J.L J.Lo against an alternative hypothesis Hl : J.L = J.Ll by means of the appropriate likelihood ratio ( LR) . To find the form of the LR, we shall use Girsanov's theorem, which we discuss in §5.7. In practice, of course, we cannot observe a Brownian path exactly over a time-interval: there would be an infinite amount of information, and our ability to sample is finite. So one must use an appropriate discretization - and then we lose the ability to pick up the diffusion coefficient with certainty. Problems of this kind are not only of theoretical interest, but also important in practice. In mathematical finance, when the driving noise is modeled by Brownian motion, the diffusion coefficient is called the volatility, the parameter that describes how sensitive a stock-price is to price-sensitive information ( or economic uncertainty, or driving noise ) . Volatility enters explicitly into the most famous formula of mathematical finance, the Black-Scholes formula. Volatility estimation is of major importance. So too is volatility modeling: alas, in real financial data the assumption of constant volatility is usually untenable for detailed modeling, and one resorts instead to more complicated models, say involving stochas tic volatility, §7.3. For recent work here, see Barndorff-Nielsen and Shephard =
(2001).
5.3.4 Brownian Motion i n Stochastic Modeling
To begin at the beginning: Brownian motion is named after Robert Brown (1773- 1858) , the Scottish botanist who in 1828 observed the irregular and haphazard - apparently random - motion of pollen particles suspended in water. Similar phenomena are observed in gases - witness the familiar sight of dust particles dancing in sunbeams. During the 19th C., it became sus pected that the explanation was that the particles were being bombarded by the molecules in the surrounding medium - water or air. Note that this picture requires three different scales: microscopic ( water or air molecules ) , mesoscopic ( pollen or dust particles ) and macroscopic ( you, the observer ) . These ideas entered the kinetic theory of gases, and statistical mechanics,
1 74
5. Stochastic Processes in Continuous Time
through the pioneering work of Maxwell, Gibbs and Boltzmann. However, some scientists still doubted the existence of atoms and molecules (not then observable directly) . Enter the birth of the quantum age in 1900 with the quantum hypothesis of Max Planck (1858-1947) . Louis Bachelier ( 1870-1946) introduced Brownian motion into the field of economics and finance in his thesis Theorie de la speculation of 1900. His work lay dormant until much later; we will pick up its influence on Ito, Samuelson, Merton and others below. Albert Einstein (1879-1955), in his work of 1905, attacked the problem of demonstrating the existence of molecules, and for good measure estimating Avogadro 's number (c. 6.02 10 2 3 ) experimentally. Einstein realized that what was informative was the mean square displacement of the Brownian particle - its diffusion coefficient, in our terms. This is proportional to time, and the constant D of proportionality, WarWt = Dt, is informative about Avogadro's number (which, roughly, gives the scale factor in going from the microscopic to the macroscopic scale) . This Einstein relation is the prototype of a class of results now known in statistical me chanics as fluctuation-dissipation theorems. All this was done without any proper mathematical underpinning. This was provided by Wiener in 1923, as mentioned earlier. Quantum mechanics emerged in 1925-28 with the work of Heisenberg, Schrodinger and Dirac, and with the 'Copenhagen interpretation' of Bohr, Born and others, it became clear that the quantum picture is both inescapable at the subatomic level and intrinsically probabilistic. The work of Richard P. Feynman (1918-1988) in the late 1940s on quantum electrodynamics (QED) , and his approach to quantum mechanics via 'path integrals', introduced Wiener measure squarely into quantum theory. Feynman's work on quan tum mechanics was made mathematically rigorous by Mark Kac ( 1914-1984) (QED is still problematic!) ; the Feynman-Kac formula (giving a stochastic representation for the solutions of certain PDEs) stems from this. Subsequent developments involve Ito calculus, and we shall consider them in §5.6 below. Suffice it to say here that Ito's work of 1944 picked up where Bachelier left off, and created the machinery needed to use Brownian motion to model stock prices successfully (note: stock prices are nonnegative - pos itive, until the firm goes bankrupt - while Brownian motion changes sign, indeed has lots of sign changes, as we saw above when discussing its zero-set Z). The economist Paul Samuelson in 1965 advocated the Ito model - geo metric Brownian motion - for financial modelling. Then in 1973 Black and Scholes gave their famous formula, and the same year Merton derived it by Ito calculus. Today Ito calculus is a fundamental tool in stochastic modeling generally, and the modelling of financial markets in particular. In sum: wherever we look - statistical mechanics, quantum theory, eco nomics, finance - we see a random world, in which much that we observe is X
5.4 Point Processes
1 75
driven by random noise, or random fluctuations. Brownian motion gives us an invaluable model for describing these, in a wide variety of settings. This is statistically natural. The ubiquitous nature of Brownian motion is the dy namic counterpart of the ubiquitous nature of the normal distribution. This rests ultimately on the Central Limit Theorem ( CLT) - known to physicists as the Law of Errors - and is, fundamentally, why statistics works. 5 .4 Point Processes
Suppose that one is studying earthquakes, or volcanic eruptions. The events of interest are sudden isolated shocks, which occur at random instants, the history of which unfolds with time. Such situations occur in financial set tings also: at the macro-economic level, the events might be stock-market crashes, devaluations etc. At the micro-economic level, they might be indi vidual transactions. In other settings, the events might be the occurrence of telephone calls, insurance claims, accidents or admissions to hospital etc. The mathematical framework needed to handle such situations is that of point processes. A point process is a stochastic process whose realizations are, not paths as above, but counting measures: random measures f..L whose value on each interval I ( or Borel set, more generally is a non-negative integer f..L ( I) . Often, each point may come labeled with some quantity ( the size of the transaction, or of the earthquake on the Richter scale, for instance) , giving what is called a marked point process. We turn below to the simplest and most fundamental point process, the Poisson process, and the simplest way to build it. Stochastic processes with stationary independent increments are called Levy processes ( after the great French probabilist Paul Levy (1886-1971» ; see §5.5 below, and for a modern textbook reference, see Bertoin (1996) . The two most basic prototypes of Levy processes are Poisson processes and Brownian motion (§5.3) . We include below a number of results without proof. For proofs and background, we refer to any good book on stochastic processes, e.g. Dur rett (1999) .
)
5.4. 1 Exponential Distribution
exponential(>')T !P(T ::; t) e->.t t ;::: JE(T) War(T) !P(T > + ti T > t) !P(T >
A random variable is said to have an exponential distribution with rate >., T if 1 for all o. Recall 1/>. and 1/>. 2 . Further important properties are or
=
=
=
-
=
Proposition 5 .4. 1 . (i) Exponentially distributed random variables possess
the 'lack of memory ' property:
s
=
s
).
176
5. Stochastic Processes in Continuous Time
(ii) Let T1 , T2 , Tn be independent exponentially distributed mndom vari ables with pammeters AI , A 2 , . . . , A n resp. Then min { TI ' T2 , , Tn } is exponentially distributed with mte Al + A 2 + . . . + A n · (iii) Let TI , T2 , Tn be independent exponentially distributed mndom vari ables with pammeter A. Then G n = TI +T2 + . . + Tn has a Gamma(n, A) distribution. That is, its density is (At) n - l lP(Gn = t) = Ae-A t for t :::: 0 (n I ) ! •
.
.
•
•
•
•
·
•
.
_
5.4.2 The Poisson Process Definition 5.4. 1 . Let tl , t 2 , . . . t n be independent exponential(A) mndom variables. Let Tn = tl , + . + t n for n :::: 1, To = 0, and define N(s) = .
max { n : Tn ::; s } .
.
Interpretation: Think of t i the time between arrivals of events, then Tn is the arrival time of the nth event and N(s) the number of arrivals by time s. as
Lemma 5.4. 1 . N ( s ) has a Poisson distribution with mean AS.
The Poisson process can also be characterised via Theorem 5.4. 1 . If {N(s) , s :::: O} is a Poisson process, then
(i) N(O) = 0, (ii) N(t + s) - N(s) = Poisson(At) , and (iii) N(t) has independent increments. Conversely, if (i), (ii) and (iii) hold, then {N(s) , s :::: O} is a Poisson process.
The above characterization can be used to extend the definition of the Poisson process to include time-dependent intensities Definition 5.4.2. We say that {N(s) , s :::: O} is a Poisson process with mte
A (r) if (i) N(O) = 0, (ii) N(t + s) - N(s) is Poisson with mean J: A(r)dr, and (iii) N(t) has independent increments. 5.4.3 Compound Poisson Processes
We now associate Li.d. random variables Yi with each arrival and consider S(t)
=
YI
+ . . . + YN( t ) ,
S(t)
=
0
if N(t) = o.
5.4 Point Processes
1 77
Theorem 5.4.2. Let ( Yi ) be i. i. d. and N be an independent nonnegative
integer random variable, and S as above. (i) If lE(N) < 00, then lEeS) = lE(N)lE(Yt ) . (ii) If lE(N 2 ) < 00, then Ware S) = lE(N) War(Yl ) + War(N) (lE(Yl ) ) 2 . (iii) If N = N(t) is Poisson(At), then WareS) = tA(lE(Yl ) ) 2 .
A typical application in the insurance context is a Poisson model of claim arrival with random claim sizes. Again we are interested in bounds on the ruin probability. Let (X (t) ) model the capital of an insurance company. The inital capital > 0, insurance payments arrive continuously at a constant rate c > 0 Xo and claims are received at random times t! , t 2 , . ' " where the amounts paid out at these times are described by nonnegative random variables Yl , Y2 , Assuming ( i ) arrivals according to a Poisson process, ( ii ) Li.d. claim sizes Yi with F(x) = IP(Yl � x) , F(O) = 0 , J1 = Iooo xdF(x) < 00 , and ( iii ) independence of arrival and claim size process, we use a model = u
•
as
X (t)
= u
+
ct - Set) .
.
. .
(5. 1)
Using the above we see that a natural requirement is c > AJ1. We are again interested in the probability of ruin. Write T=
inf {t 2': 0 : X(t) � O} ,
then the probability of ruin is JP ( T < 00) and the probability of ruin before time t is JP(T � t) . Theorem 5.4.3. Let R be the (unique) root of the equation 00
� J eTX (1 - F(x))dx = 1 . o
Then for any t and thus JP ( T
2': 0
s) = JP(X > t) (s , t > 0) , or JP(X > s + t) = JP(X > s)JP(X > t) . Writing F(x) := 1 F(x) (x :::=: 0) for the tail of F , this says that -
F(s + t) = F(s)F(t) (s, t :::=: 0) .
Obvious solutions are for some
A>
0 - the exponential law E(A). Now J(s + t) = f(s)f (t) (s , t :::=: 0)
is a 'functional equation' - the Cauchy functional equation and it turns out that these are the only solutions, subject to minimal regularity (such one sided boundedness, as here - even on an interval of arbitrarily small length!). For details, see e.g. Bingham, Goldie, and Teugels (1987) , § l . l . l . So the exponential laws E(A) are characterized by the lack-of-memory property. Also, the lack-of-memory property corresponds in the renewal con text to the Markov property. The renewal process generated by E(A) is called the Poisson (point) process with rate A, Ppp(>.. ) . So: among renewal processes, the only Markov processes are the Poisson processes. When we meet Levy processes we shall find also: among renewal processes, the only Levy processes are the Poisson processes. It is the lack of memory property of the exponential distribution that (since the inter-arrival times of the Poisson process are exponentially dis tributed) makes the Poisson process the basic model for events occurring 'out of the blue'. For basic background, see e.g. Grimmett and Stirzaker (2001) , §6.8. Excellent textbook treatments are Embrechts, Kliippelberg, and Mikosch (1997) (motivated by insurance and finance applications) and Daley and Vere-Jones (1988) (motivated by the geophysical applications mentioned above) . -
as
5.5 Levy Processes
1 79
5 . 5 Levy Pro cesses
5 . 5 . 1 Distributions The Levy-Khintchine Formula. The form of the general infinitely-divisible distribution was studied in the 1930s by several people (including Kolmogorov and de Finetti) . The final result, due to Levy and Khintchine, is expressed in CF language - indeed, cannot be expressed otherwise. To describe the CF of the general i.d. law, we need three components: (i) a real (called the drift, or deterministic drift), (ii) a non-negative (called the diffusion coefficient, or normal component, or Gaussian component) , (iii) a (positive) measure on JR (or JR \ {O}) for which
a
a
f..L 00 J00 min(1, I x I 2 )f..L (dx) J I x I 2 f..L (dx) J f..L (dx) < 00 ,
-
that is, called the
< 00 ,
Ixl < l Levy measure.
Ix l � l
< 00 ,
The result is (recall §2.1O)
Theorem 5 . 5 . 1 (Levy-Khintchine Formula) . A function ¢ is the char
acteristic function of an infinitely divisible distribution iff it has the form ¢ (u ) =
exp { - !li (u)} (u E JR) ,
where
!li (u) = u � a2 u2 J - eiux iux1C- l, 1 ) (x)f..L (dx) (5.2) 2: 0 f..L . 1. Normal N(f..L , (2). Here 0, f..L = O. 2. Compound Poisson CP(l, Here a = 0, f..L has finite total mass (far from true in general!), say, and f..L Then J� I x l df..L ( x) and J� l xf..L ( dx). 3. Cauchy. See below (under 'Stability') . ia +
for some real
a,
a
+
+
(1
and Levy measure
Examples.
F) .
l
a =
=
a=
IF.
< 00 ,
l
Recall the classical Central Limit Theorem . . . are iid with mean f..L and variance a2, Sn Xk , then X , 2 - nf..L ) /(aVn) is asymptotically standard normal: ( S:�;t � x) q>(x) := vk ] e - �y2 dy (n -+ (0) "Ix E JR. 00
The Central Limit Problem.
(CLT) . If X l , (Sn
1P
=
-+
-
E�
1 80
5. Stochastic Processes in Continuous Time
Self-decomposability. Recall that if, in the central limit problem of §2.1O, we restrict from (two-suffix) triangular arrays ( Xn k ) to (one-suffix) sequences Xn ) , we come to a subclass of the infinite-divisible laws I, called the class of self-decomposable laws SD : SD c I.
(
Stability. Suppose we now restrict to identical distribution as well as inde pendence in SD above. That is, we seek the class of limit laws of random walks Sn E� Xk with ( Xn ) iid - after an affine transformation (centering and scaling) - that is, for all limit laws of ( Sn - n ) / bn . It turns out that the class of limit laws so obtained is the same as the class of laws for which Sn has the same type as Xl - i.e. the same law to within an affine transformation, or a change of location and scale. Thus the type is 'stable' (invariant, un changed) under addition of independent copies, whence such laws are called stable. They form the class S: =
a
S
c
SD
c
I.
It turns out that this class of stable laws can be described explicitly by parameters - four in all, of which two (location and scale, specifying the law within the type) are of minor importance, leaving two essential parameters, called the index E (0, 2)) and the skewness parameter E [- 1, 1)) . To within type, the Levy exponent is
Q (Q
f3 (f3
!Ji(u) l u l O« l - if3sgn(u) tan �1l'Q) for Q =I- 1 (0 Q 1 or 1 Q 2) and !Ji(u) l u l ( l + if3sgn(u) log l u I ) if Q 1 . The Levy measure is absolutely continuous, with density of the form + dx/x H o< x 0 , M(dx) - { cc_dx/ l xI H o< x 0, with c+ , c- � 0 and f3 (c+ - c_ )/(c+ + c_ ) . For proof, see Gnedenko and Kolmogorov (1954) , Feller (1968) , XVIII.6, or Breiman (1992) , §§9.8-1 1 . The case Q 2 (for which f3 drops out) gives the normal/Gaussian case, already familiar. The case Q 1 and f3 0 gives the (symmetric) Cauchy law above. The case Q 1 , f3 =I- 0 gives the asymmetric Cauchy case, which is awkward, and we shall not pursue it. From the form of the Levy exponents of the remaining stable CFs (where the argument u appears only in l u l o< and sgn(u)), we see that, if . . . + with Xi independent copies, =
0) has Laplace transform exp { -av'2S} ( ::::: 0) ; see Rogers and Williams (1994) §I.9 for proof. This is the density of the first-passage time of Brownian motion over a level a > O. The other remarkable case is that of = 3/2, f3 = 0, studied by the Danish astronomer J. Holtsmark in 1919 in connection with the gravitational field of stars - this before Levy's work on stability. The power 3/2 comes from 3 s
a
dimensions and the inverse square law of gravity. 5.5.2 Levy Processes
Suppose we have a process X = (Xt : t ::::: 0) that has stationary indepen dent increments. Such a process is called a Levy process, in honour of their creator, the great French probabilist Paul Levy (1886-1971) . Then for each n = 1 , 2, . . . , displays Xt as the sum of n independent ( by independent increments ) , identi cally distributed ( by stationary increments ) random variables. Consequently, Xt is infinitely divisible, so its CF is given by the Levy-Khintchine formula 5.2. The prime example is: the Wiener process, or Brownian motion, is a Levy process. Poisson Processes. The increment Nt + u - Nu (t, u ::::: 0) of a Poisson pro cess is the number of failures in {u, t + u] ( in the language of renewal theory ) . By the lack-of-memory property of the exponential, this is independent of the failures in [0, u] , so the increments of N are independent. It is also identi cally distributed to the number of failures in [0, t] , so the increments of N are stationary. That is, N has stationary independent increments, so is a Levy process: Poisson processes are Levy processes. We need an important property: two Poisson processes ( on the same fil tration ) are independent iff they never jump together ( a.s. ) . For proof, see e.g . Revuz and Yor ( 1991) , XII. I.
182
5. Stochastic Processes in Continuous Time
The Poisson count in an interval of length t is Poisson pe A t ) (where the rate A is the parameter in the exponential E(A) of the renewal-theory viewpoint), and the Poisson counts of disjoint intervals are independent. This extends from intervals to Borel sets: (i) For a Borel set B, the Poisson count in B is Poisson P(A I B I ) , where 1 . 1 denotes Lebesgue measure; (ii) Poisson counts over disjoint Borel sets are independent. Poisson ( Random ) Measures. If v is a finite measure, call a random mea sure ¢ Poisson with intensity (or characteristic) measure v if for each Borel set B, ¢(B) has a Poisson distribution with parameter v(B) , and for Bl " ' " Bn, ¢(Bl ) , . . . , ¢(Bn) are independent. One can extend to a-finite measures v: if (En) are disjoint with union JR and each v(En) < 00, construct ¢n from v restricted to En and write ¢ for L ¢n ' Poisson Point Processes. With v as above a (a-finite) measure on JR, consider the product measure IL = v dt on JR x [0, 00) , and a Poisson measure ¢ on it with intensity IL. Then ¢ has the form x
¢ = L d( e(t ) , t ) , t �O
where the sum is countable (for background and details, see Bertoin (1996) , §0.5, whose treatment we follow here) . Thus ¢ is the sum of Dirac measures over 'Poisson points' e ( t ) occurring at Poisson times t. Call e = ( e ( t ) : t � 0) a Poisson point process with characteristic measure v,
e
=
Ppp(v) .
For each Borel set B, [ O, t ] ) = card { s � t : e ( s ) E B } is the counting process of B it counts the Poisson points in B - and is Poisson process with rate (parameter) v(B) . All this reverses: starting with an e = ( e ( t ) : t � 0) whose counting processes over Borel sets B are Poisson P(v(B) ) , then - as no point can contribute to more than one count over disjoint sets, disjoint counting processes never jump together, so are inde d(e ( t) , t ) is a Poisson measure with intensity pendent by above, and ¢ : = L t >o N(t, B) : = ¢(B
x
-
IL =
v
x
a
dt .
Note. The link between point processes and martingales goes back to S. Watanabe in 1964. The approach via Poisson point processes is due to K. Ito in 1970 (Proc. 6th Berkeley Symp.); see below, and - in the context of excursion theory - Rogers and Williams (2000) , VI §8. For a monograph treatment of Poisson processes, see Kingman (1993) .
5 . 5 Levy Processes
183
5 . 5 . 3 Levy Processes and the Levy-Khintchine Formula.
We can now sketch the close link between the general Levy process on the one hand and the general infinitely-divisible law given by the Levy-Khintchine formula (L-K) on the other. We follow Bertoin (1996) , § l . l . First, if X (Xt) i s Levy, the law of each Xl is infinitely divisible, so given by lE exp{iuXt } = exp{ -!li (u) } (u E JR) with !Ii a Levy exponent as in (5.2) . Similarly, =
lE exp{iuXt } = exp{ -t!li(u) } (u E JR) , for rational t at first and general t by approximation and cadlag paths. Then is called the Levy exponent, or characteristic exponent, of the Levy process X. Conversely, given a Levy exponent !Ii (u) as in 5.2, construct a Brownian motion as in §5.3, and an independent Poisson point process .1 = ( .1t t 2: 0) with characteristic measure jJ-, the Levy measure in (5.2) . Then X ( t ) at + aBt has CF !Ii
:
I
lE exp{iuXI (t) }
=
{
exp{ -t!lil (t) } = exp -t ( i a + U
.
� a 2 u2 ) } ,
giving the non-integral terms in (5.2) . For the 'large' jumps of .1 , write .1 t(2)
.
·
=
if l { 0 t else. .1
.1 t l 2: 1 ,
Then .1(2) is a Poisson point process with characteristic measure jJ-(2) (dx ) : = l ( l x l 2: 1)jJ-(dx ) . Since J min(l, I x I 2 )jJ-(dx ) < 00 , jJ-(2) has finite mass, so .1(2) , a ppp(jJ-(2) ) , is discrete and its counting process X?)
:=
� .1�2)
(t 2:
0)
s :S;t
is compound Poisson, with Levy exponent
There remain the 'small jumps', .1 t(3)
. _
.
-
if { 0.1t else.
l .1 t l
0, the 'compensated sum of jumps' =
E
184
5. Stochastic Processes in Continuous Time
X; E, 3)
:=
L 1 ( 10 < I Ll s l s� t
0, due to Ito in 1944. This corrects Bachelier's earlier attempt of 1900 (he did not have the factor 8(t) on the right - missing the interpretation in terms of returns, and leading to negative stock prices!) Incidentally, Bachelier's work served as Ito's motivation in introducing Ito calculus. The mathemat ical importance of Ito's work was recognised early, and led on to the work of Doob (1953) , Meyer (1976) and many others (see the memorial volume Ikeda, Watanabe, M., and Kunita (1996) in honour of Ito's eightieth birthday in 1995) . The economic importance of geometric Brownian motion was rec ognized by Paul A. Samuelson in his work from 1965 on (Samuelson (1965)), for which Samuelson received the Nobel Prize in Economics in 1970, and by Robert Merton (see Merton (1990) for a full bibliography) , in work for which he was similarly honoured in 1997. -
5.6 Stochastic Integrals; Ito Calculus
197
The differential equation above has the unique solution
Set) S(O) exp { (JL - �a2 ) t + adW(t)}. =
For, writing
f (t,x) : = exp { (JL - �a2 ) t + ax }, we have ft = (JL - �a2 ) f, fx af, fxx = 0'2 f, and with x Wet), one has dx dW(t), (dx) 2 dt. Thus Ito's lemma gives df(t, Wet)) ftdt fx dW(t) 21 fxx (dW(t)) 2 = f ( ( JL - �a 2 ) dt adW(t) �a 2 dt ) = f(JLdt adW(t)), so f (t, Wet)) is a solution of the stochastic differential equation, and the initial condition f (O, W(O)) S(O) as W(O) = 0, giving existence. For uniqueness, we need the stochastic ( or DolE�ans, or Doleans-Dade ) exponential ( see §5.1O below) , giving Y c(X) exp { X - ! (X)} ( with X a continuous semi-martingale) as the unique solution to the stochastic =
=
=
=
+
=
+
+
+
+
=
=
=
differential equation
1. ( for the general definition and properties see e.g. Jacod and Shiryaev ( 1987 ) , lA, Protter (2004), 11.8, Revuz and Yor ( 1991 ) , IV.3, VULl, Rogers and Williams (2000) , IV. 19 ) ( Incidentally, this is one of the few cases where a stochastic differential equation can be solved explicitly. Usually we must be content with an existence and uniqueness statement, and a numerical algo below. ) Thus above is the rithm for calculating the solution; see stochastic exponential of + Brownian motion with mean ( or drift ) In particular, and variance ( or volatility )
dY(t) Y (t-)dX(t), YeO) =
=
§5. 7 Set) JLt aW(t), 0'2 . JL log Set) log S(O) (JL - �a 2 ) t aW(t) has a normal distribution. Thus Set) itself has a lognormal distribution. This geometric Brownian motion model, and the log-normal distribution that it =
+
+
entails, are the basis for the Black-Scholes model for stock-price dynamics in continuous time, which we study in detail in §6.2.
198
5 . Stochastic Processes in Continuous Time
5 . 7 Stochastic Calculus for Black- Scholes Mo dels
In this section we collect the main tools for the analysis of financial markets with uncertainty modelled by Brownian motions. Consider first independent N(O, 1) random variables Zl , . . . , Zn on a prob ability space (n, F, lP) . Given a vector "I bl , . . . , "In ) , consider a new probability measure P on (n, F) defined by =
As exp{. } > 0 and integrates to 1 , as J expbiZildlP exp{ hn , this is a probability measure. It is also equivalent to lP (has the same null sets) , again as the exponential term is positive. Also =
P(Zi E dzi , i = l , . . . ) , n
This says that if the Zi are independent N(O, 1 ) under lP, they are indepen dent N bi 1) under P. Thus the effect of the change of measure lP -+ P, from the original measure lP to the equivalent measure P, is to change the mean, from 0 (0, . . . , 0) to "I bl , . . . , "In ) . This result extends t o infinitely many dimensions - i.e., from random vectors to stochastic processes, indeed with random rather than deterministic means. Let W (WI , . . . Wd ) be a d-dimensional Brownian motion defined on a filtered probability space (n, F, lP, IF) with the filtration IF satisfying the usual conditions. Let b (t) : 0 � t :s; T) be a measurable, adapted d dimensional process with 1:: "Ii (t) 2 dt < 00 a.s., i = 1, . . . , d, and define the process (L(t) : 0 � t � T) by '
=
=
=
L(t)
�
exp
{! -
�(,) ' dW( , ) -
�
!
l I >(s ) 1 1' ds
}.
(5.5)
Then L is continuous, and, being the stochastic exponential of - J; "I ( s ) ' d W( s ) is a local martingale. Given sufficient integrability on the process "I, L will in fact be a (continuous) martingale. For this, Novikov 's condition suffices:
5 . 7 Stochastic Calculus for Black-Scholes Models
199
We are now in the position to state a version of Girsanov's theorem, which will be one of our main tools in studying continuous-time financial market models. Theorem 5 . 7 . 1 ( Girsanov) . Let , be as above and satisfy Novikov 's con
dition; let L be the corresponding continuous martingale. Define the processes Wi , i = 1 , . . . , d by
Jo t
Wi (t) : = Wi (t) +
,i (S) dS,
(0
�
t � T) , i = 1 , . . . , d.
Then under the equivalent probability measure jp (defined on (n, FT ) ) with Radon-Nikodym derivative djp = L(T) , dIP the process W = ( WI , . . . , Wd) is d-dimensional Brownian motion.
In particular, for ,(t) constant ( ,) , change of measure by introduc ing the Radon-Nikodym derivative exp { -,W(t) - h 2t } corresponds to a change of drift from c to c - ,. If IF (Ft } is the Brownian filtration (ba sically Ft = a(W(s) , 0 � s � t) slightly enlarged to satisfy the usual condi tions) any pair of equivalent probability measures Q IP on F = FT is a Girsanov pair, i.e. =
=
'"
diJ dIP
l
= L (t) Ft
with L defined as above. Girsanov's theorem (or the Cameron-Martin Girsanov theorem ) is formulated in varying degrees of generality, discussed and proved, e.g. in Karatzas and Shreve ( 199 1 ) , §3.5, Protter (2004) , 111.6, Revuz and Yor ( 199 1 ) , VIII, Dothan ( 1990) , §5.4 (discrete time) , § 1 1 .6 (con tinuous time) . Our main application of the Girsanov theorem will be the change of mea sure in the Black-Scholes model of a financial market §6.2 to obtain the risk neutral martingale measure, which will as in the discrete-time case guarantee an arbitrage-free market model and may be used for pricing contingent claims (see § 6 . 1 ) . To discuss questions of attainability and market completeness we will need: Theorem 5 . 7 . 2 (Representation Theorem) . Let M =
(M(t))t>o be a ReLL local martingale with respect to the Brownian filtration (Ft ) . Then
5. Stochastic Processes in Continuous Time
200
t
t M (O) + J H(s)dW(s) , t � 0 o with H ( H(t))t? o a progressively measurable process such that J� H ( S) 2 ds 00, t 0 with probability one. That is, all Brownian local martingales may beas represented as stochastic integrals with respect to Brownian motion (and such are continuous). The following corollary has important economic consequences. Corollary 5.7. 1 . Let G be an FT -measurable random variable 0 T oowith G I 00 ; then there exists a process H as in theorem 5. 7. 2 such lE(I ) that T G lEG + J H (s)dW(s) . M( )
�
=
0, and boundedness of b on compact sets, one can construct a unique solution x by the Picard iteration
t
x ( O ) (t)
:=
Xo , x ( n +l) (t) := Xo +
J b(s, x ( n) (s))ds. o
See e.g. Hale (1969) , Theorem 1.5.3, or any textbook on analysis or differential equations. (The result may also be obtained as an application of Banach's contraction-mapping principle in functional analysis.) Naturally, stochastic calculus and stochastic differential equations contain all the complications of their non-stochastic counterparts, and more besides. Thus by analogy with PDEs alone, we must expect study of SDEs to be complicated by the presence of more than one concept of a solution. The first solution concept that comes to mind is that obtained by sticking to the non-stochastic theory, and working pathwise: take each sample path of a stochastic process as a function, and work with that. This gives the concept of a strong solution of a stochastic differential equation. Here we are given the probabilistic set-up - the filtered probability space in which our SDE arises - and work within it. The most basic results, like their non-stochastic counterparts, assume regularity of coefficients (e.g., Lipschitz conditions) , and construct a unique solution by a stochastic version of Picard iteration. The following such result is proved in Karatzas and Shreve (1991), §5.2. Consider the stochastic differential equation dX(t) = b (t, X (t))dt + a(t, X (t))dW(t) , X ( O) = �,
where b(t, x) is a d-vector of drifts, a(t, x) is a d dispersion matrix, W(t) is an r-dimensional Brownian motion, � is a square-integrable random d-vector independent of W , and we work on a filtered probability space satisfying the usual conditions on which W and � are both defined. Suppose that the coefficients b, a satisfy the following global Lipschitz and growth conditions: x r
I I b(t, x) - b (t, y ) II + l I a ( t , x) - a(t, y ) II
:S
K Il x - yll ,
IIb(t, x) 11 2 + Il a(t, x) 11 2 :S K 2 ( 1 + Il x I1 2 ) , for all t � 0 , x, y E JRd , for some constant K > O . Theorem 5 . 8 . 1 . Under the above Lipschitz and growth conditions,
(i) the Picard iteration X ( O ) ( t )
:=
�,
204
5. Stochastic Processes in Continuous Time
x< n +1 ) (t) : = e +
t
t
J b(s, (s))ds + J O"(s, x O. Now note that the integral is real. Differentiate under the integral sign, and use 1000 X-I sin xdx 1'0/2.) -
=
-
=
II .
=
as
-
0: =
=
=
=
6 . Mathematical Finance in Continuous Time
This chapter discusses the general principles of continuous-time financial mar ket models. In the first section we use a rather general model, which will serve also as a reference in the later chapters. A thorough discussion of the benchmark multi-dimensional Black-Scholes model is the topic of the second section. We discuss the valuation of several standard and exotic contingent claims in the continuous-time Black-Scholes model in the third section. After examining the relation between continuous-time and discrete-time models we close with a discussion of futures and currency markets. 6 . 1 Continuous-time Financial Market Models
6 . 1 . 1 The Financial Market Model
We start with a general model of a frictionless (compare Chapter 1) security market where investors are allowed to trade continuously up to some fixed finite planning horizon T . Uncertainty in the financial market is modelled by a probability space (Q, F, JP) and a filtration IF = (Ft ) O 0, where ,\ denotes Lebesgue measure, such that for all t E [0, T] we have a(t) ' cp(t) = 0 ('no exposure to risk') , ' cp(t) (b(t) - r (t) I d ) ¥= 0 ('non-zero rate') , on A . For an arbitrary constant c > 0 , construct a new trading strategy � . sgn (cp(t) ' (b(t) - r (t) Id ) ) cp(t) 'Ij;(t) =
{
For the discounted value process of the trading strategy 'Ij; we have the fol lowing dynamics: d dV", (t) = L 'lj;i (t)d Si (t) i= l �
t, ", (t) 8, (t) { (b, (t) - r(t))dt + t, u,; (t) dWj (t) } d
=
L Si (t)C Icpi (t) (bi (t) r(t)) 1 dt. i =l
-
So V", (t) � 0, 0 :::; t :::; T, hence 'Ij; is tame. In particular, V", (T) > 0 on a set B E FT with JP(B) > O. This means 'Ij; gives rise to an arbitrage opportunity.
246
6. Mathematical Finance in Continuous Time
To rule out such an arbitrage opportunity we must have that every vector in the kernel of a ' (t, w) must be orthogonal to b(t, w) - r(t, w) l d for a.e. (t, w ) . Thus b(t, w) - r(t, w) l d should belong to (kernel (a' (t, w)) .l.. = range(a(t, w)) , which is precisely the above condition. It then can be shown that 'Y(.) can be selected in Condition (6.5) to be progressively measurable. To prove (ii), recall that by Theorem 6.1 . 1 the existence of an equivalent martingale measure rules out arbitrage. The above Conditions (6.6) and (6.7) ensure that the exponential process L(t)
�
exp
{-/
�(u)' dW(u)
is a martingale, and thus JP * (A)
:=
-�/
�}
Ib(u) II' u ,
0 :5 t :5 T
lE(L(T) lA ) ' A E FT
(6.8)
(6.9)
defines an equivalent probability measure JP* with Radon-Nikodym deriva tive dJP* dJP Ft = L(t) , 0 � t � T. (Occasionally, we shall call L the Girsanov density.) JP* is the risk-neutral equivalent martingale measure . Under JP*,
1
t
W (t)
:=
W (t )
+ f 'Y ( s )ds ,
0�t�T
o
is Brownian motion. The stock-price process dynamics under JP* are
or equivalently for the discounted price processes, dB; (t)
�
B; (t)
(t,
u;;
(t)d W; (t)
),
i
�
1 , . . . , d.
So S is a local JP* -martingale and therefore JP* is an equivalent martingale 0 measure. Remark 6. 2. 2. Observe that Si are the stochastic exponentials of the pro cesses Zi = E7=1 J a j dWj (compare §5.6.1). So under the Novikov condition i
6.2 The Generalized Black-Scholes Model
247
they are martingales. As an illustration of the effect of different choices of " we consider a simple model with two securities 81 , 82 . Let the price-process dynamics be given by
+ +
d81 (t) = 81 (t) (b1 (t)dt lTl (t)dW(t) ) , d82 (t) = 82 (t) (b2 (t)dt lT2 (t)dW(t)) . Assume there i s a process , ( ) (which we might use to define a change of .
measure) such that
Observe that in the numerators we have the excess rate of return of the risky assets over the risk-free rate and in the denominators the volatility of the assets. So these quotients can be interpreted as the risk premium per unit of volatility. As in the theorem above this ratio is often called the market price of risk (compare §8.2) . We can rewrite the above as d81 ( t) = 81 (t) ( (r(t) + ,(t)O" I (t)) dt d82 (t) = 82 (t) ( (r(t) , (t) 0"2 (t) )dt
+
If, for example, "( 0, then ==
d81 (t) = 81 (t) (r(t)dt d8� (t) = 82 (t) (r(t)dt
+ 0"1 (t)dW(t) ) , + 0"2 (t)dW(t) ) .
+ 0"1 (t)dW(t)) , + 0"2 (t)dW(t) ) ,
and we see that Hi = 8d B, i 1 , 2 are (local) martingales under IP , so we are already in our usual risk-neutral setting. If we set "( = 0"2 , we get (doing some stochastic calculus) =
d
[ 8812 (t)(t) ] = (O"I (t) - 0"2 (t)) 8821 (t)(t) dW(t) .
So 8d 82 is a (local) martingale in this setting. Since the attitude towards risk is described by 0"2 (= , ) (the 'risk' in holding the asset 82 ) IP is called a risk-neutral measure with respect to 82 • (Observe that in this example we calculated the drift coefficients bi from the volatilities O"i , the risk-free rate r and the market price of risk , in order to use the original probability measure IP as a martingale measure. Our usual approach is to change the underlying measure using , determined by r, bi , O"d We now turn to the question of market completeness. Again we start by looking at the classical Black-Scholes example.
248
6. Mathematical Finance in Continuous Time
Example. Black-Scholes model (Completeness) . We already know that we have a unique martingale measure JP* (recall 'Y = (b r) l(J' in Gir sanov's transformation) . Given a contingent claim X E L l (fl, F, JP), then X E Ll (fl, F, JP* ) also, and we can define the JP* -martingale -
Using the martingale representation theorem for:. one-dimensional Brownian motion (compare Corollary 5.7.1), we know that under JP*
J h (u)dW(u) . t
M (t)
=
M(a) +
o
Now the JP* -dynamics of S are dS(t) = S (t ) dW ( t ) , (J'
so in fact M (t )
=
M(a) +
t
J c,ol (u)dS(u) , o
with c,ol (t) = hit ) . Using Remark 6.1.2 and defining (J'S(t)
c,oo (t) = M(t) - c,o l (t)S(t) = M ( t ) - h(t) , (J'
we obtain a self-financing replicating trading strategy (in terms of the dis counted values (1 , S)). We have thus shown that X is attainable, and since X was arbitrary the model is complete (in the restricted sense). Turning to the general case, we restrict consideration to contingent claims X with XI B(T) E L l (fl, FT , JP* ) . From Theorem 6.1.5 we know that unique ness of the martingale measure implies that a financial model is complete. In M B S all equivalent martingale measures are given by means of Girsanov's transformation, and hence are characterised by their corresponding Radon Nikodym derivatives with respect to JP. However, these Radon-Nikodym derivatives are characterised by functions 'Y satisfying (use Theorem 6.2.1) bet ) - r(t) l d
(J'(t)"( (t) , a ::; t
::;
T. Keeping this in mind, a characterization of completeness in M B S will involve conditions on the coefficients of the model. More precisely: =
Theorem 6 . 2 . 2 . The following are equivalent:
(i) there exists a unique equivalent martingale measure for the discounted stock price process S;
6 . 2 The Generalized Black-Scholes Model
249
(ii) the generalised Black-Scholes model MB S is complete (in the restricted sense that every contingent claim X with X/B (T) E L1 (fl, F, IP* ) is attainable) ; (iii) equality n = d holds and the volatility matrix a(t, w) is (..\ ® IP ) -a. e.
non-singular.
Remark 6. 2. 3. ( i ) If n � d and the matrix a has full rank, we can reduce the number of stocks by duplicating some of them as (t, w ) -dependent linear combinations of others. ( ii ) We discuss completeness by assuming that trading is restricted to primary securities. Completeness of financial market models with traded put and call options is discussed, e.g. in Bajeux-Besnainou and Rochet ( 1996 ) and Madan and Milne (1993). In particular the question of static hedging is treated in Carr, Ellis, and Gupta (1998) ( compare also Exercise 4.6) . Proof of Theorem 6 . 2 . 2 Again we only sketch the proof. ( i ) {:} ( iii ) follows from the special structure of the Girsanov density used to perform a change of measure in the Brownian setting ( compare §5.8) .
To show ( iii ) => ( ii ) ( which is equivalent to showing ( i ) => ( ii ) , a special case of Theorem 6.1 .5) we repeat the argument given in the classical Black Scholes model. Let IP* be the martingale measure defined by the unique Gir sanov transformation with parameter "((t) (a ' (t) ) - l (b(t) r ( t ) l d ) Using the martingale representation property for the multidimensional Brownian motion ( compare Corollary 5.7. 1), we have for any contingent claim X -
=
M(t)
:=
=
IElP*
M(O)
( B�T) 1 Ft )
+ J h' (u)dW(u) t
=
M(O)
+ L J hi (u)dWi (u) , d
'
t
,= 1 0
+
o
0 :::; t :::; T,
with W(t) = W(t) J; "((u)du . On the other hand, the IP*-dynamics of S are d dSi (t) Si (t) L O'ij (t)dWj (t) . j= l We thus have d t M(t) = M(O) L IPi(u)dSi (u), =
+
with
J
'= 1 0
250
6 . Mathematical Finance in Continuous Time
Using Remark 6.1.2 and defining CPo (t)
=
M (t) - cp' S(t) ,
cp is admissible. Then X is attainable using cp as replicating trading strategy ( the replicating trading strategy for the undiscount�d assets is of course B( t ) · cp(t» , and since X was arbitrary the model is complete ( in the restricted sense ) . To show ( ii ) implies ( i ) we use an argument like that in the first part of
the proof of Theorem 4.3.1 to show that in a complete market the mapping IEq (Y/B (T) ) is constant for any ( integrable ) contingent claim Y ( see Jacka (1992)) . If there are two equivalent martingale measures Q1 and Q 2 with Q1 =I- Q 2 , then we must have an event A E F with Q1 (A) =I- Q 2 (A) . SO we can define a contingent claim Y lA with IEql (lA/B (T» =I- IEq2 ( l A /B (T) ) , which is a contradiction to the above 0 statement. So we must have a unique martingale measure. H : P -t IR given by H(Q)
=
:=
6.2.2 Pricing and Hedging Contingent Claims
We assume now that d = n, that the coefficients of the model satisfy the inte grability conditions, that a is non-singular and that 1' ( . ) in (6.5) exists. Then the financial market model admits a unique equivalent martingale measure JP* with Radon-Nikodym derivative given by the Girsanov transformation L(')
�
exp
{i -
�(u) ' dW(u)
-
�
i dU} ' 11�(u) , , 2
0 ':; ' ':; T
and 'Y(t) = a 1 ( t ) ( b ( t) - r(t) l d ) . By Theorems 6.2. 1 and 6.2.2, the model is free of arbitrage and complete. We call such a model standard. Recall that a contingent claim X is a FT -measurable random variable such that X/ B(T) E L1 (fl, :FT , JP* ) . Using this we get ( we write IE* for IE!po in this section ) : -
Theorem 6.2.3 (Risk-neutral Valuation Formula) . Let M B S be a
standard multi-dimensional Black-Scholes model and X a contingent claim. The arbitmge price process of X is given by the risk-neutml valuation formula
Proof. Since the model is complete, any such contingent claim is attain0 able, and the result follows from Theorem 6 . 1 .4
Of course, for practioners it is of equal importance to find the replicat ing portfolio ( which exists by completeness ) . The martingale representation
6 . 2 The Generalized Black-Scholes Model
251
theorem guarantees the existence, but the explicit construction is rather in volved. We now look at the special case of the classical Black-Scholes model. By the risk-neutral valuation principle the price of a contingent claim X is given by IIx (t) = e { - r ( T - t ) } /E* [ X I Ft ] , with IE " given via the Girsanov density L(t) =
exp
{
(b - r)
( )
1 -;;b-r
- -;;- W (t) - "2
2
}
t .
Furthermore, if X is of form X = ifJ(S(T) ) with a sufficiently integrable function ifJ, then the price process is also given by IIx (t) = F(t, S(t) ) , where F solves the Black-Scholes partial differential equation Ft (t, s)
+
rsFs (t, s)
+
�
u 2 s 2 Fss (t, s) - rF(t, s)
=
0,
F(T, s)
=
ifJ(s) .
(6. 11) (6. 12)
To obtain this partial differential equation we used the Feynman-Kac repre sentation (see §5.7) of the P* -martingale (6.13)
M(t) = exp { - rT} IE* [ ifJ( S (T) ) I Ft l ;
indeed, with G (t, S (t) ) = M (t) we know from Theorem 5.7.3 that G satisfies the partial differential equation (recall that S satisfies the P* SDE dS = :
S(rdt + udW) )
with 'initial' (more accurately, final or terminal) condition G (T, s) = s-rT ifJ(s) . By the risk-neutral valuation principle IIx (t) = e r tM(t) , and so F(t, s) = e rt G (t , s ) , and computing the partial derivatives of F we obtain the repre sentation. Specialising further and considering a European call with strike K and maturity T on the stock S (so ifJ(T) = ( S(T) - K)+ ) , we can evaluate the above expected value (which is easier than solving the Black-Scholes partial differential equation) and obtain: Proposition 6 . 2 . 1 (Black-Scholes Formula) . The Black-Scholes price
process of a European call is given by C (t) = S(t) N (d1 ( S(t) , T - t) )
- K e -r ( T -t) N(d2 ( S(t) , T - t) ) .
The functions d 1 ( s , t) and d2 ( s , t) are given by
(6. 14)
252
6 . Mathematical Finance in Continuous Time _
d 1 ( s , t) -
)
2
log(s/K) + (r + T ) t O"yIit
Ii _
_
d2 (s, t - d1 ( S , t ) - O" y t -
'
log (s /K ) + (r Ii
O"y t
� )t
Observe that we have already deduced this formula as a limit of a discrete time setting in §4.6. To obtain a replicating portfolio we use Ito's lemma to find the dynamics of the JP* -martingale M(t) G (t, S (t) ) : =
dM(t)
O"S (t) Gs (t, S(t) ) d W (t) .
=
U sing this representation, we get in terms of the notation in the Black-Scholes model h(t)
=
O"S (t) Gs (t, S (t) ) ,
which gives for the stock component of the replicating portfolio using the discounted assets !P I (t)
=
Gs (t, S (t) ) B (t) ,
and using the self-financing condition the cash component is !Po (t)
=
G (t, S(t) ) - Gs (t, S (t) ) S(t) .
To transfer this portfolio to undiscounted values we multiply it by the dis count factor, i.e F(t, S(t) ) B (t) G (t, S(t)) and get: =
Proposition 6.2.2. The replicating strategy in the classical Black-Scholes
model is given by
!Po
=
F(t, S(t) ) - Fs (t , S(t) ) S(t) , B (t)
!P I
=
Fs (t, S (t) ) .
In their original paper Black and Scholes (1973) , Black and Scholes used an arbitrage pricing approach (rather than our risk-neutral valuation ap proach) to deduce the price of a European call as the solution of a partial differential equation (we call this the PDE approach) . The idea is as follows: start by assuming that the option price C(t) is given by C (t) f(t, S(t)) for some sufficiently smooth function f JR+ x [0, T] -+ JR. By Ito's formula (Theorem 5.6.2) we find for the dynamics of the option price process (observe that we work under JP so dS S (bdt + O"dW) ) =
:
dC
=
{
=
I
ft (t, S) + fs (t, S) S b + "2 fss (t , S) S 2 0" 2
}
dt + fs SO" dW.
(6. 15)
Consider a portfolio 'l/J consisting of a short position in 'l/JI (t) = fs (t, S(t)) stocks and a long position in 'l/J2 (t) = 1 call and assume the portfolio is self-financing. Then its value process is
6.2 The Generalized Black-Scholes Model
V", (t)
=
-'lfJ1 (t) S(t)
253
+ G(t) ,
and by the self-financing condition we have (to ease the notation we omit the arguments) dV", =
-
=
-
=
'l/h dS
+ dG
fs (Sbdt
( +1 ft
+ SO"dW) + ( ft + fs Sb + � fSS S2 0"2 ) dt + fs SO"dW
)
2 2 2" fss S 0" dt.
So the dynamics of the value process of the portfolio do not have any exposure to the driving Brownian motion, and its appreciation rate in an arbitrage-free world must therefore equal the risk-free rate (for a mathematically precise formulation of this equality we refer the reader to Musiela and Rutkowski (1997), §5.2), i.e. dV", (t)
=
=
rV", (t)dt
( -r fs S
Comparing the coefficients and using G(t) -rSfs
+ rf
=
ft
=
+ rG) dt.
f(t, Set» � ,
we must have
+ 2"1 0"2 S2 fss .
This leads to the Black-Scholes partial differential equation for f (6.11), i.e. ft + r s fs
Since
+ 2"1 0"2 s2 fss - rf
G(T) = (S(T) - K) + we need f(s , T) = (s K) + for all s E 1R + . Note. One point in the justification of
=
O.
to impose the terminal condition
-
the above argument is missing: we have to show that the trading strategy short '!/J l stocks and long one call is self-financing. In fact, this is not true, since '!/J l = '!/Jl (t, Set) ) is dependent on the stock price process. Formally, for the self-financing condition to be true we must have dV", (t)
Now '!/J (t)
=
=
d( '!/Jl (t) S(t) )
'!/J (t , Set» �
+
dG(t)
=
'!/Jl (t)dS(t)
+ dG(t) .
depends on the stock price and so we have
d( '!/Jl (t , S (t» S(t) ) = '!/Jl (t) dS(t)
+ S(t)d'!/Jl (t, Set)) + d ('!/Jl , S) (t) .
We see that the portfolio '!/J is self-financing, if S (t)d'!/Jl (t , Set» �
+ d ('!/Jl , S) (t)
=
O.
It is an exercise in Ito calculus to show that this is not the case (see Exercise 6. 1).
254
6 . Mathematical Finance in Continuous Time
6.2.3 The Greeks
We will now analyse the impact of the underlying parameters in the standard Black-Scholes model on the prices of call and put options. The Black-Scholes option values depend on the ( current ) stock price, the volatility, the time to maturity, the interest rate and the strike price. The sensitivities of the option price with respect to the first four parameters are called the Greeks and are widely used for hedging purposes. We can determine the impact of these parameters by taking partial derivatives. Recall the Black-Scholes formula for a European call (6. 14) : C(O)
=
C(S , T, K, r, a)
=
SN(dI ( S, T) )
-
K e - rT N(d2 (S, T) ) ,
with the functions dI e S , t) and d2 (s, t) given by _
d1 ( s , t ) -
2
log(s/ K) + ( r + T ) t ' avfi.t
. fi. _ log(s/ K) + ( r d2 ( s , t ) - d l ( S, t ) - av t avfi.t _
� )t
One obtains . ac V =
.
e :=
p
:=
r
'
.
=
aa
ac aT ac ar a2 c aS 2
= TKe -rT =
N(d2 ) >
0,
n(dd > °. Sa VT
( As usual N is the cumulative normal distribution function and n is its density. ) From the definitions it is clear that ..:1 - delta - measures the change in the value of the option compared with the change in the value of the underlying asset, V - vega - measures the change of the option compared with the change in the volatility of the underlying, and similar statements hold for - theta - and p - rho ( observe that these derivatives are in line with our arbitrage-based considerations in §1.3). Furthermore, ..:1 gives the number ofrshares in the replication portfolio for a call option ( see Proposition 6.2.2) , so measures the sensitivity of our portfolio to the change in the stock price.
e
6.2 The Generalized Black-Scholes Model
255
The Black-Scholes partial differential equation (6.1 1 ) can be used to ob tain the relation between the Greeks, i.e. ( observe that 8 is the derivative of C, the price of a European call, with respect to the time to expiry T t, while in the Black-Scholes PDE the partial derivative with respect to the current time t appears ) -
1 rC = '2 s 2 a 2 r + rsLl - 8.
Let us now compute the dynamics of the call option's price C(t) under the risk-neutral martingale measure JP* . Using formula (6.15) we find dC(t)
=
rC(t)dt + aN(d1 (S(t) , T - t)) S(t)dW (t) .
Defining the elasticity coefficient of the option's price as ry C (t)
=
Ll (S(t) , T - t)S(t) C(t)
=
N(d 1 (S(t) , T - t)) C(t)
we can rewrite the dynamics as dC(t) = rC(t)dt + m( (t)C(t)dW (t) .
So, as expected in the risk-neutral world, the appreciation rate of the call option equals the risk-free rate r. The volatility coefficient is a ryC , and hence stochastic. It is precisely this feature that causes difficulties when assessing the impact of options in a portfolio. 6.2.4 Volatility
One of the main issues raised by the Black-Scholes formula is the question of modelling the volatility a ( for a discussion of further sensitivities in using the Black-Scholes model see Dana and Jeanblanc (2002)) . Before we can implement the Black-Scholes formula to price options, we have to estimate
a. V
Because the formula is explicit, we can, as noted in §6.2.3, determine the the partial derivative
-
V
=
8C/8a,
finding The important thing to note here is that vega is always positive. This mathe matical consequence of the explicit Black-Scholes formula is reassuring, as it is evident on financial grounds: increasing volatility does not make us more likely to 'win' ( possess an option in the money at expiry ) , but does mean that when we win, we 'win bigger', hence that our option - the right to make this one-way bet - is worth more.
256
6. Mathematical Finance in Continuous Time
Next, since vega is positive, C is a continuous - indeed, differentiable - strictly increasing function of a . Thrning this round, a is a continuous ( differentiable ) strictly increasing function of Cj indeed, 1 aa V = ac aa ' so V = ac ' Informally, one can get the graph of a against C from that of C against a thought of as drawn on tracing paper - by turning the paper over and rotating through a right angle. Formally, one needs explicit proof of the relevant result on inverse functions, for which see a text on analysis ( e.g. Burkill (1962) , §3.9) . Thus the value a = a(C) corresponding to the actual value C C(a) at which call options are observed to be traded in the market can be read off. The value of a obtained in this way is called the implied volatility. First, note that the accuracy with which the ( implied ) volatility a can be inferred from the observed price C depends on the vega V, or rather its reciprocal l/V, by the formulae t5C Vt5a, t5a V - 1 t5Cj if vega is small, i.e. V - 1 is large, small errors t5C in call-option prices are magnified into large errors t5a in the implied volatility, and one loses accuracy. But when vega is large, one gains accuracy: since implied volatility depends on sensitivity analysis, one expects it to work best when prices are sensitive to a! Next, note that determination of implied volatility in this way is well adapted to the familiar and classical Newton-Raphson iteration procedure of numerical analysis. If the transcendental equation C = C(a) is to be solved for a, and an is the current approximation, then the next is - C an - C(an) - C an +l := an - C(Sn) V(an) . C' (an) For background, see a text on numerical analysis, e.g. Jacques and Judd ( 1987) . But the most important thing to note about implied volatility is that here one is at the mercy of the model. One of the great merits of the Black-Scholes analysis is that it uses a simple, parametric model and gives explicit formulae in its conclusions. All parametric models - in statistics, finance or any other field - are wrong in the sense that they represent at best idealised or simpli fied descriptions of reality, and yield correspondingly idealised or simplified conclusions. One cannot expect the geometric Brownian motion model un derlying the Black-Scholes analysis to be any more than a rough description of reality. Thus, one cannot expect the above derivation of implied volatil ity, which depends explicitly on the details of this model, to give anything =
/'oJ
/'oJ
=
6. 2 The Generalized Black-Scholes Model
257
more than a rough description of actual volatility of actual prices ( see Shaw (1998) , Chapter 1 , for the pitfalls arising when using implied volatility in the context of exotic options ) . Since volatility is so important - just as the Black-Scholes analysis is so important - it has been studied in great detail, and much empirical knowledge has been built up about implied volatility in a range of empirical studies. Now volatility is a parameter of the model, and so should be constant - it represents a property of the particular underlying stock in question. However, volatility is observed to vary, most notably with the underlying stock price S - or equivalently, with the strike-price K. In particular, for values of S much higher than the strike price K - that is, for options deeply in the money - the volatility is higher. The resulting upward curve in the graph of against K is called the volatility smile. Since volatil ity is observed in practice to vary with the price of the underlying, which is random - a stochastic process - an obvious way to refine the model is to accept that the volatility is itself random and treat it accordingly. Such stochastic volatility models ( on which we touched in §5.1O) are undoubtedly important, but as they are much more complicated than their counterparts with deterministic volatility, we postpone a treatment of them in the context of incomplete market models to §7.2 For references to the recent literature, see e.g. Campbell, Lo, and MacKinlay (1997) , §§9.3.6, 12.2. 1, 12.2.2, Frey (1997) and Hobson (1998) . Returning to the Black-Scholes model with its geometric Brownian motion dynamics, the other obvious way to estimate volatility is to use, not the option price observed now, but the underlying asset price observed over time. The resulting volatility estimate is called the historic volatility. To estimate it, given data available in practice - finitely many values of the stock price at finitely many time points tl < . . . < tn - one may follow the usual statistical procedure for parameter estimation, the method of maximum likelihood ( see §5.9 and e.g. Rao (1973)) . One can avoid explicit use of a parametric model as simple as geometric Brownian motion: motivated by the important problem of volatility estima tion, there has been an upsurge of recent interest in estimation of diffusion coefficients for diffusion processes. For details, background and references, see e.g. Dohnal (1987) , Florens-Zmirou (1989) and Genon-Catalot and Jacod (1994). We now have two types of volatility estimate: implied and historic. Were the model exact, these two would agree ( approximately, to allow for sampling error ) , as they would simply be different ways of measuring the same thing. However, empirical studies reveal more disagreement between implied and historic volatility estimates than can be accounted for merely by sampling error - which of course reflects, in another guise, the deficiencies of the un derlying model - geometric Brownian motion - in the Black-Scholes analysis. One obvious way to seek to avoid the limitations of this - or any other parametric model is to avoid choice of a parametric model altogether and a
.
258
6 . Mathematical Finance in Continuous Time
resort instead to the methods of non-parametric statistics. Non-parametric volatility estimates have been studied recently, but to discuss them in detail here would take us too far afield. We refer for details to Campbell, Lo, and MacKinlay ( 1997) , § 12.3.4, Ghysels et al. ( 1998 ) , Fouque, Papanicolaou, and Sircar (2000) .
6 . 3 Further Contingent Claim Valuation
We generally consider a Black-Scholes setting. 6.3.1 American Options
As in discrete time ( § 1 .3.3) , these are equivalent to Eu ropean calls - there is no advantage in exercise before expiry. One can also handle extensions, including ) which goes back to McKean in 1 965; ( i ) the perpetual call (T ( ii ) options on stocks paying dividends. For a full account, we refer to Karatzas and Shreve ( 1998) , §2.6. American Puts. We saw in §4.8, in discrete time, the link between Ameri can options and the Snell envelope. We also saw, in the binomial model; how to price American options by backward induction ( dynamic programming ) , and how the stopping and continuation regions S and C - the parts of the tree where it is optimal to exercise now and when it is optimal to hold for exercise later - emerge a corollary of this procedure. We now consider American options ( puts, above ) in more detail, following the excellent sur vey by Myneni ( 1 992) and the textbook account Karatzas and Shreve ( 1998) to which we refer for proofs, background and further references. We content ourselves with formulating and discussing the main results. We stress that, American options are much harder to handle than European ones, the avail able results are much less explicit than in the European case. In particular, explicit pricing formulae - and explicit descriptions of the regions S, C or the optimal stopping boundary S * separating the two - are not known, and are probably inaccessible in general. Early-exercise Decomposition. We use our standard approach, risk neutral valuation. Writing Tt, T for the set of stopping times 7 taking values in [t, T] , the optimal stopping problem is to maximise the risk-neutral expectation of our discounted payoff (K S (7) )+ over all stopping times 7 E Tt, T . Write American Calls.
= (0
,
as
as
as
-
-
6 . 3 Further Contingent Claim Valuation
259
for this optimal expected payoff over all stopping times (optimal expected gain on stopping without foreknowledge of the future) . Then as in discrete time, J(t) is the Snell envelope (least supermartingale majorant of the dis counted payoff process). When the current stock price at time t is x, write P(x, t) sup{lEx [e-r( r- t ) (K - S(7) ) + ] : 7 E 1t,r } for the value function of the option (so P(x, t) =: pet) = ert J(t) , using that Set) = x contains as much relevant information as Ft ) . Then P(x, t) ;::: ( - x) + , with equality if and only if stopping now is optimal: S = { (x, t) E IR+ [0, T] : P(x , t) = - x) + }, :=
K
X
(K [0, T] : P(x, t) > (K - x) + } .
C = { (x, t) E IR+ Then P is continuous (Myneni (1992), Prop. 3.1) , whence the stopping region S is closed and its complement, the continuation region S, is open. Also Set) := {x : (x, t) E S} and C (t) := {x : (x, t) E C} are intervals, the graph of S* (t) := sup{x : x E Set) } is contained i n S , and for each t, S * (t) gives the price level at or below which exercising now is optimal (as the option is a put - giving us the right to sell at price K - the optimal exercise region will clearly be of this form) . The Snell envelope, being a supermartingale, may be (a.s.) decomposed uniquely into a martingale part and a potential part (the Riesz decomposition; see below) , as follows: X
J(t) lE [,-"T (K - S(T)) + IF,j + lE �
[1 ,-""rKl{s(u) <s- (u) } du l
F' '
(Myneni (1992) , Theorem 3.2; the result is due to El Karoui and Karatzas) . Because pet) ert J ( t) , there is an analogous decomposition of the value function, P(x, t) = p(x, t) + e(x, t) , where p(x, t) = lEx [e-r(T - t ) (K - S(T)) + is the price of the corresponding European option, which can be thought of as the intrinsic value of the American option - what it is worth if the possibility of early exercise is removed. Accordingly, the second term =
]
260
6. Mathematical Finance in Continuous Time
is the early-exercise premium - the extra value conferred by the right to ex ercise early. It is proportional to the strike price K, as it should be; it also shows very clearly the role of the indicator process l {s( t) <s* (t)} , which tells us whether or not exercise now is optimal. Because of this, the result is often called the early-exercise decomposition. The proof involves the Ito-Tanaka extension of Ito's formula ( and so local time ) , dual predictable projections ( compensators ) , and the Doob-Meyer decomposition in continuous time. For background on the Riesz decomposition - the probabilistic counterpart of the classical decomposition of F. Riesz of 1930 of a superharmonic function into a harmonic function and potential - see Doob (1984) and Neveu (1975). The theory may be extended in various ways, including perpet ual options, options on dividend-paying stocks, restrictions on borrowing ( of cash ) , restrictions on short selling ( of stocks ) , higher interest rates for bor rowing than for lending etc. For a detailed account, see Karatzas and Kou (1998) . Convergence of American Options. Given a sequence of discrete-time financial models approximating a continuous-time model, the weak conver gence theory of §6.4 below ensures that the price of, say, a European option in the discrete setting converges to that of its continuous-time counterpart. This argument does not, however, apply to the more complicated setting of American options, as here the early exercise possibility introduces a control to which standard weak-convergence theory does not apply. Nevertheless, under mild conditions the prices of American options in the discrete-time setting do indeed converge to their continuous-time counterparts. For details, see Amin and Khanna (1994) . By appropriate choice of topology, weak convergence theory can be ex tended to American options, giving convergence results for price processes, their Snell envelopes and hedging strategies. For formulation and proof of results of this type, see Lamberton and Pages (1990) . Extensions.
6.3.2 Asian Options
Here the payoff is a function of the average price of the underlying between contract time and expiry time. Asian options are widely used in practice for instance, for oil and foreign currencies. The averaging complicates the mathematics, but, e.g., protects the holder against speculative attempts to manipulate the asset price near expiry. For details and references, see e.g. Geman and Yor (1993), and Rogers and Shi (1995) .
6.3 Further Contingent Claim Valuation
261
Pricing options typically involves calcu lating an expectation of the form JE (X + ) , where X is a random variable such as S K (European call) , K - S (European put) or Y K with Y JOT Sdp, (Asian option above) . Rogers and Shi (1995), §3, point out a general method yielding a lower bound, which may be surprisingly accurate. Splitting X into its positive and negative parts X + and X - (X X + X , IXI X + + X - ) , we have X + 2: X, whence
The Rogers-Shi Lower Bound. -
-
=
as
=
=
and also JE (X + I Z) 2: 0, so Thus using iterated conditional expectations and the above, The accuracy of this lower bound may be estimated from the quite general bounds The success of the method hinges on finding a suitable choice of condition ing variable Z for which JE (X I Z) , and hence the expectation of its positive part, is easier to handle than JE (X + ) but the two are still close. In the case of a fixed-strike Asian option considered above, a suitable choice is given by T Z := W (u)du
J a
(a zero-mean Gaussian variable) . For the details (which are quite compli cated) , and numerical illustration, we refer to Rogers and Shi (1995) . We now outline the approach to Asian options of Geman and Yor (Geman and Yor (1993) ; Yor (1992a) , Yor (1992b) , Chap ter 6) in terms of Bessel processes and Laplace transforms. For b 2: 0 and W a standard B M (IR) , consider the SDE The Geman-Yor Method.
dp(t)
=
bdt + 2 -/p (t) dW (t) , Po
=
a 2: O.
This SDE has a solution p = ( p (t)) t > o , or p" , called a Bessel-squared pro cess with index �b - 1, BES Q li� As for the name: the square root R" of a Bessel-squared process p" is a Bessel process BESli , a diffusion whose v :=
262
6. Mathematical Finance in Continuous Time
generator involves the Bessel operator giving rise to the classical Bessel func tions (see Watson (1944) ) . The semigroup of and the transition probability of involve the Bessel function of imaginary argument. The link with mathematical finance arises because the exponential of a drifting Brownian motion (the price process in the lognormal or geometric Brownian motion model of Black-Scholes theory) is a time-changed Bessel process. Specifically (Geman and Yor ( 1993) , Prop. 2.3) :
BESQ Iv v ,
BES6,
exp(W(t) + vi)
�
R"
(i
)
exp(2(W(s) + VS) } dS .
For the fixed-strike Asian option, the payoff is 1 (A (T) - K) + , A(x) := __ x to -
x
jS(U)dU,
to and risk-neutral valuation gives the price at time t as (writing IE for the risk-neutral expectation) Define c(") (x, q)
,�
DE
[ (l
exp{2(W( u ) H u ) } du
-
q
fl
Use of the Markov (indeed, independent increments) property of Brownian motion at time t and Brownian scaling leads to e - r ( T- t ) 4 (t) ( ) Ct, (K) = C ( h , q) , T a2 T - t0 where t a2 a2 2r T to) = 1 , t)j q := K(T h := 4 ( : lJ 2 a 4S(t)
(S )
-
(
v
-
-
[ S(U)dU) .
Thus q is a random variable whose value is known at time t. Negative values of q reflect high stock prices over the time already expired. Geman and Yor (1993) , (3. 1 ) , obtain the following contingent pricing formula (which applies only when q :::; 0):
6 . 3 Further Contingent Claim Valuation
263
This explicit pricing formula - the Geman- Yor formula - somewhat resem bles the classic Black-Scholes formula in structure. It does not involve the volatility a explicitly (only implicitly, via the stock price S(t) and its integral to date, It: S(u)du). From the Geman-Yor formula, one may make comparisons between Asian option prices and those of their European counterparts. Geman and Yor showed that: (i) for v + 1 ;?: 0 (that is, for ;?: 0) the Asian option price is less than that of its European counterpart, for any strike price K; (ii) for v < 0 the reverse is true, at least for K close enough to zero. Note that for options on domestic stock the risk-neutral drift r is the domestic spot rate, which will be positive, so case (i) applies. For options on foreign stock, or currency options, the risk-neutral drift is the difference of the foreign and domestic spot rates, and can have either sign. In particular, the wide spread belief that Asian options are cheaper than European ones - which partly accounts for the popularity of Asian options - is only partly true. As above, our ability to price Asian options is tantamount to our knowl edge of the functions CCv) (h, q). Geman and Yor show that the Laplace trans form (in h) of this function can be identified in terms of confluent hyperge ometric functions (for background on these, see Slater (1960)) . They find that, writing J.L := V2A v 2 , r
oo J
+
e - >'hCCvl (h , q)dh
o
=
fl/C2q) e-x x ! CJL-Vl - 2 ( 1 - 2 q x) ! (JL + v ) + l dx . A(A 2 2v)r( -21 ( J.L v) - 1)
Jo
-
-
-
Numerical inversion of the Laplace transform may now be applied to the right-hand side to obtain numerical values of CCvl (h, q) . The fast Fourier transform (FFT) is applied to this problem in Eydeland and Geman ( 1995) . 6.3.3 Barrier Options
The question of whether or not a particular stock will attain a particular level within a specified period has long been an important one for risk managers. From at least 1967 - predating both CBOE and Black-Scholes in 1973 practitioners have sought to reduce their exposure to specific risks of this kind by buying options designed with such barrier-crossing events in mind. As usual, the motivation is that buying specific options - that is, taking out specific insurance - is a cheaper way of covering oneself against a specific danger than buying a more general one. One-barrier options specify a stock-price level, H say, such that the op tion pays ('knocks in') or not ('knocks out') according to whether or not level H is attained, from below ('up') or above ('down') . There are thus four pos sibilities: 'up and in', 'up and out', 'down and in' and 'down and out'. Since
264
6. Mathematical Finance in Continuous Time
barrier options are path-dependent (they involve the behaviour of the path, rather than just the current price or price at expiry), they may be classified as exotic; alternatively, the four basic one-barrier types above may be regarded as 'vanilla barrier' options, with their more complicated variants, described below, as 'exotic barrier' options. Note that holding both a knock-in option and the corresponding knock-out is equivalent to the corresponding vanilla option with the barrier removed. The sum of the prices of the knock-in and the knock-out is thus the price of the vanilla - again showing the attractive ness of barrier options being cheaper than their vanilla counterparts. A barrier option is often designed to pay a rebate a sum specified in ad vance - to compensate the holder if the option is rendered otherwise worthless by hitting/ not hitting the barrier. We restrict attention to zero rebate here for simplicity. Consider, to be specific, a down-and-out call option with strike K and barrier H (the other possibilities may be handled similarly) . The payoff is (unless otherwise stated min and max are over [0, T] ) as
-
(S(T) - K )+ l {min S( .) ;:::: H } = (S(T) - K) l {s (T);:::: K,min S( .);:::: H } ,
so by risk-neutral pricing the value of the option is DOCK, H IE [e-rT (S(T) - K) l{ s ( T) ;:::: K ,min S (. );:::: H } ] ' where S is geometric Brownian motion, S(t) = po exp{ (/L - � a 2 t) aW(t) } . Write c /L - �a 2 fa; then min S(.) 2: H iff min (ct + W(t) ) 2: a - 1 log (H/po ) . Writing X for X (t) := ct + W(t) drifting Brownian motion with drift c, m , M for its minimum and maximum processes m (t) min{X( s ) : s E [0, tn , M(t) := max{X(s) : s E [0, tn , the payoff function involves the bivariate process (X, m ) , and the option price involves the joint law of this process. Consider first the case c = 0: we require the joint law of standard Brown ian motion and its maximum or minimum, (W, M) or (W, m ) . Taking (W, M) for definiteness, we start the Brownian motion W at the origin at time zero, choose a level b > 0, and run the process until the first-passage time (see Exercise 5.2) r(b) := inf{t 2: 0 : W(t) 2: b} at which the level b is first attained. This is a stopping time, and we may use the strong Markov property for W at time r(b) . The process now begins afresh at level b , and by symmetry the probabilistic properties of its further evolution are invariant under reflection in the level b (thought of as a mirror) . This reflection principle leads to the joint density of (W(t) , M(t) ) as lPo (W(t) E dx , M(t) E dy ) :=
+
:=
-
:=
=
{I
x) 2 v'2ifi3 exp - 2 (2Y - X) /t
2(2 y
-
}
(0 :::;
x
:::;
y) ,
6.3 Further Contingent Claim Valuation
265
a formula due to Levy. ( Levy also obtained the identity in law of the bivariate processes (M(t) - W (t) , M (t ) ) and ( ! W(t) ! , L(t) ) , where L is the local time process of W at zero: see e.g. Revuz and Yor The idea behind the reflection principle goes back to work of Desire Andre in and indeed further, to the method of images of Lord Kelvin then Sir William Thomson, of on electrostatics. For background on this, see any good book on electromagnetism, e.g. Jeans Chapter Levy's formula for the joint density of (W(t) , M (t) ) may be extended to the case of general drift c by the usual method for changing drift, Girsanov's theorem. The general result is
(1991), VI. 2 ). (1824-1907), 1887, (1925), VIII.
1848
IPo (X (t) E
dx, M (t) E dy) 2(2y - x) exp { - (2Y - X) 2 + ex - 1 t } (O :s; x :s; y) . 2t 2 27ft3 See e.g. Rogers and Williams (1994), I, ( 13. 10), or Harrison (1985), §1. 8 . As an alternative to the probabilistic approach above, a second approach to this formula makes explicit use of Kelvin's language - mirrors, sources, sinks; see e.g. Cox and Miller (1972), §5.7. Given such an explicit formula for the joint density of (X ( t ) , M(t)) - or equivalently, (X (t) , m(t) ) - we can calculate the option price by integration. The factor S(T) - K, or S - K, gives rise to two terms, in S and K, while the integrals, involving relatives of the normal density function may be =
� V
-c
2
n,
obtained explicitly in terms of the normal distribution function N - both features familiar from the Black-Scholes formula. Indeed, this resemblance makes it convenient to decompose the price DOCK ,H of the down-and-out call into the ( Black-Scholes ) price of the corresponding vanilla call, CK say, and the knockout discount, K O D K , H say, by which the knockout barrier at H lowers the price:
The details of the integration, which are tedious, are omitted; the result is, writing A - � a2 , := r
where Po is the initial stock price as usual and Cl , C2 are functions of the price P Po and time t T given by ) log (H jpK ) + ( r ± � a 2 ) t Cl , 2 p, t = Vt =
=
2
(
a
is that of the excellent text Musiela and Rutkowski (1997), §9.(the6 , notation to which we refer for further detail) . The other cases of vanilla barrier
266
6. Mathematical Finance in Continuous Time
options, and their sensitivity analysis, are given in detail in Zhang (1997) , Chapter 10. Geman and Yor ( 1996) adapt the Laplace-transform method they devel oped for the Asian options of §6.3.2 to two-barrier options. They show (their Appendix 2) how, and why, the distribution of (X(t) , m(t) , M (t) ) becomes simpler if the fixed time t is replaced by a random time To , independent of X and exponentially distributed with parameter �02 (such exponentially distributed times lead directly to Laplace transforms) . We refer to Geman and Yor ( 1996) for details of their proof, which involves Girsanov's theo rem, Brownian scaling, Wiener-Hopf factorization and excursion theory. The Geman-Yor method gives numerical results for pricing after numerical in version of the Laplace transform. It also gives numerical results for hedging to the same accuracy - which is a rare occurrence for path-dependent op tions. In particular, this approach completely avoids the dangers inherent in Monte-Carlo simulation applied to barrier-crossing problems. Before sim ulating, we must discretize to pass from a diffusion in continuous time to a random walk in discrete time, which reduces calculating probabilities to counting paths. However, the path counts inevitably depend on the specifics of the discretization used (irrelevant to the original problem) , to a degree that leads to unavoidable loss of accuracy. Many other, exotic, types of barrier options have been developed. These include: (i) moving-boundary (or 'floating-barrier') options, where the barrier at con stant level H is replaced by a function H (t) of time; (ii) Asian barrier options, where the boundary crossing is by an average of the price process rather than the price process itself; (iii) barrier options where the barrier operates only for part of the time interval [0, T] : forward start, for an interval of the form [u, TJ , early end ing, for one of the form [0, J window for one of the form [u, ] C [0, T] ; (iv) outside barrier options, where the barrier and the payoff function relate to different underlying assets (for instance, a stock and a currency, etc.). v ,
v
For details, see e.g. Zhang (1997) , Chapter 1 1 . 6.3.4 Lookback Options
In everything we have encountered so far, uncertainty has unfolded with time, and our task has been to make optimal use of the information available to date. For options, at expiry T the investor is in possession of the history of the price evolution over the time interval [0, T] of the option's life, and it may well be - will be - that with hindsight he could have done better than he actually did without hindsight. It is only natural to look back with regret at what inability to foresee the future has cost one. If only one could buy at the low, and sell at the high . . .
6 . 3 Further Contingent Claim Valuation
267
Goldman, Sosin, and Gatto (1979) made two key realizations: i this natural wish on the part of investors would provide a potential mar ket for options providing them with the right to do just that, looking back in time; ii the relevant mathematics - risk-neutral valuation, and the distributional results on drifting Brownian motion and its maximum and minimum, used above in our treatment of barrier options - existed to price such options. It thus became as practical matter to develop, and price, such lookback options as other option types we have already encountered, and this was accordingly done. Of course, the term 'option' is applied here by extension rather than strictly: the options described above will always be exercised! They will accordingly be more expensive than options encountered before. We write S for the price process, Ml, mr for its maximum and minimum over [0, t , M� , v] ' mru , v] for its maximum and minimum over [u , and simi larly with S replaced by other processes X . The two basic types of standard, vanilla lookback options are the lookback call, with payoff
() ()
a
)
]
v], (
LC(T) (S(T) - mf)+ S(T) mf, giving one the right at time ° to buy at the low over [0, T] , and the lookback put, with payoff LP(T) (M,j. - S(T)) + M,j. S(T) , =
:=
:=
-
-
=
giving one the right at time t to sell at the high over [0, TJ . Risk-neutral pricing gives the arbitrage price at time t E [0, T) of the lookback call as LC(t) e-r (T - t ) IE [(S(T) - mf ) IFd e - r ( T - t ) IE (S(T) IFt) e-r ( T - t ) IE ( m f l Ft) II - 12 , say. Now =
-
=
=
as the discounted price process is a JP* -martingale. Also
S(t)
=
Po exp
{ (r
-
� a2 ) t
+
}
a W (t) ,
with W a standard JP*-Brownian motion, so for t :::; u :::; T,
(
S u)
=
( )}
S(t) exp {X(u) X t , -
268
6. Mathematical Finance in Continuous Time
where
1 2
X (t) = ct + O'W(t) , c := r - _ 0' 2
is a drifting ]P* -Brownian motion. Now
S ( . ) = S(t) exp {m;T mf, T = min ' }' [t ,TJ and
' { mtS , mts T } . mTS = mIn ,
By stationary independent increments of X, m;T is independent of Ft and equal in law to m ; , where T := T - t . To evaluate I2 , it thus suffices to find f(s, m ) : = .IE [min { m, s exp {m ; } }) ,
as then
I2 = e-r ( T -t ) f(S(t) , m f ) , giving 12 in terms of data known at time t. Now f(s, m ) - m = .IE [min { m, s exp{m; } } - m] = .IE [ (s exp { m; } - m) l{ m:; �log ( m/ s ) }] .
Since we already have an explicit expression for the density of the minimum of a drifting Brownian motion, we can calculate the expectation on the right by multiplying the function on the right by the density and integrating. As before, the calculations are simplified by reducing to the drift less case c = 0 by Girsanov's theorem. We omit the details of the calculation, which are quite tedious; the result is, writing s for S(t) , m for m f , for T - t as before, and r I , 2 := r ± � 0' 2 ,
T
with a similar formula for the put, P LP(t) . For details, see Musiela and Rutkowski (1997) , Proposition 9.7. 1. The options above are floating strike lookback options, in that the role previously played by the strike K is now played by a random variable. But one can also have fixed strike lookback options, with payoffs max ( M� - K, O ) , max ( K - mf, 0)
6.3 Further Contingent Claim Valuation
269
for calls and puts respectively. One can have partial lookback options, with payoffs max{S{T) - )..mf , 0)
for calls,
max{).. Mi - S{T) , 0)
for puts, where ).. E (0, 1] is the degree of partiality: one only partially elim inates regret - and pays less for the option accordingly. And one can have American lookback options, which may be exercised early. For details and background on such exotic lookback options, see Zhang (1997) , Chapter 12. The theory above supposes that prices are monitored continuously. In practice, prices are only monitored discretely - and as we saw in our treat ment of barrier options, discretization can produce appreciable errors, partic ularly if Monte-Carlo simulation is resorted to. For a contemporary account of discrete lookback ( and barrier) options, see Levy and Mantion (1997). 6 . 3 . 5 Binary Options
A binary ( or digital ) option is a contract whose payoff depends in a discon tinuous way on the terminal price of the underlying asset. The most popular variants are: Cash-or-nothing options. Here the payoffs at expiry of the European call resp. put are given by BCC{T) = C l{s ( T » K} resp. BCP{T) = C l {s( T) 1 ( see §2.6 and Williams ( 1991) , Chapter 13 ) . Example. As an example one can consider pricing a European put option in a continuous-time model. The theorem tells us that we can price the put by pricing the corresponding put in the appropriate discrete-time setting and then computing the limit of the discrete-time model prices ( a European call is then priced by put-call parity ) . The next theorem explores the consequences for trading strategies for contingent claims. Theorem 6.4.2. Let M ( n ) (s( n ) , L( n ) ) be a finite market approximation of a
continuous-time market model M (S, L) , and assume that { s( n ) } is a good se quence of semi-martingales. Assume that the sequence of discrete-time trading strategies cp( n ) converges weakly to a continuous-time trading strategy cpo Then the (discrete-time) gains Gcp(n) and value Vcp( n ) processes converge weakly to their continuous-time counterparts Gcp and Vcp .
The proof is a direct consequence of the definition of the processes and the goodness property of semi-martingales. Of course the real work in applying Theorem 6.4.2 is to show the goodness of the underlying approximating process { s( n ) } and the weak convergence of the trading strategies. Having established these facts, one can apply the the orem to justify 'discrete-time hedging of continuous-time claims'. Suppose we construct a continuous-time hedging strategy cp to hedge a contingent claim X, then the theorem tells us that under appropriate conditions on the ap proximating discrete-time models, we can hedge ( in the limit ) the contingent claim with the induced discrete-time strategies. Given that in practice trad ing is discrete, this is very reassuring for risk managers trying to cover their positions with trading strategies deduced from continuous-time models.
274
6. Mathematical Finance in Continuous Time
6.4.3 Examples of Finite Market Approximations
We consider a Black-Scholes type continuous-time financial market model with a diffusion setting within the framework of §6.2. Let us consider a securities market consisting of d risky stocks and one locally risk-free bond. The d-dimensional vector of stock prices, S, and the bond price, B, are described by the stochastic differential equations 1,
dB(t)
=
B(t)r(S(t) )dt, B (O)
dS(t)
=
b(S(t)) dt + a(S(t) )dW(t) , Si (O)
=
=
Pi
E (0, 00) ,
(6 . 16)
where W (WI , . . . , Wd ) is a d-dimensional standard Brownian motion de fined on a complete probability space (il, :F, JP). We assume that the ap preciation rate vector b JRd � JR, the volatility matrix a JRd � JRd x d and the interest-rate process r JRd � JR are continuous. Furthermore, r is assumed to be bounded and a to be non-singular. Define the covariance matrix a(x ) a(x)a' (x) (aij (x))i.j = I . . . . d . This is necessarily non-negative definite; we assume that it is positive definite: then for some number 8 > 0, =
:
:
:
=
=
(6. 1 7)
In addition, we assume that b and satisfy the uniform Lipschitz condition, that is, there exists a constant C > 0, such that for all x , y E JRd , a
j b(x) - b( y ) j + jj a(x) - a( y ) jj � C j x - yj .
(6.18)
Given this setting, it follows by the results of §6.2 that the market M = M (B, S) described by the above bond process B and the d-dimensional vec tor S with the bond price process used as a numeraire is arbitrage-free and complete ( in the restricted sense). In particular, there exists a unique mar tingale measure given by its Radon-Nikodym derivative L( ' )
�
exp
with ( Recall I d equation
{-i
, ( S (u)) ' dW(u)
0 ", t '" T,
a(x) - I (b(x) - r (x) ld ) . ( 1 , . . . , 1)'.) Observe that L satisfies the stochastic differential , (x)
=
- � i 1 I, ( 8 (u)) II ' dU } ,
dL(t)
=
-1' (S(t) )L(t)dW(t) , L (O) = 1 . ( L i s the stochastic exponential as defined in §5. 1O, compare §6.2.2) . Our =
aim now is to construct a sequence of discrete-time financial markets M ( n ) , sharing the properties of no-arbitrage and completeness, which approximates the above continuous-time financial market.
6.4 Discrete- versus Continuous-time Market Models
275
We do this by utilising a finite Markov-chain approximation scheme ( for an excellent overview of such methods, see Kushner and Dupuis (1992) ) , and start by approximating the stock price processes. Let h > 0 be a scalar approximation parameter. We wish to find a se quence of Markov chains on finite state spaces that converges in distribution ( h -+ 0) to the process defined in (6. 16) over the time interval [0, T] . The basis of the approximation is a discrete-time parameter, finite state space Markov chain {�� , n < } whose 'local properties' are consistent with those given in ( 6 . 1 6 ) . The continuous-time parameter approximating process will be a piecewise constant interpolation of this chain, with appropriately chosen interpolation intervals. To make the above precise, for each h > 0 let {�� , n < } be a discrete parameter Markov chain on a discrete state space Sh E ]Rd . Suppose we have an interpolation interval Ll th ( x ) > 0, and define Llt� Ll th (�� ) . Let sUPx Ll th ( x ) -+ 0 as h -+ 0, but infx Llth ( x ) > 0 for each h > O. Define the difference Ll�� = �� +1 - �� and let IE�, n resp. Cov� , n de note the conditional expectation resp. covariance given {�f , i ::; n, �� x } . Suppose that the chain obeys the following 'local consistency' conditions: sup 1I�� +1 - �� I I -+ 0 (h -+ 0) , as
oo
,
oo
=
=
n
IE�, n (Ll�� ) = b h ( x ) Ll t h ( x ) + 0 (Ll t h ( x ) ) , Cov�, n (Ll�� - IE�, n Ll�� ) = a h ( x ) Llt h ( x ) = a ( x ) Llt h ( x ) + 0 (Llth ( x ) ) . Note that the chain has the 'local properties' of the diffusion process (6. 16) ,
in the following sense:
IE [ S(t + Llt) - S(t) I S(t)] '" b(S(t) ) Ll t, Cov [S(t + Llt) - S(t) I S(t)] '" a (S(t)) Ll t.
We outline the construction of suchT a Markov chain ( the reader should consult He ( 1990) for details ) . Assume = 1 and set h lin , hence our grid . h t ( n ) kl n , k 0 . . . , n. T } WIt n n n I n = {O = t o( ) < t (l ) < . . . < t n( ) k Now construct a sequence of triangular arrays of independent, identically distributed, d-dimensional random vectors (E( n ) ) k::; n , with components E� ; ) uncorrelated, but possibly dependent. Each component takes exactly d + 1 different values and =
�
=
=
=
,
We now define the d-variate, ( d + l ) -nomial approximation process for the diffusion, s ( n ) , as the solution of the stochastic difference equation ( we omit the index n for the grid ) s( n ) (tk + d
=
�
s (n Ek s( n ) (tk ) + b( ) (tk ) ) + O' ( s( n ) (tk ) ) , n yn
(6. 19)
276
6. Mathematical Finance in Continuous Time
and s( n) (0) = S(O) . From (6.19) , we see that the random vector €�n) is used to approximate the random increment of the Brownian motion from time tk to time tk +l , and that s( n) is a Markov chain. We check the consistency requirements: for the drift condition we use that €�n) have expectation vector zero:
for the covariance it follows that
Hence the constructed Markov chain has the local consistency property. Define s( n) (t) = s( n) (tk) for tk � t < tk + l , then the sample paths of S are piecewise constant and only have jumps at tk (of course s( n) (T) = s( n) (T) ) . In the same way, replacing differential equations by difference equations we model bond price processes B( n) as
and the Radon-Nikodym processes L( n) as
We refer the reader to He (1990) for the actual construction of the processes L( n) , starting with the state-price vectors (compare § 1 .4) . In our current set ting, we know from §4.7.2 that the discrete-time markets are free of arbitrage and complete, so L( n) (T) defines the unique martingale measure. We have the following convergence theorem. Theorem 6 . 4 . 3 . The sequence of discrete-time financial markets M ( n) ((B( n) , s( n » ) , L( n » ) is a structure-preserving finite market approxima tion with respect to dynamic completeness of M ( (B, S) , €) . Proof.
We already know that completeness is shared by each of the
M ( n) , s and M . So it only remains to show that x( n) = ((B( n) , s( n » ) , L ( n » ) converges weakly to X = ( (B, S) , L ) . This is done by applying the martingale
6.4 Discrete- versus Continuous-time Market Models
277
central limit theorem ( Ethier and Kurtz (1986) , Chapter 7, and again see He 0 (1990) for the computational details ) . We now show weak convergence of contingent prices. Here a contingent claim is defined to be a random variable of the form Y = for some measurable, bounded function 1] --+ 1R+ Define by the time 0 values of the contingent claim in the continuous-time market and the nth approximating market.
t
g(S) n IIy, II� )
g : Dd[O,
=
Proposition 6.4. 1 . We have the following convergence:
nlim-too IIy(n) IIy. Proof. Since the discount factors 1/ B resp 1/ B( n ) are bounded we only have to show uniform integrability of L( n ) (T) to be able to apply Theorem 6.4.1 . Using Remark 6.4. 1 , it is enough to show uniform boundedness of n) (T) in L2. The continuity of b, and together with the condition on the L(covariance matrix ( 6.17 ) ensure that "Y is continuous. Furthermore, mimicking the uniform boundedness proof of the successive approximations from any =
a
r
existence proof for strong solutions of stochastic differential equations ( see §5.8 or Kloeden and Platen ( 1992), proof of Theorem 4.5.3) , we can show that the sequences are uniformly bounded, i.e. for = 1, 2, . . .
s(n) L2 n sup IE ( Is; n\t k ) 1 2 ) C 00, i = 1 , . , d O� k � n ( recall the initial conditions are Si(O) Pi E [0, 00) , i = 1 , . d and that the components of the tkn ) have second moment equal to 1 ) . So "Y(s( n ) (t k )) is uniformly bounded for n 1 , 2, . . . O�supk � n IE ( l "Yi(s(n)(tk )) 12 ) C'Y 00, i 1 , . . . , d (6.20) for some constant C'Y ' Now using successively the Cauchy-Schwarz inequality, independence of the tkn) and S(n)(t k ), zero correlation between the coeffi cients of the tkn ), and condition (6.20): ::;
0,
where J1 Q is a constant drift ( which may have either sign ) , uQ is a positive vector of volatilities, W(t) is a standard d-dimensional Brownian motion. Solving, one obtains
where 11 . 1 1 denotes the Euclidean norm in JR d . The value of our foreign savings account Bf (t) is Bf (t) Q (t) in domestic currency, and Bf (t) Q (t)/ Bd (t) when discounted by the domestic interest rate. We write this process as Q * : its dynamics are given by d Q * (t)
with solution
=
Q * (t) ( ( J1Q
+ rf - rd )t + uQ dW(t ) ) ,
To avoid arbitrage - between the domestic and foreign bond markets ( the only markets presently in play ) - we need to pass to an equivalent mar tingale measure eliminating the drift term in the dynamics above. We have
286
6. Mathematical Finance in Continuous Time
a d-dimensional noise process, and cannot expect uniqueness of equivalent martingale measures unless there are d independent traded assets available. When this is the case, there exists a unique equivalent martingale measure JP* , called the domestic martingale measure, giving the dynamics as dQ ( t ) Q (t) ( (rd rf ) dt + UQ . d W ( t )) , Q (O) 0 =
>
-
with W a JP* -Brownian motion. This, as its name implies, is the risk-neutral probability measure for an investor reckoning everything in terms of the do mestic currency. Risk-neutral valuation gives the price process of a contingent claim X as 7l"x ( t ) e - rd ( T -t ) JE* (X I Fd · Consider now an agent involved in international trade, wishing to limit his exposure to adverse movements in the exchange-rate process Q. He will seek to purchase an option protecting him against this, in the same way that an agent dealing in a stock S will purchase an option in the stock. Of course, an exchange rate is not a tangible asset in the sense that a stock is; nevertheless, it is possible, and helpful, to treat options on currency in a way closely analogous to how we treat options on stock. For this purpose, consider the forward price at time t of one unit of the foreign currency, to be delivered at the settlement date T, in terms of the domestic currency. It is natural to call this the forward exchange rate, FQ (t , T) say. In terms of the bond-price processes, one has =
a relationship known as interest-rate parity. For, absence of arbitrage requires that the forward exchange premium must be the difference rd rf between the two exchange rates. Currency options may now be constructed, and priced, analogously to options on stock. For example, a standard currency European call option may be constructed, with payoff -
GQ (T)
:=
(Q (T) - K)+ ,
where Q (T) is the spot exchange rate at the option's delivery date and K is the strike price. Finding the arbitrage price of this option is formally the same as finding the price of a futures option, with the forward price of the stock replaced by the forward exchange rate. We already have the solution to this problem, in Black's futures options formula. Adapted to the present context, this yields the following currency options formula, due to Garman and Kohlhagen (1983) and to Biger and Hull (1983) independently. Proposition 6 . 5 . 2 . The arbitrage price of the currency European call option
above is
Exercises
where F ( t )
=
FQ ( t , T) is the forward exchange rate, log ( F/ K) ± � a� t d 1 , 2 (F, t)
:=
aQ Vit
287
'
Exercises
6 . 1 In the setting of the classical deduction of the Black-Scholes differential equation ( compare §6.2.2 ) , show that the trading strategy short fs stocks, long a call is not self-financing. Show furthermore that the additional cost associated with this trading strategy up to time T can be represented by a random variable with zero mean under the risk-neutral martingale measure in the standard Black-Scholes model. 6 . 2 1. Use the put-call parity ( compare §1 .3) to compute the pricing formula for the European put option in the classical Black-Scholes setting: P(O) P(S, T, K, r, a ) K e -rT N( -d2 (S, T)) - SN(-d1 (S, Tn 2. Compute the 'Greeks' for the European put option ( with the help of a computer -programme such as Mathematica, if available ) . 6.3 Use the parameters of the example in §4.8.4 to compute Black-Scholes prices of the European put and call options. Construct discrete ( using the tree ) and continuous ( using the Black-Scholes . 0). An investor with such a utility function U and initial endowment x trading only in the underlying assets So , . . . , Sd forms a dynamic portfolio t,p, whose value at time t is (we need to keep track of the initial endowment) . His objective is to maximise expected utility under the original probability measure of his final wealth at time given that he is allowed to choose his trading strategy t,p from a suitable subset iPa of the set of self-financing trading strategies. We write =
=
=
=
=
=
e-cx ,
Vcp , x (t)
T
U(x)
=
cpsupE<Pa IE [U (Vcp,x (T)) ]
for the maximal utility. Now suppose that a contingent claim X (a sufficiently integrable random variable) is made available for trading with current pur chase price p. We ask the question whether the maximal utility above can be increased by doing so. To find a 'fair' price p for a contingent claim we follow a 'marginal rate of substitution argument' quite commonly used in pricing: p is a fair price for the contingent claim if diverting a little of his funds into it at time zero has a neutral effect on the investor's achievable utility. More precisely, if we set W(8, x, p)
=
sup
¢E<Pa
IE
[U ( Vcp,x -o (T) + P�X) ] ,
then we can state: Definition 7. 1 . 2 . Suppose that for each fixed (x, p) , W (8, x, p) is differen tiable as a function of 8 for 8 = 0, and that there is a unique solution p(x)
of the equation
8W (O , p, x) = O. 88 Then p(x) is the fair option price at time t = O.
Before we can go on, we have to check whether this pricing approach is consistent with the arbitrage price for attainable contingent claims (otherwise the new approach would introduce arbitrage opportunities in the market and
7. 1 Pricing in Incomplete Markets
291
X
we would have to reject it ) . To this end let be an attainable contingent claim with arbitrage price ( found as the unique initial endowment of any self-financing replicating trading strategy ) . Suppose that is offered for p and that the investor buys contingent claims with the amount of cash and so has a remaining endowment of which can be used to set up a dynamic trading strategy cp E CPa . Since is increasing it is optimal for the investor to sell the replicating portfolio short ( so no exposure results from the contingent claim at time and invest 1) optimally. This results 1)). The marginal rate of in attaining an expected utility of substitution is therefore
Po 81p
X
x 8, U x + 8(� U(x + 8(P;
8,
-
T)
-
-
:8 U [X + 8 ( � - 1 )] ! 6=0 = ( � - l) U' (X) , which is only zero for p Po . In general it is very hard to check that the function U is differentiable =
( to justify our rather informal deduction above ) . Fortunately only a 'weak' form of differentiability, quite similar to a 'viscosity solution'-like approach to optimal control problems, suffices ( see Fleming and Soner (1993) and Karatzas (1996) for details on such weak approaches) . For the purpose of our current outline we will take differentiability of for granted, and as sume additionally the existence of a trading strategy cp * E CPa such that = IE Under these assumptions we obtain a general pric ing formula based on Definition Theorem 7. 1 . 1 (Davis) . Suppose that is differentiable at each x E IR +
U
[U (Vcp*,x (T))].
U(x)
and that
7.1. 2 .
U of Definition 7.1.2 is given by U' (x) > O. Then the fair price fi(x) IE l U ' (Vcp* , x (T)) X] p (7.1) UI(X) . •
Proof.
_
-
Assume that cp * E CPa is such that
W(8, x, p)
[U ( Vcp* , x (T) + �X) ] for 8 0, then one can show ( see Davis (1997) , Lemma 2) that !W (0,X, P) ! 6 = 0 :8 IE [U ( Vcp * , x (T) + �X) ] ! 6=0 ' So we have to differentiate IE ( U (.)) with respect to Using a Taylor expan sion of U around Vcp*, x (T) we have =
IE
=
=
O.
IE
[U (VCP* ,X -6(T) + �X)]
[U (Vcp*, x -6(T))] + P�IE lU' (V,CP' .x - 6(T)) X] + ( 0) . IE
0
292
7. Incomplete Markets
Neglecting the (J)-term and differentiating with respect to 15, we obtain 0
:J IE [U (v� x -o (T) � X ) ] 1 0=0 :J IE [U (V� x - o (T))) 1 0=0 + P� IE [U' (V� x (T)) X) . IE [U(V� x (T))) the first-order condition for optimality is -U ' (x) � IE [U' (V� x (T)) X ] 0 +
. .
•
Since U(x)
=
•
•
.
.
+
and from this
.
• .
(7.1) follows.
=
o
We ask for the reader's patience until cation of this pricing approach.
§7. 3 , where we will give an appli
7. 1 . 2 The Esscher Measure
S(t)
t
Let denote the price at time of a non-dividend-paying stock. Assume that there is a stochastic process {X(t)} with stationary and independent increments (Le. a Levy process, §5.5) such that
S(t) S(O) eX(t) , t � O. For 'IjJ a measurable function and Y a random variable, write IE ['IjJ(Y) ; h] IE ['IjJ(Y) ehY] lIE [ehY] . Assume that the moment-generating function M(h, t) IE [e hX(t) ] =
=
=
of X (t) exists; then The process
t) [M(h, 1 )] t. { ehX(t) M(h, 1)- t } t �O M(h,
=
is a positive martingale and can be used to define a change of probability measure, Le. it can be used to define the Radon-Nikodym derivative dQldlP of a new probability measure Q with respect to the original probability mea sure IP; Q is called the Esscher measure of parameter h. Gerber and Shiu introduced the risk-neutral Esscher measure: the Esscher measure of parameter h h * such that the process
(1995)
=
is a martingale. The condition
7. 1 Pricing in Incomplete Markets
293
lE [e - rt S(t); h*] S(O) yields ert lE [eX(t) ; h* ] = lE [ e X(M(ht ) *h,.lX() tt) ] [ M(lM(h+*h*, 1), 1) ] t or l + h* , 1 ) . e r = M(M(h * , l) This equation then uniquely determines the parameter h * . Because for t 0 h , e hX (t) M(h , l ) - t - -lE--=-e[he-XhX-(t()""'t)�] lES(t) [S(t)h] =
=
+
=
2::
_
we have the following:
( For a measurable function and h, k and t real numbers, with t 0, lE [S(t)kg (S(t» ; h] lE [S(t)k; h] lE [g (S(t») ; k + h].
Lemma 7. 1 . 1 Factorization Formula ) . 2::
9
=
Proof.
lE [S( t)k g (S(t)); h] lE [S(t)k g (S(t))e hX (t) M(h, 1 ) - t ] _ lE [ S(t)k+ h g (S(t))] - lE [S (t)h] _ lE [S (t)k+ h ] lE [ S(t) k+ h g ( S (t)) ] - lE [S(t)h] lE [S (t) k +h] = lE [S(t)k ; h] lE [g (S(t)) ; k + h]. =
o
We apply the above technique to the valuation of a European call with maturity T and strike K on the underlying with price dynamics By the risk-neutral valuation principle, we have to calculate
S(t).
lE [e-rT (S(T) K)+ ; h* ] lE [e - rT (S(T) K ) l{ s (T» K } ; h* ] e- rT { lE [S ( T) l { S ( T » K } ; h* ] K lE [ l { S ( T » K } ; h* ] }. -
=
=
-
-
To evaluate the first term, we apply the factorization formula with k and = l{ x> K} and get
h* g (x)
=
1,
h
=
294
7. Incomplete Markets
IE [ S( T) I {S( T » K} ; h * J
= IE [S( T) ; h *] IE [1{S( T » K} ; h * + 1 J = IE [e - rTS ( T) ; h * J erT JP [S(T)
>
K; h * + 1]
>
= S(O)erT JP [S(T) K; h* + 1] , where we used the martingale property of e- r t S (t) under the risk-neutral Esscher measure for the last step. Now the pricing formula for the European call becomes S(O)JP [S ( T)
>
K; h * + 1] - e - rT KJP [S( T)
>
K; h *] .
For X (t) a Brownian motion, the above formula recovers the Black-Scholes pricing formula. It is possible to generalise the above approach to the multi-asset case. Let X (t) = (X l ( t) , . . . , Xd ( t) ) ' . Then M ( z , t) = IE e zf X (t ) , z E IRd is the moment-generating function of X (t) . Assuming that { X ( t) } t >o is a Levy process (has stationary independent increments: §5.5) , we have
[
For h = (h 1 ' . . . ' hd ) E
]
M ( z , t) = [ M ( z, l) ] t , t ::::: O. IRd for which M (h , 1 ) exists the e hf X (t) M (h , l ) - t t�O
{
}
positive martingale
can be used to define a new measure, the Esscher measure of parameter vector h. Again h * is defined as the parameter h = h * such that e-r t Sj (t) , j = I , . . . , d are martingales. We can rewrite these conditions as er =
M (lj + h * , l) . , = l, . . . M (h * , l ) )
, d,
where I j = (0, . . . , 0 , 1 , 0, . . . , 0) with the jth coordinate being l . As in the one-dimensional case we have a factorization formula. For k = (k1 , . . . , kd )' let S( t)k = S l ( t) k1 . . . Sd ( t) kJ , then IE [S (t) k g (S(t) ) ; hJ = IE [ S( t) k ; hJ IE [g(S( t) ) ; k + h] . Remark 1. 1 . 1 . (i) The choice of the Esscher measure may be justified by a utility-maximising argument for a representative agent along lines similar to those leading to the pricing formula in §7. l .2. However, the class of possible trading strategies for the agent is very restrictive in this case (see Gerber and Shiu ( 1995) for details) . (ii) The Esscher measure offers a very attractive way to find at least one equivalent martingale measure in many incomplete market situations (see for instance its use in the hyperbolic Levy model Eberlein and Keller ( 1995) ) . X
X
7.2 Hedging in Incomplete Markets
295
7 . 2 Hedging in Incomplete Market s
In Chapter 6 we used general martingale representation theorems to prove existence of hedging strategies and to construct them. The existence of non attainable contingent claims implies in that context that there exist ( suffi ciently ) integrable, Fr-measurable random variables X for which we cannot find an integral representation ( with respect to an equivalent martingale mea sure ) in terms of the ( discounted ) price processes S (So , . . . , Sd) ' . There fore such claims carry an intrinsic risk, and our aim can only be to reduce the remaining risk to this minimal component. We will now describe sev eral criteria to quantify the remaining risk and the related construction of 'optimal' hedging strategies. To avoid technicalities, we assume that price processes are already discounted ( i.e. use So 1 as numeraire ) , and that the prices of the risky assets are given by continuous, square-integrable semi martingales, i.e. S P + M + A with P (P I , . . . , Pd ) ' , Pi E [0, 00 ) M J ad ( M ) , (MI , . . . , Md) ' a square-integrable martingale under 1P and A with a ( 0. 1 , . . , ad )' a predictable process. One of the most prominent problems in the contemporary mathematical finance literature is the pricing and hedging of contingent claims ( random cash flows at a prespecified time point ) in an incomplete market. In case of a complete market, the pricing and hedging of the contingent claim can be done via the unique martingale measure and martingale representation results. To price a contingent claim, we calculate an expected value with respect to the martingale measure, to hedge a contingent claim perfectly, we can obtain a hedging strategy by using the integrand of an appropriate stochastic integral, as in Chapter 6, or Harrison and Pliska ( 1981) . In the incomplete setting however, there are infinitely many martingale measures, each leading to a different pricing formula, and so the question of finding a 'price' of a contingent claim and a suitable hedging strategy is more involved. One stream of thought originating in the work of Follmer and Sonder mann (1986) and subsequently considerably improved by, to name a few, Follmer and Schweizer (1991 ) , Schweizer (1991) , Schweizer (1994) , Pham, RheinHinder, and Schweizer (1998) , Schweizer (2001b) ( an excellent review ) , is to use quadratic hedging approaches: local-risk-minimization and mean variance hedging ( see §7.2.2 below ) . Typically, one tries to find a hedging strategy that minimises a suitably defined ( quadratic ) risk function. In case an optimal hedging strategy exists it could be used to define a dynamic value process for the contingent claim; however, there is typically a positive probability for the wealth process of an investor following this strategy to be negative at the end, and thus this approach seems not to be suitable for pricing purposes ( see Korn ( 1997b) for a detailed discussion ) . Alternative approaches relying on utility-based arguments for pricing con tingent claims in incomplete markets are Aurell and Simdyankin (1998) , Davis (1994) , Karatzas and Kou (1996) , Rouge and EI Karoui (2000) . ( For a detailed study for the use of utility function and optimal investments, see =
==
=
=
=
=
=
.
296
7. Incomplete Markets
Kramkov and Schachermayer (1999) .) In all of these papers, links to hedging problems and thus martingale measure can be established. 7.2.1 Quadratic Principles
Quadratic principles have a distinguished history in finance and insurance, arising from their even more distinguished history in statistics and mathemat ics. The fountainhead in statistics is the method of least squares, introduced by Gauss and Legendre in the early 19th century, and its development into the Linear Model of statistics (see e.g. Plackett (1960)). The fountainhead in finance is the work of Markowitz, from 1952 on. Markowitz emphasised that one should think in terms of risk, as well as (expected) returns. Since expectations about returns can be summarized by means (mean vectors, in the multi-dimensional case), risks by variances (covariance matrices) , this gives one the mean-variance framework, still one of the cornerstones of con temporary investment theory. Mean-variance portfolio theory fits naturally with any developments that depend on distributional assumptions (on asset returns) only through the first two moments, such as normality assump tions or investors maximising quadratic utility. In an equilibrium setting, mean-variance theory leads to capital asset pricing models and the concept of diversification: one should hold a basket of assets to mimic the (efficient) market portfolio in order to be exposed only to systematic risk and diversify away specific risk exposure (which according to this theory is not rewarded anyway). For further aspects of mean-variance theory, see e.g. Bodie, Kane, and Marcus (1999) , Elton and Gruber (1995), Ingersoll ( 1986) . A third aspect is the use of quadratic loss functions. In many areas decision-theoretic statistics, optimization theory etc. - one introduces a loss function to quantify choice between alternatives, and is guided by the prin ciple of minimising expected loss. For background, see e.g. Robert (1997), Chapter 2 (esp. §2.5. 1) (Bayesian and decision-theoretic statistics) , Whittle (1996) (LQG: linear dynamics, quadratic cost, Gaussian noise). It is as natural to think of maximising expected utility as to minimize expected loss. To quantify this, one needs to select a utility function, u, say. The classic treatment here is von Neumann and Morgenstern (1953); for a more recent treatment see Elton and Gruber ( 1995) , Ingersoll (1986) (finance and investment) , or Robert (1997), Chapter 2 (Bayesian statistics). Here quadratic loss min iE [(W(O) - L) 2 ] (J E 8 where W(.) is the wealth function and L the target for final wealth, e an appropriate space of strategies - corresponds in the utility formulation max iE [u(W(O » ] (J E 8
to a quadratic utility function of the form ( w ) = w - cuP . (Note while the former seems very natural, the latter seems less so, as one needs to restrict u
7.2 Hedging in Incomplete Markets
297
in order to avoid negative wealth, and it implies unrealistic attitudes of investors towards absolute risk aversion, again see Elton and Gruber (1995) or Ingersoll ( 1986)) . Relevant here is the contrast between economic or financial theories that do, or do not, depend on the attitude to risk of individual agents. On the one hand, the assumption of non-satiation (preferring more to less) plus the condition of absence of arbitrage lead in complete markets (informally: there are at least as many traded assets as sources of risk) via the arbi trage pricing technique to pricing and hedging results including the classic Black-Scholes-Merton theory. One the other hand, incomplete market situa tions (more sources of risk than traded assets) are by definition characterised by the nonexistence of a perfect hedge for some securities, and thus require further assumptions on, e.g. the agent's risk preferences as expressed in his utility function or loss function to solve the hedging and pricing problems. Premium-principles, well-established in insurance, might also be used in this framework, see Schweizer (2001a) for application of quadratic valuation prin ciples, such as the variance and the standard-deviation principle. From the mathematical point of view, one uses martingale theory, as mar tingales model random phenomena without systematic drift - fair games in gambling, absence of arbitrage in financial market models etc. (see Musiela and Rutkowski (1997)). A mean-variance framework as above necessitates second moments, so one uses the L2 -theory of martingales, in particular the Kunita-Watanabe inequalities and decomposition (Rogers and Williams (2000), IV.4, Dellacherie and Meyer ( 1978), VII.2). They allow one to use Hilbert space methods, and the geometric language of projections and or thogonality. This is very natural in situations where all the essential features of that situation are modelled by the mean and (co-)variances. The classical situa tion of this type is, of course, the Gaussian (or multivariate normal) case, where means and variances determine - in the presence of Gaussianity the structure. The downside is that one cannot then model other features asymmetry and heavy tails, for instance - not present in the Gaussian case. One way to proceed is to retain means and variances - for convenience, their economic interpretation in the Markovitz case etc. - but replace the Gaus sian assumption by something more general. One is led to a semi-parametric model, with means and variances forming the parametric component and a non-parametric component describing other aspects. See for instance Bing ham and Kiesel (2002) and the references quoted there for background. w
7.2.2 The Financial Market Model
We use the setting outlined in e.g. Heath, Platen, and Schweizer (2001), Pham, Rheinliinder, and Schweizer (1998) and Schweizer (2001b) to which we also refer for further details.
298
7. Incomplete Markets
Let ( fl, F, JP) be a probability space with a filtration IF = ( Ft ) O:5t:5T satisfying the usual conditions of right-continuity and completeness, where T E (0, 00] is a fixed time-horizon. All processes considered will be indexed by t E [0, T] . We consider a market with d + 1 assets with price processes (St ) = (Sf , Sf , . . . st ) available for trading. One of the assets, So , is used as a numeraire, that is Sf is assumed to be strictly positive and all other assets are discounted with So . (In order to reflect the time value of money, it is natural when following asset values over time to work in discounted terms. Indeed, we have a choice as to which numeraire - the basic asset to take as our unit of accounting - to use, and to discount this to make its value constant.) We denote the discounted assets by ( 1 , Xd , with Xt (Xl , · . . , xt ) = (Sf / Sf , . . . , Sf / Sf ) , and consider only the discounted values in the sequel. We assume that X satisfies the structure condition (SC) ; this means X admits the decomposition (se) X = Xo + M + A , where M E M (JP) � is an JRd -valued locally square-integrable local JP martingale null at 0, 'and A is an JRd -valued adapted continuous process of fi nite variation null at 0. Furthermore, we denote by (M) = ( (M)ij ) t,:J. l , , d = = ( (M i , MJ» ) j = l d the matrix-valued covariance process of M. We assume i , , , continuous with respect to M, in the sense that that A is absolutely ,
=
lac
.
...
...
A� =
(J
d (M) ) \s
o
)
, �
t j X�d (M' , M'). , 0
J=1 0
'"
t " T, i
�
1, 2, . . , d, .
with a predictable JRd -valued process >" such that the variance process of the stochastic integral J >"dM , namely
J >..;rd t
Kt =
o
(M) s >" s =
J 'J= 1 0 d
.2:
t
>"� >"� d (M i , Mj) s '
is JP - almost surely finite for t E [0, T] . We fix an RCLL version of K and call this the mean-variance tradeoff (MVT) process. As shown in Delbaen and Schachermayer (1995a) and Schweizer (1995) , the structure condition (SC) is equivalent to a rather weak form of the no arbitrage condition (no free lunch with bounded risk) . Heuristically speaking, we can say that the MVT process K 'measures' the extent to which X devi ates from being a martingale (consult Schweizer (1994) , Schweizer (1995) for a precise statement and an explanation of the terminology 'mean-variance tradeoff') . If X is continuous (as it will be throughout) and admits an equiv alent local martingale measure (see §7.2.3 below) , the structure condition is automatically satisfied.
7 . 2 Hedging in Incomplete Markets
299
By construction, XI are discounted price processes that together with the constant price process 1 form a financial market model. We now introduce as a further traded asset a European contingent claim, i.e. a random payoff at time T, in our market. Formally, a contingent claim H is an (TT ' IP) random variable. A standard example would be a European call option On Xi with strike K and payoff H max{X i K, O } . If X is continuous and admits an equivalent local martingale measure (see below) , the structure condition is automatically satisfied. =
-
7.2.3 Equivalent Martingale Measures
The martingale property is evidently suitable for financial modelling because it captures unpredictability - absence of systematic movement or drift. The basic insight - due to Harrison and Kreps (1979) and Harrison and Pliska (1981) in the financial setting of Chapters 4 and 6 - is that, although dis counted prices are not martingales in general, they become martingales un der a suitable change of measure. One passes to an equivalent measure (same null sets: same things possible, same things impossible) , under which dis counted processes become martingales. Such a measure is called equivalent martingale measure (EMM); localization may be needed - working with local martingales rather than martingales - leading to equivalent local martingale measures (ELMM) . Mathematically, this change of measure technique is Gir sanov'S theorem, §5.7. Although this idea dates in the financial context from around 1980, it may be traced back much further in the actuarial and insur ance settings. The intuition is that it amounts to shifting probability weight between outcomes - giving more weight to unfavourable ones - to change the risk-averse into a risk-neutral environment. Returning to the mathematical formulation, we denote by P the set of equivalent local martingale measures (ELMM) . We assume that X admits an equivalent local martingale measure Q E P, and hence that our market model is free of arbitrage opportunities. For our subsequent analysis, two equivalent martingale measures will prove to be important. We denote by pE
=
{ Q P I �� E
E
L 2 (IP)
}
the set of all ELMMs with square-integrable density. Now define the strictly positive continuous local lP-martingale
(this is the stochastic exponential of Theorem 5. 10.4; see Musiela and Rutkowski (1997) , §1O.1.4 and recall the definition of K). If Z is a square integrable IP-martingale, then
300
7. Incomplete Markets
diP := dIP
A
ZT
E L 2 (IP)
defines an equivalent probability measure iP IP under which X is a local martingale, i.e. iP E P E . iP is called minimal equivalent local martingale measure for X (see Follmer and Schweizer ( 1991) , Schweizer ( 1995)) . As our second important ELMM we need the variance-optimal ELMM iP. The formal definition is �
Definition 7.2 . 1 . The variance-optimal ELMM jp is the unique element in
p 2 that minimizes
1 + War./p
[��]
over all Q
E
p2 .
The existence of iP for continuous X was shown by Delbaen and Schacher mayer (1996). As mentioned above, the first formal definition of the minimal martingale measure was given in Follmer and Schweizer (1991) , where the terminology was motivated by the fact that a change of measure to the minimal martingale measure disturbs the overall martingale and orthogonality structure as little as possible. Subsequently the minimal martingale measure was found to be very useful in a number of hedging and pricing applications (see Schweizer (2001b) for discussion and references) . Furthermore Schweizer (1995) pro vides several characterizations of the minimal martingale measure in terms of minimizing certain functional over suitable classes of (signed) equivalent local martingale measures (see also Schweizer (1999)) . 7.2.4 Hedging Contingent Claims
We now turn to the problem of hedging, that is, covering oneself against future losses associated with possession, or sale, of a contingent claim. The problem of hedging requires the introduction of trading strategies, i.e. selection of a covering portfolio in the underlying assets, and loss functions, i.e. how and when to value the success of a strategy. Informally, we can think of this problem as follows. By using our initial capital and following a certain trading strategy, we can generate a value process, which we use to cover our exposure. That is, at time T we have a certain value VT from trading; we need to cover H and evaluate the difference H - VT . Rewriting this, we get a decomposition
LT
=
H = VT + L T ·
(7.2)
As we shall see, decompositions like this under different measures play a decisive role in the solution of various hedging problems.
7.2 Hedging in Incomplete Markets
301
The corresponding mathematical tool in our setting is the basic decompo sition for L 2 -martingales, the (Galtchouk-)Kunita-Watanabe decomposition (see Rogers and Williams (2000) , IV.4) . L 2 -martingales null at 0 are if 0 for all stopping times T (equivalently, if is a uniformly integrable martingale). Such an possesses an or thogonal decomposition into its continuous part and its purely discontinuous part. In the financial context the continuous part is the stochastic integral - seen as a gains from trading process - and the purely discontinuous part orthogonal to it is the unhedgable risk L. Thrning to the problem of exact definitions of trading strategies, we note that P =I- 0 implies that X is a semi-martingale under !P. Thus we can introduce stochastic integrals with respect to X and associate the integrands with trading strategies.
strongly orthogonal lE(MT NT ) MN
M, N M
=
d -valued pre the linear space of all JR Denote by L(X) dictable X -integrable processes iJ. A self-financing trading strategy is any pair (Vo, iJ) such that Vo is an .'Fo -measurable random variable and iJ L(X ) We can think of iJ; as being the numbers of shares of asset i held at time t and Vo the initial capital of an investor. We associate a value process vt ( Va , iJt ) with it given by (7.3) vt (Vo, iJt ) : = Va + f iJu dXu · o Definition 7.2.2.
E
.
t
Here, we call Gt (iJ) = J� iJ u dXu the gains from trading process. The process is self-financing since as soon as the initial capital is fixed no further in- or outflow of funds is needed. We now turn to the problem of identifying the risk associated with the hedging problem, focusing first on the problem of covering our exposure to H at time T. Starting with an initial capital Vo and following a trading strategy iJ, we obtain a final wealth VT (VO , iJ) . Our assessment of the quality of our trading strategy for the problem at hand will thus rely on the difference between these quantities. Several approaches are possible: We could use a quadratic loss function; this leads to the notion of mean-variance hedging, and is described in further detail below. Other possibilities are Value-at-Risk (VaR) principles to cover against extreme losses, as in Follmer and Leukert (1999) and Follmer and Leukert (2000) , Or value-preserving strategies as in Korn (1997c) . In order to be able to consider quadratic loss functions, we restrict the class of trading strategies using Definition 7.2.3. iJ E L(X) GT (iJ) Gt (iJ) Q L 2 (!P)
the spaceprocess of all ()' betrading and the gainsLetfrom
which for iseachin is a formartingale
302
7. Incomplete Markets
E p 2 . A pair ( vo , iJ) such that Vo E 1R and iJ E () ' is called a mean-variance optimal strategy for H, if it solves the optimization problem
Q
(7.4) over iJ
E
() ' .
The solution of the problem can b e given i n terms of a decomposition (7.2) under the variance-optimal measure iP. From Gourioux, Laurent, and Pham (1998) we know
[ l] -
- diP :Ft = Zo + Zt : = IE dIP
t
Xu , I0 (ud
for some ( E 8' . We have (see Schweizer (2001b)) Theorem 7.2 . 1 . Let H E L 2 (IP) be a contingent claim and write the Galtchouk-Kunita- Watanabe decomposition of H under iP with respect to X as H = iE(H)
with
v,.H,jp
+
T
I ��, jp dXs + L!f.' jp o
:= iE[H I:Ft l
=
vi!' jp ,
(7.5)
t
=
iE(H) +
I ��, jp dXs + Lf' jp · o
Then the mean-variance optimal strategy for H is given by Vo* iJ * t
iE(H)
- cH,jp (v.H,jp
=
0, solve the optimization problem (7. 18) c
=
c
(G
2 and P solved by a Hilbert-space projection argument: cp * E P is optimal if and only if IE [(X - c - G K } lE [S(T) l{s (T» K} ; ()* ] = lE [S (T) ; () *] lE [ l{S (T» K} ; () * 1] = lE [e-r T S(T); () * ] erT JP [ S (T) > K ; () * = S(O)erT JP [ S (T) > K; () * where we used the martingale property of e - r t S(t) under the risk-neutral Esscher measure for the last step. Now the pricing formula for the European
To evaluate the first term, we apply the factorization formula with k and and get
h
=
=
+ 1] ,
call becomes
K; +
+
+ 1]
K;
K
S(O)JP [S(T) > () * 1] - e - rT JP [S(T) > () *] . (7.48) We now can use formula ( 7.48) to compute the value of a European call with strike K and maturity T. Denote the density of .C(Zf., 8 (t)) by ftf. , 8 ( compare (2.8) for the exact form ) . Then 00
E [e - r T (ST - K)+ ; () * ] S(O) J f�, 8 (x ; ()* l )dx +
=
c
K
00
_e-rT J f�, 8 (x ; ()*)dx , c
( 7.49)
326
7. Incomplete Markets
where c = log ( Kj S ( O )) . Eberlein and Keller (1995) and Eberlein, Keller, and Prause (1998) com pare option prices obtained from the Black-Scholes model and prices found using the hyperbolic model with market prices. They find that the hyper bolic model provides very accurate prices and a reduction of the smile effect observed in the Black-Scholes model.
8 . Interest Rate Theory
In this chapter, we apply the techniques developed in the previous chap ters to the fast-growing fixed-income securities market . We mainly focus on the continuous-time model (since the available tools from stochastic calcu lus allow an elegant presentation) and comment of the discrete-time analogue (Jarrow ( 1996) gives a splendid account of discrete-time models) . As we want to develop a relative pricing theory, based on the no-arbitrage assumption, we will assume prices of some underlying objects as given. In the present context we take zero-coupon bonds as the building blocks of our theory. In doing so we face the additional modelling restriction that the value of a zero-coupon bond at time of maturity is predetermined ( = 1 ) . Furthermore, since the entirety of fixed-income securities gives rise to the term-structure of inter est rates (sometimes called the yield curve) , which describes the relationship between the yield-to-maturity and the maturity of a given fixed-income secu rity, we face the further task of calibrating our model to a whole continuum of initial values (and not j ust to a vector of prices) . A first attempt at ex plaining the behaviour of the yield curve is in terms of a continuum of spot rates of maturities between T and T, where T is the shortest (instantaneous) lending/borrowing period, and T the longest maturity of interest . We model these rates as correlated stochastic variables with the degree of correlation decreasing in terms of the difference in maturity. Discretizing the maturity spectrum, we are tempted to start with a generalized Black-Scholes model (as in §6.2) d
dri (t) = ai (t) + L bij (t)dWj (t) , i j=l
=
1, .
..,d
with W = (Wi , . . . , Wd) a standard d-dimensional Brownian motion. The de gree of (instantaneous) correlation of the different rates can then be described in terms of their covariation
d (ri , rj) (t)
=
d
L bik (t)bjk (t)dt = pij (t)dt.
k=l
Empirical investigations in the degree of correlation show that the Pij are usually very large (close to 1 ) . In fact , using principal component analysis
328
8. Interest Rate Theory
(Mardia, Kent, and Bibby ( 1 979) , Chapter 8, Krzanowski ( 1 988) , §2.2) one can show that the first principal component often explains 80 - 90% of the total variance, and that the first three components taken together describe up to 90-95% of the total variance. Therefore, especially in early approaches, it has been tempting to choose one specific rate, usually the short rate, r { t) , as a proxy for the single variable that principal component analysis indicates can best describe the movements of the yield curve. The standard model is then of the form dr{t)
=
a{t, r { t ) ) dt + b{t, r { t))dW { t) .
(8. 1 )
After developing the general features of a model using zero-coupon bonds as building blocks, based on the general theory in §6. 1 and §6.2, we will discuss various such models of form (8. 1 ) , taking the short rate as state variable, and explore their implications and shortcomings in §8.2. By contrast, the modern, evolutionary approach pioneered by Heath, Jarrow, and Morton ( 1 992) takes a continuum of instantaneous forward rates as the building blocks to describe the dynamics of the whole term structure. We will discuss this approach in detail in §8.3. Finally, we apply the different models to contingent claim pricing in §8.4. Modern approaches to the pricing of interest derivatives are discussed in sections §8.5 and §8.6. The structure of our approach below follows the treatment of Bjork ( 1997) . For additional textbook accounts, see Part II of Musiela and Rutkowski ( 1 997) and Brigo and Mercurio (200 1 ) . 8 . 1 The Bond Market
8 . 1 . 1 The Term Structure of Interest Rates
We start with a heuristic discussion, which we will formalize in the follow ing section. The main traded objects we consider are zero-coupon bonds. A zero-coupon bond is a bond that has no coupon payments. The price of a zero-coupon bond at time t that pays, say, a sure £ at time T 2: t is denoted p{ t, T) . All zero-coupon bonds are assumed to be default-free and have strictly positive prices. Various different interest rates are defined in connection with zero-coupon bonds, but we will only consider continuously compounded interest rates ( which facilitates theoretical considerations) . Using the arbitrage pricing technique, we easily. obtain pricing formulas for coupon bonds. Coupon bonds are bonds with regular interest payments, called coupons, plus a principal repayment at maturity. Let Cj be the pay ments at times tj , j 1 , F be the face value paid at time tn . Then the price of the coupon bond Be must satisfy =
. . . , n,
n
Be
=
L Cjp{O, tj ) + Fp{ O, tn ) .
j= 1
(8.2)
8 . 1 The Bond Market
329
Hence, we see that a coupon bond is equivalent to a portfolio of zero-coupon bonds. The yield-to-maturity is defined as an interest rate per annum that equates the present value of future cash flows to the current market value. Using continuous compounding, the yield-to-maturity Ye is defined by the relation n Be = L Cj exp{ Ye tj + F exp{ - Yetn}. j=l
- }
If for instance the tj , j = 1 , are expressed in years, then Ye is an annual continuously compounded yield-to-maturity ( with the continuously compounded annual interest rate, defined by the relation = exp The term structure of interest rates is defined as the relationship between the yield-to-maturity on a zero-coupon bond and the bond's maturity. Nor mally, this yields an upward sloping curve ( Fig. 8 . 1 ) , but flat and downward sloping curves have also been observed.
{-r(T) (T/365)}).
. . . , n
r (T) ,
p(O , T)
Yield
Maturity
Fig. 8 . 1 .
Yield curve
In constructing the term structure of interest rates, we face the additional problem that in most economies no zero-coupon bonds with maturity greater than one year are traded ( in the USA, Treasury bills are only traded with maturity up to one year ) . We can, however, use prices of coupon bonds and invert formula (8.2) for zero-coupon prices. In practice, additional compli cations arise, since the maturities of coupon bonds are not equally spaced and trading in bonds with some maturities may be too thin to give reli able prices. We refer the reader to Edwards and Ma ( 1992) and Jarrow and Turnbull (2000) for further discussion of these issues.
330
8. Interest Rate Theory
8 . 1 . 2 Mathematical Modelling
We use a standard setting as described in §6. 1 and §6.2 (we follow Bjork ( 1 997) and El Karoui, Myneni, and Viswanathan ( 1 992a) ) . Let ([2, F, IP, IF) be a filtered probability space with a filtration IF = (Ft)t � T ' satisfying the usual conditions (used to model the flow of information) and fix a terminal time horizon T * . We assume that all processes are defined on this prob ability space. The basic building blocks for our relative pricing approach, zero-coupon bonds, are defined as follows.
Definition 8 . 1 . 1 . A zem-coupon bond with maturity date T, also called a T - bond, is a contract that guarantees the holder a cash payment of one unit on the date T. The price at time t of a bond with maturity date T is denoted by p(t, T) . Obviously we have p(t , t ) = 1 for all t . We shall assume that the price process p(t , T ) , t E [0 , TJ is adapted and strictly positive and that for every fixed t , p(t, T) is continuously differentiable in the T variable. Based on arbitrage considerations (recall our basic aim is to construct a market model that is free of arbitrage) , we now define several risk-free inter est rates. Given three dates t < Tl < T2 the basic question is: what is the risk-free rate of return, determined at the contract time t , over the interval [Tl ' T2J of an investment of 1 at time Tl ? To answer this question we consider the arbitrage Table 8 . 1 below (compare § 1 . 3 for the use of arbitrage tables) .
t
Time
Buy
Sell Tl bond p ( t , Tt l T2 bonds p ( t , T2 )
Net investment Table 8 . 1 .
0
Tl
T2
Pay out 1 Receive
-1
p ( t ,Tt l p ( t , T2 )
+ p ( t , Tt l p(t , T2 )
Arbitrage table for forward rates
To exclude arbitrage opportunities, the equivalent constant rate of interest
R over this period (we pay out 1 at time Tl and receive e R(T2 - T, ) at T2 ) has
thus to be given by
We formalize this in:
Definition 8 . 1 . 2 . (i) The forward rate for the period [Tl ' T2J as seen at time at t is defined as
8 . 1 The Bond Market
R ( t ., T1 , 'T' .L 2 )
=
331
_ log p (t , T2 ) - log p (t , T1 ) . To2 - T1
(ii) The spot rate R (T1 , T2 ) , for the period [T1 , T2J is defined as
(iii) The instantaneous forward rate with maturity T, at time t, is defined by f (t , T)
=
_
a log p (t , T) ' aT
(iv) The instantaneous short rate at time t is defined by r (t )
=
f (t , t ) .
Definition 8 . 1 . 3 . The money account process is defined by B (t)
�
exp
{j } . r(s ) d'
The interpretation of the money market account is a strategy of instan taneously reinvesting at the current short rate. An immediate consequence of the definitions is
Lemma 8 . 1 . 1 . For t �
s
p (t , T)
� T we have �
p( t , , ) exp
and in particular
{ -1
f(t, u ) d"
}.
In what follows, we model the above processes in a generalized Black Scholes framework. That is, we assume that W ( WI , . . . , Wd ) is a standard d-dimensional Brownian motion and the filtration IF is the augmentation of the filtration generated by W ( t ) . The dynamics of the various processes are given as follows: =
Short-rate Dynamics: dr (t)
=
a (t) dt + b(t) d W ( t ) ,
(8. 3 )
Bond-price Dynamics: dp(t , T )
=
p ( t , T ) {m( t , T ) dt + v (t , T ) dW ( t ) } ,
(8.4)
8. Interest Rate Theory
332
Forward-rate Dynamics: df (t, T)
=
a(t, T)dt + a(t, T)dW (t) .
(8.5)
We assume that in the above formulas, the coefficients meet standard con ditions required to guarantee the existence of the various processes - that is, existence of solutions of the various stochastic differential equations; see Furthermore, we assume that the processes are smooth enough to allow differentiation and certain operations involving changing of order of integra tion and differentiation. Since the actual conditions are rather technical, we refer the reader to Bjork ( 1997 ) , Heath, Jarrow, and Morton ( 1 992 ) and Prot ter (2004) ( the latter reference for the stochastic Fubini theorem ) for these conditions. Following Bjork ( 1 997 ) for formulation and proof, we now give a small toolbox for the relationships between the processes specified above.
§5 . 8 .
dynamics we have(i) If p(t, T) satisfies (8.4 ), then for the forward-rate
Proposition 8 . 1 . 1 .
df (t, T)
=
a(t, T)dt + u(t, T)dW(t) ,
where a and u are given by
{
VTV(t,(t,T)T)v (t , T) mT (t , T) , T . (ii) If f (t, T) satisfies (8. 5 ), then the short rate satisfies a (t, T) u(t, T)
-
=
=
dr (t)
-
=
a(t) dt + b(t) dW (t) ,
where a and b are given by { b(t)a (t) u(t, fr (t, t) + a (t, t) , (8. 6 ) t) . (iii) If f (t, T) satisfies (8. 5 ), then p (t, T) satisfies dp (t, T) p( t , T) { ( r (t) + A(t, T) + � I I S(t, T) 1 1 2 ) dt S (t, T) dW (t) } , where T A(t, T) J a (t, s)ds , S(t, T) J u (t, s)ds. (8. 7) =
=
+
=
T
=
-
=
t
-
t
Proof. To prove ( i ) we only have to apply Ito's formula to the defining equation for the forward rates and we leave this as Exercise below. To prove ( ii ) we start by integrating the forward-rate dynamics. This leads to
8. 7
8 . 1 The Bond Market
J( t, t) = ret) = J( O, t) + Writing also
t
t
! a(s, t)ds + ! a(s, t)dW(s) . o
0
a and a in integrated form a (s, t) = a(s, s) + a (s, t)
333
( 8.8 )
t
! aT (s, u)du, s
t
!
a ( s, s) + aT (s, u)du,
=
s
and inserting this into ( 8.8 ) , we find
ret)
=
J( O, t) + +
t
t t
t
! a (s, s)ds + ! ! aT (s, u)duds o
0 t t
s
! a(s, s)dW(s) + ! ! aT ( s, u)dudW(s) . o
0
s
After changing the order of integration we can identify terms to establish ( ii ) . For ( iii ) we use a technique from Heath, Jarrow, and Morton (1992) . By the definition of the forward rates we may write the bond-price process as
p(t, T) where
Z is given by
=
exp {Z(t ,
T) }
,
T
Z ( t,
T) ! J( t, s)ds. =
-
( 8 . 9)
t
Again we write ( 8.5 ) in integrated form:
J( t, s) = J( O, s) +
t
t
! a(u, s)dt + ! a(u, s)dW(u) . o
We insert the integrated form in ( 8 . 9 ) to get
T
Z ( t,
t
0
T
t
T
T) ! J (O, s)ds ! ! a(u, s)dsdu ! ! a(u, s)dsdW(u). =
-
t
-
O t
-
0 t
Now, splitting the integrals and changing the order of integration gives us
334
8. Interest Rate Theory
Z(t, T)
=
T
t
T
t
T
J 1(0, s)ds J J a(u, s)dsdu J J a(u, s)dsdW(u) + J f(O, s)ds + J J a(u, s)dsdu + J J a(u, s)dsdW (u) Z(O, T) J J a(u, s)dsdu J J a(u, s)dsdW (u) + J 1(0, s)ds + J J a(u, s)duds + J J a(u, s)dW (u)ds. -
o
-
0
t
o
-
t
0
t
o
t
0
-
t
0
0
t
u
T
u
u
-
s
0
t
0
t
0
t
u
T
u
u
t
0
s
0
The last line is just the integrated form of the forward-rate dynamics (8.5) over the interval [0, sJ . Since r(s) f(s, s) , this last line above equals J� r(s)ds. So we obtain =
Z(t, T)
=
Z(O, T) +
t
t
T
t
T
J r(s)ds J J a(u, s)dsdu J J a(u, s)dsdW(u) . -
o
0
u
-
0
u
Using A and S from (8.7) , the stochastic differential of Z is given by
dZ(t, T)
=
(r(t) + A(t, T))dt
+ S(t, T)dW(t).
Now we can apply Ito's lemma t o the process p(t, T) complete the proof.
=
exp{Z(t, T) } to 0
8 . 1 . 3 Bond Pricing, Martingale Measures and Trading Strategies
We will now examine the mathematical structure of our bond-market model in more detail. As usual, our first task is to find a convenient characterization of the no-arbitrage assumption. By Theorem 6 . 1 . 1 , absence of arbitrage is guaranteed by the existence of an equivalent martingale measure Q. Recall that by definition an equivalent martingale measure has to satisfy Q '" JP and the discounted price processes ( with respect to a suitable numeraire ) of the basic securities have to be local Q-martingales. For the bond market this implies that all zero-coupon bonds with maturities 0 :=:; T :=:; T* have to be local martingales. More precisely, taking the risk-free bank account B (t) as numeraire we have
Definition 8 . 1 .4 . A measure Q
'" JP defined on (il, F, JP) is an equivalent martingale measure for the bond market, if for every fixed 0 :=:; T :=:; T * the process
8. 1 The Bond Market
p(t, T ) , B (t)
335
o � t � T
is a local Q -martingale. Assume now that there exists at least one equivalent martingale measure, say Q . Defining contingent claims as FT-measurable random variables such that XI B (T) E L l ( FT , Q) with some 0 � T � T * ( notation: T-contingent claims ) , we can use the risk-neutral valuation principle (6. 1) to obtain:
Proposition 8 . 1 . 2 . Consider a T-contingent claim X . Then the price pro cess IIx (t) , 0 � t � T of the contingent claim is given by
In particular, the price process of a zero- coupon bond with maturity T is given by T p(t, T) = IEQ e - ft r( s )ds Ft .
1 )
[
Proof. We j ust have to apply Theorem
6.1.4.
We thus see that the relevant dynamics o f the price processes are those given under a martingale measure Q. The implication for model building is that it is natural to model all objects directly under a martingale measure Q. This approach is called martingale modelling. The price one has to pay in using this approach lies in the statistical problems associated with parameter estimation. Turning now to market completeness, we define trading strategies - as with our models in Chapters 4 and 6 as dynamic investment portfolios involving some or possibly all of the ( basic ) traded securities ( in the present case zero-coupon bonds ) . In contrast to our previous market models, we now have a continuum of bonds with different maturities available for trade. So a portfolio could in clude an infinite number of financial securities ( consider e.g. the rolling-over strategy to define a risk-free savings process involving always only bonds of the shortest possible maturity ) . We shall, however, in accordance with prac tical restrictions only consider portfolios involving an arbitrary, but finite, number of financial securities ( see Bjork, Di Masi, Kabanov, and Runggaldier ( 1 997) and Bjork, Kabanov, and Runggaldier ( 1 997) for infinite portfolios ) . For any collection of maturities 0 < Tl < T2 < . . . < Tk = T* we then consider bond-trading strategies
K} and Q T resp. Q* the T- resp. T*-forward risk neutral measure. Now T*) T) = T) has Q-dynamics (omitting the arguments and writing S* for
t
=
=
=
{w :
Z (t,
p(t, p(t,
dZ Z { S(S - S*)dt - (S - S*)dW(t) } , =
S(t, T*»
352
8. Interest Rate Theory
so a deterministic variance coefficient. Now Q* (P (T, T* )
2::
(��,r;;
K) = Q *
2::
K
)
=
Q* (Z (T, T)
2::
K) .
Since Z(t , T) is a Q T -martingale with Q T -dynamics
dZ(t , T) -Z(t , T) (8(t , T) - 8(t, T* ))dW T (t) , we find that under Q T (use the stochastic exponential, compare §5.7 and §5.8) (again 8 = 8(t, T) , 8* = 8(t, T* ) ) =
Z(T, T)
�
:��.'�:i exp
{j
- ( S - S' ),nvT (t)
- � 1 <s - S')'dt }
(with W T a Q T -Brownian motion). The stochastic integral in the exponential is Gaussian (compare §5.2) with zero mean and variance T 2 E (T) (8(t, T) - 8(t , T*)) 2 dt . =
J o
So with
d2 =
log
( i')(��' » ) - � E2 (T) . JE2 (T)
Similarly, for the first term
Z* (t , T)
=
p(t, T) p(t, T * )
has Q-dynamics ( compare (8.22))
dZ*
=
{
Z* 8* (8* - 8)dt + (8 - 8*)dW(t) } ,
and also a deterministic variance coefficient. Now
Q * (p (T, T* )
2::
K)
=
Q*
( P(/T* ) �) ::;
=
Q* ( Z* (T, T)
Under Q * Z * (t, T) is a martingale with
dZ* (t , T) = Z* (t , T) (8(t , T) - 8(t, T*))dW* (t) , so
::;
�).
8.4 Pricing and Hedging Contingent Claims
353
Again we have a Gaussian variable with the ( same ) variance E 2 (T) in the exponential. Using this fact it follows ( after some computations ) that :
Q * (p(T, T* ) � K)
=
N(dI ) ,
with So we obtain:
Proposition 8 . 4. 1 . The price of the call option defined in (8.23) is given by
C(O)
=
(8.24)
p(O, T* ) N(d2 ) - Kp(O , T)N(dI ) ,
with parameters given as above. Remark 8.4 . 1 . ( i ) Observe that the above reasoning generalizes to assets Set) in place of pet, T * ) as long as t ( t , T) = S( t ) /p ( t , T) has a deterministic volatility coefficient . ( ii ) Since we have a closed formula for the option's price above, we can com pute sensitivities with respect to the underlying in much the same way as in the standard Black-Scholes model. In particular, we can construct .d-neutral hedging portfolios. 8.4.3 Swaps
This section is devoted to the pricing of swaps. We consider the case of a forward swap settled in arrears. Such a contingent claim is characterized by: • • • •
a fixed time t, the contract time, dates To < TI , · · · < Tn , equally distanced R , a prespecified fixed rate of interest , K, a nominal amount.
Ti +1 - Ti
=
0 holds. Clearly, the payer-swaption-value in To is So ,{3 ( t )
_
•
-
as
=
(3
(So,{3(To) - K) + L Tip(To, Td , i= o + l and the receiver-swaption-value is {3 (K - So, {3(To)) + L TiP (To, Ti) . i=o + l Now we turn to the problem of modelling the dynamics of the swap rate, and the related question of deriving a pricing formula for swaptions. First, we observe that AO,{3(t )
(3
L Tip(t, Ti) i=o + l is the t-price of a a portfolio of bonds ( i.e. a traded asset ) . Consequently, A O, {3 (t) , which if known as accrual factor or present value of a basis point, can be used as numeraire. Now note that =
366
8. Interest Rate Theory
Sa,{3 (t)
=
p(t, Ta ) - p(t, T(3 ) , A a , {3 (t)
where the numerator can be regarded as the price of a traded asset as well. We conclude that , in order for our model to be arbitrage-free, the swap rate Sa,{3 (') has to be a martingale under the numeraire pair (Qa,{3 , A a,{3 ( ' ) ) ' Qa , {3 is the so-called forward swap measure. To proceed, we assume that Sa,{3 (' ) follows a lognormal martingale:
where u is a deterministic function and Wa ,{3 (') is a standard Qa,{3-Brownian motion. The fact that the forward swap rate Sa,{3 (t) is lognormally distributed under Qa,{3 motivates the name lognormal forward swap model. This leads us to the following
Theorem 8.5.4. The price of a payer swaption as specified above in a log
normal forward swap model is consistent with Black 's formula for swaptions and is thus given by with
and d2
=
d 1 - E(Ta ) with E 2 (Ta )
J U(s) 2 ds. To
=
o
The price of a receiver-swaption is given by Proof. For the payer-swaption, we have
PS ( 0, { Ta , Ta , . . . , T{3 } , K ) = Aa,{3 ( O ) lElQo,{3 = =
( (Sa,{3 (TaA) a,{3- K)(Ta+)Aa,{3 (Ta ) )
A a , {3 (O)lElQo,{3 ( (Sa , {3 (Ta ) - K ) + ) A a,{3 (O) ((Sa,{3 (0 ) N(d 1 ) - KN(d2 ) ) ,
and it is evident that our judicious choice of the numeraire simplifies the calculations considerably. The same argument goes through for the receiver 0 swaption.
8.5 Market Models of LIB OR- and Swap-rates
367
So far, we have only been concerned with the evolution of one swap-rate. While this is sufficient for the purpose of swaption-pricing, the pricing of more complicated derivatives whose value depends on more than one swap rate necessitates the modelling of the simultaneous evolution of a set of swap rates under one probability-measure. This leads us to the class of Swap Market Models (SMMs), which are arbitrage-free models of the joint evolution of a set of swap rates under a common probability measure. The modeler has some freedom of choice when it comes to determining the set of swap-rates that is to be modelled. Examples are: 1 . The set of swap-rates {Sa,a + ! (t) , Sa+!,a + 2 (t) , . . . , S(3 - 1, (3 (t) , } . As this is exactly the set of LIBOR rates {L(t, Ta) , L (t, Ta + l ) ' . . . ' L(t, T(3 - 1 ) } ,
this choice leads us back to the LMM framework. 2. The set of swap-rates {Sa,a + ! (t), Sa,a + 2 (t) , . . . , Sa, (3 (t) } . 3. The set of swap-rates {Sa ,(3 (t) , Sa +l ,(3 (t) , . . . , S(3 - 1 , (3 (t) } . All of the above choices have in common that they fully describe the LIBOR structure from Ta to T(3 , so that all products that can be priced in a LMM can also be priced in a SMM ( and vice versa, as the LIBORs also determine the swap rates ) . Which of the above sets is chosen of course depends on the concrete application or the interest-rate derivative to be priced. The ideas and tools underlying the construction of a SMM are very similar to those of a LMM, and so are the results ( e.g. measure relationships ) . The interested reader might want to consult Hunt and Kennedy ( 2000 ) , Pelsser ( 2000 ) and Rutkowski (1999) for further details. 8.5.6 The Relation Between LIBOR- and Swap-market Models
As already remarked, all products that can be priced in a LMM-framework can in principle also be priced in a SMM-framework and vice versa. This is due to the fact that swap rates can be expressed as weighted sums of LI BOR rates, and LIBOR rates can be expressed as functions of swap-rates. However, this does not mean that these model-classes are equivalent. As we have seen, one of the main advantages of the lognormal LMM is that it prices caplets with Black's caplet formula, which is the market standard. Accord ingly, the lognormal SMM prices swaptions with Black's swaption formula, which is also standard among practitioners. At this stage, a natural question is whether lognormal LMMs yield swaption-prices that agree with Black-prices, and whether lognormal SMMs give Black-consistent caplet prices. In general, the answer is negative. The reason for the incompatibility of lognormal LMMs and lognormal SMMs lies in the fact that if LIBOR rates are lognormal under their respective forward-measures, swap rates ( that are determined by these LIBOR rates ) cannot be lognormal under their respective forward-swap mea sures and vice versa. Even though this might seem disappointing at first, as it
368
8. Interest Rate Theory
implies that neither the lognormal LMM nor the lognormal SMM can price both caplets and swaptions according to the market standard, it does not constitute a major drawback or practical limitation. The incompatibility is mostly of a theoretical nature, as it can be shown ( for example by simulation studies ) that swap rates in the lognormal LMM are "almost" lognormal, and therefore swaption-prices in the lognormal LMM ( which usually have to be obtained by Monte Carlo simulations, as no closed formulas exist ) are very similar to those one would obtain by the Black formula. For more information on the incompatibility problem, the interested reader is referred to Rebonato ( 1999 ) and Brigo and Mercurio (2001). 8 . 6 Potential Models and the Flesaker-Hughst on Framework
The aim of this section is to present the basic ideas of the potential approach to term structure modelling as set out in Rogers ( 1997) , and to develop connections with the positive interest rate models formulated by Flesaker and Hughston ( 1997) and Flesaker and Hughston ( 1996b ) . Our discussion builds on Rogers ( 1997 ) , Jin and Glasserman (2001) and Hunt and Kennedy (2000) . 8.6.1 Pricing Kernels and Potentials
Our starting point is a filtered probability space ( il, F, ( Ft ) , IP ) . We assume that ( Ft ) is the IP-augmentation of the natural filtration generated by a continuous and positive process process ( at ) . Define t At = a ( t) dt .
J o
As A ( At ) is an increasing process, Aoo = limt-+oo At exists. We assume that lE ( A ;" ) < 00 holds. Define the pricing kernel ( Zt ) as Zt = lE ( Aoo I Ft ) - At , t E [0, 00] . Without loss of generality, we may assume that Zo lE ( Aoo ) = 1 . Because lE ( Zt I Fs ) S Zs for 0 S s S t S 00, ( Zs ) is a supermartingale. Furthermore, limt-+oo lE ( Zt ) = 0 for obvious reasons. Stochastic processes characterized by these two properties are called potentials. =
=
We are now in the position to define zero coupon bond prices by (8.31)
8.6 Potential Models and the Flesaker-Hughston Framework
369
Observe that bond prices tend to zero with increasing time to maturity as
1 0. Tlim -+oo -+oo p (t, T) Tlim Zt IE(A:;o - A T I Ft ) For the bond prices in t 0, we get =
=
=
p(O, T } lE(ZT ) �
� l - lB(Ar}
�
l - JE
(1 ) � l 0,
d'
l-
lB(o.} d'
where we have used Fubini's theorem. Differentiating both sides with respect to T leads to
fJ fJT P (O , T)
=
- IE(aT ) '
The above relation can be used to calibrate the model to an observed bond price structure. As is well-known from the treatment of the HJM framework, the specifica tion of bond-price dynamics implicitly gives rise to the dynamics of forward rates and vice versa. Using the well-known relation between bond-prices and instantaneous forward rates, we can derive the forward-rate-dynamics in the above model:
Here we have used that
by Fubini 's theorem for conditional expectations; differentiating both sides then gives
IE(aT IFt)
=
fJ fJT IE(AT I Ft) .
Obviously, the forward rates f(t, T) are strictly positive, so that the bond prices p (t, T) = exp f ( t , s ) ds are strictly decreasing in T. Further more, using r(t) = f ( t , t), the short rate is simply r(t) = a(t)/Zt .
( - It
)
370
8. Interest Rate Theory
Before proceeding, let us briefly touch on the question of absence of ar bitrage in our framework. From equation (8.31), it is immediately clear that all normalized bond price processes (p ( t , T)Zt ) are lP-martingales. However, this does not imply that the model is arbitrage-free, as the process (ljZt) does not necessarily represent the price process of a traded asset. Neverthe less, Rutkowski ( 1999 ) is able to prove that in our framework, there exists an implied money market account B (Bt) , B being an increasing process of finite variation, and a corresponding risk-neutral measure lP* , such that all bond price processes discounted by B are lP* -martingales, which proves that the model excludes arbitrage opportunities. =
8.6.2 The Flesaker-Hughston Framework
This section presents the basic ideas of the positive interest rate framework introduced by Flesaker and Hughston ( 1997) and Flesaker and Hughston ( 1996b ) , and shows how their approach relates to the potential approach de scribed above. Flesaker and Hughston introduce a general class of term structure models by postulating that bond prices are of the form
f; ¢(s)Mts ds , 0 :::; t :::; T < 00 , (8.32) Jroo t ¢(s)Mts ds where ¢ is a deterministic and positive function, and (MtT k�o is a family of positive martingales indexed by T with respect to a probability measure lP and a filtration (Ft). Both integrals in (8.32) are assumed to be finite almost surely. From the definition, it is evident that bond prices are strictly decreas ing in T, Le. interest rates are always positive. The potential approach and the Flesaker-Hughston approach are closely related as the following theorem shows. p
( t, T)
=
Theorem 8.6. 1 . (i) Given a Flesaker-Hughston-model, define the pricing
kernel (Zt) as
00
Zt
=
J ¢(s)Mts ds. t
Then (Zt) is a potential and the zero coupon bond prices defined by
are of the form of Equation (8. 32}.
8.6 Potential Models and the Flesaker-Hughston Framework
371
(ii) Conversely, given a (9t ) -adapted, positive and continuous process (at ) and a positive and increasing process (At) with A t = f; as ds, define a potential ( Zt) by setting Zt = IE(A oo I9t ) - At . Then there exist a deterministic and positive function fjJ, and a family of positive ( !P, Wt ) ) martingales (MtT k:�o indexed b y T with M( O, T ) 1, such that Zt can be represented as
=
00
Zt
=
! fjJ( s )Mts ds . t
Consequently, bond prices are of the form of Equation (8. 32}.
Proof. ( i )
With Fubini's theorem for conditional expectations, we get
=!
00
T
and therefore
00
fjJ( s ) IE (Mts IFt ) ds
=
! fjJ( s)Mts ds
T
The potential-property of ( Zt) is apparent. ( ii ) Observe JE ( ZT I9, )
�
JE (A oo 19, ) - JE (A TI9, )
�
JE
and set MtT
=
(1 ) l a . ds 9,
IE (aT I 9t) and fjJ( t ) IE(aT )
=
�
JE ( . '9, ) ds
IE(at ) .
Then MOT 1 and (MtT k::o is a family of strictly positive ( !P, ( 9t)) martingales indexed by T, while fjJ is a deterministic and positive function. This leads to =
00
Zt
=
which completes the proof.
IE( Zt I 9t)
=
! fjJ( s )Mts ds, t
o
372
8. Interest Rate Theory
Remark 8. 6. 1 . Observe that (i) above, together with the comment on the no arbitrage property of potential models in the preceding section, shows that Flesaker-Hughston models are arbitrage free. Probably the most popular subclass of term structure models in the Flesaker Hughston framework is the class of rational models. The building blocks in this case are a positive (IP, (Ft))-martingale (Nt } t? o with No 1 and ( Ft ) the IP-augmentation of the natural filtration generated by (Nt) , two positive and deterministic functions f, 9 : � + -+ �+ with f (t) + g ( t ) 1 Vt 2: 0 and a positive and deterministic function ¢ � + -+ �+ defined by ¢(t) - itp(O, t) , where we assume that p(O, t) is strictly decreasing in t. We now introduce a family (Mt T )t?o of strictly positive martingales by setting =
=
=
:
Mt T
=
f (T) + g (T)Nt .
Then Formula (8.32) immediately yields the bond prices p(t , T)
=
�
I ¢(s) Mts ds t ¢(s)Mts ds
I
=
�
I ¢(s) ( f (s) + g (s)Nt) ds It ¢ (s) ( f ( s ) + g ( s) Nt) ds
=
F (T) + G ( T ) Nt F (t) + G(t) Nt
with positive and decreasing functions F and G defined by 00
F(T)
=
00
J ¢(s) f ( s) ds and G (T) J ¢(s) g (s) ds . =
T
T
A nice feature of this class of models is that the consistency with the initial term structure is already guaranteed by its very construction, because p(O, T) (F(T) + G(T) ) j (F(O) + G(O) ) , and one easily checks that F(T) + G(T) p(O, T) and F(O) + G(O) 1. It can also be shown, see e.g. Flesaker and Hughston (1997) , that both bond prices and interest rates are bounded above and below: =
=
=
G (T) G(t)
T) . Now Y (t) Ys ,r (t) ;�::�� . Check that Y has deterministic volatility, and find the parameters for the pricing formula. 8.4 .d-hedge a zero-coupon bond p( t, T) using two other zero-coupon bonds that mature on dates Tl and T2 . Use the self-financing condition and compute the sensitivity of changes in dx ( consider only first-order terms ) in the bonds in order to set up equations to find the portfolio weights. Is your portfolio insensitive to large changes in x(t)? Here, ( dX ) 2 must be considered. Include a third zero-coupon bond with maturity T3 in your portfolio to obtain a Gamma-neutral position. =
=
374
8.5
by
8. Interest Rate Theory
In the Gaussian HJM framework, assume that the volatility is specified O' (t, T)
exp{ ->" (T - tn , where 0' > 0 and >" 2: O. This specification leads to a deterministic, time stationary volatility, dependent on T - t and not T, t separately, which in creases time to maturity decreases. The HJM Condition (8.21) implies = 0' X
as
a(t, T)
=
0' 2
A
exp { - >" (T - t n ( 1 exp { - >" (T - tn) · -
1 . Verify that bond prices are given by p(t, T)
where
=
���,�1 exp {S(t, T)x(t) - a(t, Tn ,
x (t)
=
r(t) - 1(0, t) ,
S(t, T)
=
1 - >: { I - exp{ ->" (T - t n } ,
a(t, T)
=
(0' 2 / 4 >" )S(t, T) 2 ( 1 - exp { -2 >.. t } ) .
2 . Since x (t) i s not dependent on the maturity T , it can b e used as a single factor in a factor model (replacing e.g. r(t)) . Use x (t) to construct ..1- and
r-neutral portfolios of T-bonds. 3. Find the specific form of European call option prices in this setting (use (8.24) ) and compute sensitivities with respect to x(t) . 8.6 Verify the caplet Formula (8.25) in the setting of §8.4.4. (Repeat the deduction of (8.24) in §8.4.2.) 8 . 7 Prove Proposition 8. 1 . 1 part (i) .
9 . Credit Risk
Approaches to modelling financial assets subject to credit risk can roughly be divided into two types of models: reduced-form and structural models. While reduced-form models typically use a point process to model the default event ( exogeneously ) , structural models try to describe the default triggering event within the framework of all traded assets. The structural approach goes back to Merton (1974) , where the dynam ics of the value of the assets of a firm are described by a standard geometric Brownian motion and the default event is triggered by this value process crossing a default boundary given by the value of a single bond issued. Al though this model gave valuable insight into the default process, shortcomings have subsequently been raised: the liabilities of the firm are supposed to con sist only of a single class of debt, the debt has a zero coupon, bankruptcy is triggered only at maturity of the debt, bankruptcy is cost less and inter est rates are assumed to be constant over time. Thus, the assumptions of the Merton model are highly stylized versions of reality and are not able to account for the magnitude of yield spreads. This motivated several gener alizations of Merton's model: Black and Cox ( 1976) incorporate classes of senior and junior debt, safety covenants, dividends, and restrictions on cash distributions to shareholders, Geske (1977) considers coupon bonds by using a compound options approach and provides a formula for subordinate debt within this compound option framework. Leland (1994) extends the model further to incorporate bankruptcy costs and taxes, which makes it possible to work with optimal capital structure. Zhou (2001) and Madan (2000) use Levy processes to model the value of the firm process. While in most pa pers using option pricing frameworks bankruptcy is triggered as the moment when the value of the firm reaches the value of the debt, they model default as the time when the value of the debt reaches some constant threshold value K that serves as a distress boundary, i.e. the default time T can then be expressed formally as T inf {t � 0 V (t) ::; K} , the first passage time for V (t) ( the value of the firm's assets at time t) to cross the lower bound K. If the value of the assets breaches this level, default is triggered, some form of restructuring occurs and the remaining assets of the firm are allocated among the firm's claimants. Implicit in this formulation is the assumption that once this level is reached, default occurs on all outstanding liabilities at the same =
:
376
9. Credit Risk
time. Thus, contrary to Merton's model, default can occur prior to maturity. Nielsen, Saa-Requejo, and Santa-Clara (1993) , Briys and de Varenne (1997) and Hsu, Saa-Requejo, and Santa-Clara (1997) allow for stochastic default boundaries and deviation from the absolute priority rule. Because of the com plexities of all these extensions, often a closed-form solution can no longer be obtained and numerical procedures must be used. Literature related to credit risk in book form includes Sch6nbucher (2003) , Bielecki and Rutkowski (2002) and Duffie and Singleton (2003) . Madan (2000), Rogers (1999) and Lando (1997) are overview papers. 9 . 1 Aspects of Credit Risk
9 . 1 . 1 The Market
According to the International Swaps and Derivatives Association, the credit derivatives market grew 37%, with total notional outstandings reaching $2. 15 trillion during the first half of 2002. Notional outstanding volume in interest rate and currency derivatives increased 20%, to $99.83 trillion, in the first half, while equity derivatives outstanding volumes rose to $2.45 trillion up 6%. This growth demonstrates the importance of credit derivatives as a mechanism for mitigation and dispersion of credit risk. 9 . 1 . 2 What Is Credit Risk?
We can distinguish between individual risk elements: 1. Default Probability. The probability that the obligor or counterparty will default on its contractual obligations to repay its debt. 2. Recovery Rates. The extent to which the face value of an obligation can be recovered once the obligor has defaulted. 3. Credit Migration. The extent to which the credit quality of the obligor or counterparty improves or deteriorates, and portfolio risk elements: 1. Default and Credit Quality Correlation. The degree to which the default or credit quality of one obligor is related to the default or credit quality of another. 2. Risk Contribution and Credit Concentration. The extent to which an individual instrument or the presence of an obligor in the portfolio contributes to the totality of risk in the overall portfolio. Since credit risk focuses on default probabilities, recovery in default, identity of the counterparty - all factors that are not directly relevant to market risk - the modelling of credit risk requires the development of new techniques. In particular, since the underlying risk variable in credit risk - occurrence
9 . 1 Aspects of Credit Risk
377
or otherwise of default - is not normally distributed, we find credit portfolio distributions which are asymmetric with fat tails ( limited upside potential with remote possibilities of severe losses ) . In addition, when implementing models, we face the problem of sparse data. Data on credit events are much more limited than information on market risk. Credit events are infrequent and many credit instruments are not marked-to-market on a daily basis, so parameter estimation is difficult. Finally, market risk tends to focus on a relatively short time horizon - credit risk analysis is concerned with a much longer horizon. 9 . 1 . 3 Portfolio Risk Models
Banks implement credit risk models, which may be Ratings Based (Cred itMetrics) . So we need:
1. the
definition of the possible states for each obligor's credit quality, and a description of how likely obligors are to be in any of these states at the horizon date - Ratings and Transition Matrix, 2. the revaluation of exposures in all possible credit states - using term struc ture of bond spreads and risk-free interest rates, 3. the interaction and correlation between credit migrations of different oblig ors - use of an unseen driver of credit migrations. Ratings are in principle supplied by commercial firms, so-called rating agencies. Rating agencies evaluate the creditworthiness of corporate, mu nicipal, and sovereign issuers of debt securities. In the USA, where capital markets are the primary source of debt capital , rating agencies have assumed enormous importance. The importance of these agencies has been increased through the internal rating approach in the new Basel capital accord ( Basel II ) , which requires established rating agencies as benchmark. Of course, this increased importance triggers the question of how reliable rating agencies are. Several empirical studies ( some by the agencies them selves ) find: •
•
•
Moody's ratings have a high predictive power for defaults; spreads are generally higher for lower rated bonds, bond and equity values move in the expected direction when issuers' ratings change, ratings do help predict financial distress and bond spreads.
However, recent studies show the equity-based default probabilities ( see be low ) change well before ratings when firms fall into financial distress ( rating stickiness ) and bond prices based on average spreads show inconsistencies with actual prices. Equity-Based Models (Moody's KMV, Credit Grades) rely on the argument that a firm defaults when its asset value drops to the value of
378
9. Credit Risk
its contractual obligations (or a critical threshold - the default point). One can use an option pricing framework to derive the value of debt and equity. Each model contains parameters that affect the risk measures produced, but which, because of a lack of suitable data, must be set on a judgmental basis. Empirical studies such as Gordy (2000) and Koyluoglu (1998) show that parameterization of various models can be harmonized, but use only default-driven versions. 9 . 2 Basic Credit Risk Modeling
A complete mathematical framework is developed in Bielecki and Rutkowski (2002) . For our purpose, a simplification of their framework suffices. To in troduce our basic model, we specify a time horizon T* > 0 and assume an underlying stochastic basis (fl, F, JP, IF) with IF (Ft) O �t� T ' ) a filtration that supports the following objects: the firm's value process V , thought of as the total value of firm's assets, the barrier process (signalling process) which will serve to specify default time, promised contingent claim X, representing the firm's liabilities to be re deemed at time T � T* (other notation D , L) , 7, a default time, which - in the structural approach is defined as 7 := inf{ t > 0 : Vi < vt} so 7 is a IF stopping-time; - in the intensity-based approach is not a stopping time for the market filtration, recovery claim X , represents recovery payoff received at T, if default occurs prior or at the claim's maturity date T ( recovery at maturity), recovery process Z, specifies recovery payoff received at time of default, if default happens prior to or at T (recovery at default) . Technical Assumptions. V, Z, A , are progressively measurable with re spect to IF, X and X are FT measurable. All processes are assumed to satisfy further suitable conditions. Suppose there exists an equivalent martingale measure (EMM) JP* (im plying that the financial market model is arbitrage-free) . So discounted price processes of tradeable securities, which pay no coupons or dividends, follow IF-martingales under JP* . Let r be the short-term interest rate process and use as the discount factor the savings (bank), account, which we assume to exist: =
• •
v,
•
•
•
•
v
9.3 Structural Models
379
Let H
= (Ht ) = ( l{r$ t } ) be the indicator process of the default event, D = (Dt) cash flows -received by the owner of the defaultable claim and X d ( T ) = X1{r> T } + X1{ r $ T } . Then the process D of a defaultable claim, which settles at time T, equals
Dt
=
X d (T) 1 { t � T } +
J Zu dHu ,
(O , t]
where the first term takes care of the payoff at T ( if any ) and the second term captures the payments in case of premature default. Observe that D is of finite variation over [0, T] . Now let Xd (t , T) be the price process of a default able claim. Thus, Xd (t, T) represents the current value at time t of all future cash flows associated with a given default able claim. We have Definition 9 . 2 . 1 (Risk-neutral Valuation Formula) . The price process
of a defaultable claim which settles at T is given as
X d (t, T )
=
BtIE*
(J
( t , T]
B; / dDu Ft
)
Vt E [0, T] .
Use of the formula depends on the attainability of a defaultable claim, which is not obvious. Usually one argues that pricing the defaultable claim accord ing to the above formula does not introduce an arbitrage opportunity into a previously arbitrage-free market. Example.
In case of recovery at maturity, we have Z
=
o.
So
Then the valuation formula is X d (t , T )
=
BtlE*
( (X1{r> T } + X1{r$ T } ) BT l I Ft ) ,
and the discounted price process follows an .IF-martingale under JP* ( given some integrability conditions ) . 9 . 3 Structural Models
9.3.1 Merton's Model
The basic foundations of structural models have been laid in the seminal paper Merton (1974) . Here it is assumed that a firm is financed by equity
380
9. Credit Risk
and a single zero-coupon bond with notational amount ( face value) F and maturity T. The firm's value is given by dV(t)
=
(r - 8)V(t)dt + o"V (t)dW(t)
under an equivalent martingale measure 1P'* , with r, a constant, W Brownian motion and constant payout ( dividend ) rate 8, which may be negative ( i.e. pay-in ) . Default is only possible at maturity. There are two possibilities; Vr
or
:::::
0 and the ( l +Ei) are log-normally distributed with parameters 2 lE* log(l + E ) i - 82 , War* log(l + ) 8 2 , lE*E k ei' - l under JP * . We find for the put price =
E
=
=
=
with parameters A'
=
5.(1 + k) ,
Theorem 9.3. 1 . The price of a credit risky bond is given by
p(O, T) = p(O, T)F
( ( X T) exp{ -A' T} �1 t}. T
9.3 Structural Models
= JE * ( L e -r(T - t) l {f�T,vT � L } 1 Ft ) + JE* ( ,61 VT e- r(T - t) l {T�T,vT < L} 1 Ft ) + JE* ( K,62 e - I'(T - f) e -r(T - t ) 1 { t ( - IOg(Vt/ii(t)) + v(s - t) ) , ii (t )
2
a
a�
386
9. Credit Risk
-
with a V/U 2 (r "I - u 2 /2)/u 2 . By Equation (9.3) , we find that for every t < s :::; T and x � v ( s ) we have on {T > t} (9.5) =
=
{p
=
_
K,
log(x / v (s) ) v (s - t) ) ( log( Vt /V (t)) - u� log(x/ v (s) ) + 1/ (s - t) ) . ( VVt(t) ) 2a {p ( - log( Vt /V (t)) -u� +
-
Proposition 9.3.2. Set i/ = 1/ "I and a = i/u - 2 . Assume that i/ 2 + 2u 2 (r > O. Then the price process of a defaultable bond on {T > t} equals
"I )
pd(t, T)
Lp(t, T) ({P(h l ( Vt , T - t)) - R�ii{p (h 2 ( Vt , T - t)) )
=
+ i31 Vt e - It( T - T ) ( {P (h3 ( Vt , T - t)) - {P ( h4 ( Vt , T - t))) + i31 vt e-It(T - T ) R�lL+ 2 ( {P (h5 ( vt , T - t)) - {P (h6 ( vt , T - t) ) )
(
where Rt
=
)
+ i32 Vt RfH{p (h 7 ( Vt , T - t)) + RfH{p ( hs ( Vt , T - t)) , v (t)/ Vt , () a + 1 , ( u - 2 Ji/2 + 2u2 (r "I ) and =
h 1 (V, T t) t,
_
h2 ( V,t , T t) _
_
h3 (V,t , T t) h
4
(V,t , T
h
5
(V,t , T
=
=
=
_
t)
=
_
t)
=
h6 ( V,t , T t )
=
h 7 (V,t . T t)
=
_
_
hs ( V,t , T t ) _
-
_
-
=
log( Vt /L) + 1/ (T - t) ' uvT t log v 2 (t) - log(L Vt ) + v(T - t) ' uVT t log(L/ Vt ) - ( 1/ + ( 2 ) (T - t) ' uVT - t log(K/ Vt ) - ( 1/ + ( 2 ) (T - t) ' uVT - t log( v 2 (t)) - log(L Vt ) + ( 1/ + ( 2 ) (T - t) ' uVT t log v 2 (t) - log(K Vt ) + ( + ( 2 ) (T - t) ' uVT t log( v (t)/ Vt ) + ( u 2 (T - t) ' uVT t log( v (t)/ Vt ) - ( u2 (T - t) . � uvT -t _
_
_
1/
_
_
9.3 Structural Models
Proof (Outline) .
387
We need to evaluate
D1 ( t, T)
=
Le - r (T - t ) JP* ( r 2 T, VT 2 L I Ft ) ,
We consider only t = ° for convenience (for the general case apply the strong Markov property). Now the formula for D1 (O , T) follows from Equation (9.5) . To compute D2 (O, T) we observe again using (9.5) L
=
J XdJP*(VT X, r 2 T)
t} the value of senior debt is
L
L,
Lj
388
9. Credit Risk
p� (t, T) = pd (t, T ; Ls , v)
and at time of default it is min{v (T) , Lsp(T, T) } for T < T. Now on the value of junior debt is
{ T > t}
pj (t, T) = pd (t, T) - P s (t, T) = p d (t, T ; L, v) - pd (t, T ; Ls , v) and at T < T it equals min{ v (T) - Lsp(T, T) , Lj p(T, T) } . If v (t) = Kp(t, T) for some constant K :::; L , we get if K = L, Ljp(t, T) pj (t, T) = p d (t, T) - Lsp(t, T) if Ls :::; K < L,
{
p d (t, T) - p� (t, T)
if K < Ls .
9.3.4 Structural Model with Stochastic Interest Rates
We now introduce stochastic interest rates into the modelling framework, as e.g in Black and Cox (1976) , Longstaff and Schwartz (1995) and Briys and de Varenne (1997) . We assume the same dynamics of the firm value process as before, i.e. (9.6) where rt , Kt and u(t) are suitable processes. rt models the stochastic short rate, K t is the dividend-rate and u (t) specifies the volatility. The dynamics of default-free bond prices are given by dp(t, T) p(t, T) (rt dt + b(t, T)dWt ) . (9.7) We consider the forward value of the firm Fv (t, T) Vt /p(t, T) under the T - forward measure jpT . Now (9.8) dFv (t, T) Fv (t, T) ( - K t dt + (u(t) - b(t, T) ) dWt) , T where W T is a JP Brownian motion. Now Fv (t, T) Fv (t, T)e - It I« u) du has the dynamics (9.9) dFv (t, T) FV (t, T) (u(t) - b(t, T) )dWr If (u( t) -b( t, T) ) is deterministic, Fv is a Gaussian process. In case of a bound ary function of the form Kp(t, T)e- It I« u) du one can exploit this property to again obtain an analytic expression for the forward value of a defaultable bond. The main steps are an application of the change-of-numeraire tech nique to transfer the valuation problem to the forward values (as outlined above) , and in case (u(t) - b(t, T) ) is constant, calculation of first-passage times similar to the ones performed in the last section. If (u( t) - b( t, T) ) not constant one has to perform a deterministic time change to be able to use the technique above. For background on time-changes of Brownian motion, we refer to e.g. Revuz and Yor (1991), V. l . =
=
=
=
=
9.3 Structural Models
Note.
389
In Equation ( 9.8 ) , the key component is the 'volatility' coefficient,
(u(t) - b(t, T)) , which we need to be deterministic in order to find explicit
formulae. This is a strong restriction, but does include models with both volatility and interest rates being stochastic. 9.3.5 Optimal Capital Structure - Leland's Approach
The basic idea is that equity holders ( = owners of the firm ) can choose the bankruptcy policy in such a way that the value of the equities will be maximized ( or the value of debt will be minimized ) . We assume the standard model with 0 and r > 0 constant. Also, the outstanding debt is a consol bond, i.e. a bond with infinite maturity, which pays continuously at a constant rate c. Its price Dc (t) at any date t E 1R+ equals /'i, =
D, (t)
,�
(1
T� E'
ce-> ( · - 'l l{,,,) dB
(
:F')
+ )�moo JE * K,82 el'(r- t ) e-r(r- t ) 1 { t t) > O. The associated jump process H ( Ht ) = (1 { r 9 } ) is called the default process. Define 1H as the filtration generated by H, i.e. 1it = a(Hu u S t) a({T S u } : u S t). Consider the enlarged =
=
=
=
:
=
392
9 . Credit Risk
filtration G IF V IH with gt Ft V 'Nt a(Ft , 'Nt ) . T is not necessarily a stopping time w.r.t. IF, but T is a stopping time w.r.t. G. We want to value default able claims within the framework of a financial market model for which we assume that IF contains the market information (and IH contains the information on the default time) . In this market, we assume the existence of a savings account =
=
=
where is the short-term interest rate process. We assume the existence of an equivalent martingale measure JP * (risk-neutral measure) such that the discounted price process of any tradeable security, which pays no dividends or coupons, follows a G-martingale under JP* . Recall that the cash-flow process Dt (payments from time t on) of a default able claim equals r
Dt
=
X d (T) l{t>T} +
J
(O, t ]
ZudHu ,
where the pay-out at maturity T is Xd(T) X l { o t } + Zl { r::; T} and the pay out in case of premature default is modelled by Z . Let X d (t, T) be the price process of the defaultable claim. Using the risk-neutral valuation formula we obtain =
Xd (t
,
T)
=
BtIE*
(J
B;; l dDu gt
(t ,T)
)
' t} B n {T > t}. The usual measure-theoretic approximation argument implies then that for a gt-measurable random variable Y there exists an Frmeasurable random variable Y such that Y = Y on {T > t}. Furthermore, we find that for any g-measurable random variable and any t E 1R+ we have
Ft
=
Y I Ft) IE * ( l {T> t } Y I gt ) l {T> t } IE*IP*(l{T>t} (l{T>t} J Ft ) =
t
(9.16)
One then only needs to use Equation (9. 16) together with the tower prop0 erty of conditional expectation to prove the claims. The final step is to establish a representation for the pre-default value of a defaultable claim in terms of the hazard process ( or its intensity 'Y in case it exists ) of the default time. In doing so we will be able to find convenient formulations for the valuation equation (9. 1 1 ) in important applications.
r
9.4 Reduced Form Models
395
Theorem 9.4. 1 . The value process X (t, T) of a defaultable claim has the following representation for t E [0, T] .
1. X d (t , T )
=
l {r>t} Bt IE*
(J
B;; l e rt - ru Zu dru + BT l X e rt -rT Ft
( t ,Tj
)
.
2. In case r admits an intensity 'Y X d (t, T)
=
l {r>t } IE*
(J
e - ft ( r . + 'Y. ds ) 'Yu Zu du Ft
( t , T]
(
l )
)
+ l {r>t} IE* e- f,T ( r s + 'Y. ds ) X Ft .
For proof and further discussion, we refer the reader to Bielecki and Rutkowski (2002) , Chapter 8. Valuation of General Defaultable Claims. We can now use Theorem 9.4.1 to value various defaultable claims. We will always assume that r admits an intensity 'Y. Fractional Recovery of Par (Face) Value. Let V represents the claim's constant par value and 8 the claim's recovery rate. Thus the pre-default value of the claim has no influence on the recovery in ca..c;e of default. We set Zt 8 · V, 0 ::; t ::; T and Theorem 9.4.1 yields =
X6 (t, T )
=
l{r>t} IE*
( J 8V
(
( t ,Tj
e - ftU (r s + 'Ys ) ds 'Yu du Ft
I )
)
+ l {r >t } IE* e- f,T ( r' + 'Ys )d S V Ft .
Fractional Recovery of No-default Value. Here, it is assumed that in case of default a fixed fraction of an equivalent non-defaultable security is received. In case of a default able bond this scheme is known as fractional recovery of treasury value. By the risk-neutral valuation formula, the time t value of the non-defaultable equivalent security xe is given by the discounted expectation ( under JP* ) of the promised payoff X, so
Now assume that Zt 8xe (t, T) with 8 the recovery rate. The valuation equation ( 9.1 1 ) , Theorem 9.4.1 and an application of Fubini's theorem yield =
396
9 . Credit Risk
Xe , O (t, T )
= (1
-
(
8 ) 1 { r> t} 1E *
+ 81{ r> t } 1E*
(
e
-
e
-
l )
It Crs + 'Ys ) ds X Ft
)
It r s ds X l gt .
For corporate bonds, one can find a convenient pricing formula. Let L 1 8 be the stochastic loss rate and L t = lE* ( L I Ft ) the risk-neutral mean fraction of market value if default occurs at time t. Thus Lt captures all information about recovery that is important for bond pricing. The pricing formula =
-
(9.17)
where S t 'Yt L t is now the risk-neutral conditional expected rate of loss of market value, and the necessary technical conditions under which it holds, are given in Duffie and Singleton ( 1 999) . Building on its specific form, one can build tractable models for defaultable bond pricing models parallel to the default-free models. In particular, HJM-type forward spread models can be constructed. One observes the initial default-free forward rates 1(0, T ) and the initial forward spread rates s(O, T) . After specification of the volatilities a f (t, u) for forward rates and as (t, u) for spreads and fractional loss at default L, default able bond prices are given as =
In addition to the standard HJM-drift condition, the spread dynamics ds(t, T )
=
f..L s (t, u)du + as (t, u)dW(u)
have to satisfy a drift condition (under recovery of market value) f..Ls (t, u)
=
as (t , u)
T
J af (t, v )dv t
+af (t, u)
T
J as (t, v )dv , t
see Schonbucher ( 1 998) and Schonbucher (2003) for additional details. Reduced-form Models with State Variables. In many applications, it is useful to think of underlying state variables that drive the economy (economic cycle) and subsequently have an influence on the default intensity. To model such state variables, we assume that Y is a d-dimensional stochastic process defined on (n, g , IF, IP* ) and follows a IF-Markov process under IP* .
9.4 Reduced Form Models
397
One can then model T as the first jump time of a Cox process, which has an intensity of form At A(yt) for some function A JRd -+ JR+ , as e.g. in Lando (1998). Under the further assumptions that the promised payoff X at T of the default able claim is FT -measurable, that the recovery process Z is JF predictable and finally, that the s.hort-rate process satisfies rt r(yt) for some function r JRd -+ JR, we can apply Theorem 9.4.1 to obtain =
:
=
:
Proposition 9.4. 1 . The price process of a defaultable claim with the above
specifications is given by
Motivated by the pricing formula (9.17), Duffie and Singleton (1997) , Duffie and Singleton (1999) and Duffee (1999) used an econometric model for the term structure of credit spreads. They modelled the short-rate process and the short-spread process using underlying square-root process state variables Xl , X2 , X3 • They as sumed that the state variables satisfy dX1 (t)
=
[lI:u ( lh - X1 (t)) + 11:1 2 (02 - X2 (t) )]dt + JX1 (t)dW1 (t) ,
dX2 (t)
=
11: 22 (02 - X2 (t) )dt + 0"22 VX2 (t)dW2 (t) ,
dX3 (t)
=
11:33 (03 - X3 (t) ) + 0"3 2 VX2 (t)dW2 (t) + 0"33 VX3 (t) dW3 (t) .
Conditions on the coefficients are needed to ensure that s et) > 0 (positive affine function of correlated square-root diffusions) and the correlation be tween s and r is negative. 9.4.2 Rating-based Models
Usually there is a deterioration of credit quality until risky debt goes into default mode - this is called credit migration. Credit quality corresponds to the probability that a firm will be able to meet its contractual obligations. Ordering firms according to their default probabilities leads to rating systems
398
9. Credit Risk
Year Rating AAA AA A BBB BB B CCC
1
3
2
4
0.00 0.05 0 .00 0. 1 1 0.00 0.02 0.07 0 . 1 5 0.04 0.12 0.21 0.36 0.24 0.54 0.85 1 . 52 1.01 6.32 9.38 3 . 40 5.45 12.36 19.03 24.28 23.69 33.52 4 1 . 1 3 47.43 Table 9 . 1 .
5
6
7
8
9
10
0 . 1 7 0.31 0.47 0 . 76 0.87 1 .00 0.27 0.43 0.62 0.96 0 . 77 0.85 0 . 56 1.01 1 .69 0.76 1 .34 2.06 4.55 2.19 2.91 3 . 52 4.09 5.03 12.38 15.72 17.77 20.03 22.05 23.69 28.38 31 .66 34.73 37. 58 40.02 42.24 54.25 56.37 57.94 58.40 59.52 60.91
Standard and Poor's cumulative Default Rates ( Percent)
(discrete) . These rating systems allow empirical calculation of transition ma trices. The table below shows Standard and Poor's cumulative Default Rates (Percent) ordered according to the rating classes of S&P. One can now assume that the credit quality of a firm is a continuous time Markov chain M on a finite state space E = ( 1 , . . . , K, K + 1) (the rating classes) with transition probability matrix pet) . We assume that the state K + 1 corresponds to bankruptcy and that a bankrupt firm remains in that state. As usual, we call Pi , j the probability that at time t the process, which started in state i, is in state j. Thus P k + l, i (t) = 0, i = 1, . . . K and P K + l, K+l (t) 1. The semigroup of M is =
pet) = exp(tA)
=
00 (tA) k
L �'
(9. 18)
k=O
where A is the generator matrix -AI , l Al ,2 A2 ,l -A2 , 2 A= AK- l ,l AK- I , 2 . . . -AK- l ,K-l AK- I , K o
o
o
where by definition
o
Pi,j (t) - Pi,j (O) A 'O ,J = tlim . --+O t We have Ai,j � 0, i =f. j and Lj A i,j 0, Vi. Now assume that a firm has rating k(t) at time t. Let A i,j (Xt) > 0 be the state-dependent, risk-neutral transition intensity from rating i to j (where Xt is a suitable process for the state variable) . With ret) = r(Xt) default able bond prices are as usual 0
=
9.5 Credit Derivatives
Pd (Xt , kt , t, T)
=
IE
399
( e- It r (u)du l{ r > T} I 9t ) ( e - It r (u)du 81 {r�T} 1 9t ) ,
+IE
where 8 is the recovery rate. Now assume fractional recovery of market value, so 8 = (1 - L)Pd (r- , T) with a constant L. Then Pd (Xt , kt , t, T)
=
IE
( e - It (r (u)+h (u) L)du I 9t ) ,
where h(t) Llk t , K+1 (Xt ) is the rate of transition from current rating into default. For further details on rating-based models, we refer the reader to Jarrow, Lando, and Turnbull ( 1 997) , Lando ( 1 998) , Lando (2000) and Bielecki and Rutkowski (2002) , Chapters 1 1 and 12. =
9 . 5 Credit Derivatives
We only discuss two specific examples. Credit Default Swaps, CDS. A credit default swap is an exchange of a periodic payment against a one-off contingent payment if some credit event occurs on a reference asset. The basic cash flow is shown in Figure 9 . 2 contingent payment Protection Buyer
f----
�
Protection Seller
periodic fee Table 9 . 2 .
Cash flow of a credit default swap
The ingredients of the basic structure are specification of 1 . maturity T: usually from one to ten years, 2. underlying: corporate or sovereign, 3. credit event: default, bankruptcy, downgrade. Let c(T) be the fixed coupon that the protection buyer pays. The payment continues until either default or maturity. In case of default, assume that the payment from the protection seller to the protection buyer is equal to the difference between the notational amount of the bond and the recovery value 8. The fixed side of the payment is set so that contract value is zero at initiation. Thus, since the cash flow at coupon date i for the protection buyer is c(T) l { r> i } and the payment for the protection seller at time of default r is ( 1 - 8 ) 1 { r � T} ' we obtain
400
9. Credit Risk
where we assume constant interest rates. Since both ]E* (e - r T l { r$T } ) and !P* (l { r>i}) are readily available in intensity-based form models (or can be inferred from market data assuming such a model) , these models are typically used to price CDS. Extensions of CDS include: 1. contingent credit swaps, which require an additional trigger, i.e. a credit event with respect to another entity or movement in equity prices or interest rates, 2. total (rate of) return swaps, which transfer an asset's total economic performance including - but not restricted to - its credit-related performance. First-to-default Swap (FtD) and Basket Default Swap. Now several assets are bundled together, and a credit swap is created on the whole basket. The default event is defined in terms of default on any of the assets in the basket, e.g. the first default of any asset in case of a first-to-default swap or the ith-to-default, or any other similar contract structure. The additional modelling component to be considered now is the dependence of defaults (e.g. clustering of defaults) . For the FtD, recall that for intensity-based models the default time T min{ T1 , . . . , Td } has intensity A A 1 + . . . + A n , where Ai is intensity of Ti , the default time of asset i. Thus, with an affine model for Ai , tractable models for FtD can be obtained. Further details on credit derivatives can be found in Bielecki and Rutkowski (2002) and Sch6nbucher (2003) . =
=
9 . 6 Portfolio Credit Risk Mo dels
We will only consider a rating-based credit portfolio model, such as Credit Metrics. A formal description of such a model consists of n + 1 rating cate gories (of which the first one corresponds to the default state) , a transition matrix of probabilities of rating changes within the time horizon of interest, and some re-evaluation procedure for the exposures within each rating class. To introduce dependencies of the individual exposures, a latent factor driving the transitions is assumed. Model Description. Assume the portfolio consists of bonds (obligors) that we consider at discrete time periods t = 0, 1 , . . . , T corresponding to the coupon payments. Let Ri (t) be the state indicator (i.e. rating class) at time t. We assume n + 1 rating classes i.e. the state space is {O, 1 , . . . n } with class o corresponding to default. m
9.6 Portfolio Credit Risk Models
Let
Furthermore, let
S =
- 00
401
(SI , . . Sm ) ' be a m-dimensional random vector. .
= C- l < Co < C l < . . . < Cn
be a sequence of cut-off levels. We assume
=
00
We call (Si (t) , ( Cj ) j E { - l , o , . . . , n } ) i E { l , . . . m } a dynamic latent variable model for the state vector R = (R1 , , Rm ) ' . Dynamic Modelling. In general Merton-type ( or asset-based ) models the value of Si (t) may be interpreted as the value of the assets of the firm. Indeed, a variety of distributions is possible for Si corresponding to various Levy-type specifications for stock price modelling ( e.g. the hyperbolic model, Eberlein (2001), the Variance-Gamma model and relatives, Carr, Geman, Madan, and Yor (2002), Carr, Chang, and Madan (1998)) . In particular, jump-diffusion models are easily incorporated, see Hamilton, James, and Webber (2001), for such a model. The relation to the standard approach is seen by recalling the standard Black-Scholes model defined via the SDE .
•
.
with constant coefficients and a standard Brownian motion W . The solution of the SDE is A t Ao exp �2 t + Wt
{,.tt
=
-
a
},
hence motivating the use of a normal return distribution. One can now con sider a general exponential Levy process model for asset values with a Levy process L. Now the solution of the SDE can be written as At
=
Ao exp { Lt }
with Lt a related Levy process ( which can be computed explicitly ) . Hence we may assume that the return distribution St log ( A d is generated by a Levy process. See §5.5, §7.4, and Bingham and Kiesel (2002) for an overview and further discussion of Levy-type models. Factor Modelling. In a typical credit portfolio model, dependencies of in dividual obligors are modelled via dependencies of the underlying latent vari ables S. In light of the typical portfolio analysis, the vector S is embedded in a factor model, which allows for easy analysis of correlation, the typical measure of dependence. One assumes that the underlying variables Si are driven by a vector of common factors. Typically, this vector is assumed to be normally distributed ( see e.g. JP Morgan (1997) ) . Let Z N(O, E) be =
rv
402
9. Credit Risk
a p-dimensional normal vector and € = ( 10 1 , 10m ) ' independent normally distributed random variables, independent also of Z. Define •
•
•
,
p
. . . m.
L aij Zj + Uifi, i 1 , j=l Setting Yi = Si generates a Gaussian factor model. However, such a set ting corresponds to the standard Brownian motion return structure. To cap ture Levy-type equity models, we use a normal mean-variance mixture model ( compare Bingham and Kiesel (2002) ) . To define such a model, let W be a further positive random variable, independent of € and Z and define Si ai + biW2 + WYi, i 1 , . . with ai, bi constants. Then S has a (p + I ) -dimensional conditional indepen dence structure. Now S inherits ( in principle ) the correlation matrix of Y . The individual returns Si are heavy-tailed and the vector S exhibits tail dependence. Such a model could alternatively be generated by using heavy-tailed factors and heavy-tailed idiosyncratic risk. An advantage of such a model is that it allows analytic approximations, in the sense that the loss distribution can be approximated as a function of the factor risk only. Approximation of Loss Distribution. Assume that 1 , i.e. we only have two rating classes, one of which corresponds to default. We calculated the distribution of number of defaults for one period. Assume the collateral consists of bonds, all from the same rating class implying a common default boundary Assume one normally distributed common factor ( as in Gordy (2000) ) , i.e. Yi pZ + V I - p2 fi and Si a + bW2 + WYi a + bW 2 + pWZ + VI - p2 Wfi. So the conditional distribution of Si given (Z, W) is normal. Also by condi tional independence, we get for the number M of defaults in one period ( let W have density g). Yi
=
=
=
=
.m
n =
c.
=
=
7r1
=
ooJ Joo (m)tP (c 00 (
= JP (M = l) =
o
-
l
xtP
) - )
a - bw 2 - pwz l V I - p2 w m -l c - a - bw 2 Pwz ¢J(z)g(w)dzdw. 1 - p2 w � -
9.6 Portfolio Credit Risk Models
403
Using the above formula, we can obtain the loss distribution analytically; see also Croughy, Galai, and Mark (2001), Ong ( 1 999) and the article Frey and McNeil (2003) and references. Multi-period losses: here we simply assume independence of the subse quent periods. Hence the only change in the above calculation is an adjust ment of the number of bonds in the collateral. The distribution of losses in the second period is given as m
IP(M2 l) jL=O IP(M2 = ll M1 j)IP(M1 j) jL=O IP(M(m - j) = lI M (m) = j)7rj(m), with the notation that M (m) is the number of defaults given that m bonds are in the collateral, and 7rj(m) = IP(M(m) j). An application of a strong law of large numbers ( Hall and Heyde ( 1 980 ), =
=
=
=
m
=
=
Theorem 2. 19) yields a convenient approximation: 11. m
.!.M( )
n -+ oo n
n
_
- A'o �
(c -
b 2 - pwz 1 - p2
a - W � V W
)
a.s.
(9. 1 9)
Approximation (9.19) allows us to compute the one-period number of losses given the joint distribution of (Z, W) . See Lucas, Klaassen, Spreij , and Straetmans ( 1999) for such an analytic approach. Copula Modelling. Multi-variate Normal models don't show tail depen dence, i.e. for bivariate normal with correlation p E (-1, 1), we have oX = 0, where
(Xl, X2 )
The degree of tail dependence depends on the copula of to which we now turn. A is a multivariate distribution with standard uniform marginal distributions. That is, is a mapping [0, l]d ---+ [0, 1 ] with - 1, . is increasing in each component = for all i E { I , . . . , d} , E . . , 1 , 1, . For all ( a I , . . E [0, l]d with a � we have: • •
•
(Xl, X2 )' copula C C C(ul ,U ) Ui,Ui [0 , 1]' C( , Ui, d, a ), ,(b1,1) . . Ui. , b ) i bi d d 2 2 L · · · L (-1) i 1 + . . + id C(U1, i 1, . . . , Ud ,tLd ) :::: 0, i1 =1 id= l where Uj , l and Uj , 2 = bj for all j E { I , . . , d } .
. .. . . . =
aj
.
404
9. Credit Risk
If X
FI , .
. .
= (X l , . . . , Xd )' has joint distribution F with continuous marginals , Fd , then the distribution function of the transformed vector
is a copula C and Thus copulas can be used to link marginal distributions (alternative to using the joint distribution) . We can now consider a general latent-variable model. That is, we model each of the underlying variables independently and subsequently use a copula to model the dependence structure. If we assume that the copula function is symmetric (in all variables) , the random vector S will be exchangeable: d
(8I , . . . , 8m ) = (8p(I) , . . . , 8p(m) )
for every permutation p of { I , } In this case, all possible k-dimensional marginal distributions are identical, and in particular, we have for the default probabilities 0, . . 0) , V {i l , . . . , i k } C { I , . } 1 � k � m. 7rk In this case, the distribution of the number of defaults can be computed via an application of the inclusion-exclusion principle: . . . , m
=
JP(Ril
JP(M
=
=
k)
=
.
,Rik
.
=
. . , m
(7)JP(RI 0, . . . ,Rk =
=
O, Rk + 1
#
0, . .
,
. ,Rm
#
0) (9.20)
If we fix the individual default probabilities of a group of k obligors to be we get (9.21) 7rk = C1 , . . . k ( . . . , 7f ) where C1 , ... , k is the k-dimensional margin of C. Using copulae with positive tail dependence for the factors (e.g. t distribu tion) leads to heavier tails of loss distribution, i.e. increased VaRs (compare results from the study of Frey and McNeil (2003) ) . 7r,
,
7r ,
9 . 7 Collateralized Debt Obligat ions ( CD O s )
9.7. 1 Introduction
Collateralized Debt Obligations (CDOs) are an important example of asset backed securities (ABS) and as such are backed by a pool of assets. Basic
9 . 7 Collateralized Debt Obligations (CDOs)
405
information on ABS and CDOs is given in Bowler and Tierny (1999) , Lucas (200 1 ) and Rayre (200 1 ) or in textbook form Bluhm, Overbeck, and Wagner (2003 ) . In the case of CD Os we distinguish two basic types, based on the type of debt used as the collateral: Collateralized Loan Obligations ( CLOs ) , backed by a pool of loans, and Collateralized Bond Obligations ( CBOs ) backed by a pool of bonds. The typical st.ructure is shown below: Collateral
----+
----+ ----+
Spy
----+
----+
----+
Notes
So we start with a pool of credit risky assets. This pool is then transferred to a special purpose vehicle (SPV) , which is a company set-up only for the purpose of the transaction. The SPY then issues securities or structured notes backed by the cash flow of the asset pool. Thus interest and principal of the notes are paid from interest and principal proceeds from the pool. The notes are divided in several classes according to their credit quality: senior notes and mezzanine, which usually carry ratings from triple-A to single-B. There is frequently an unrated equity class. The holders of the notes are paid interest and principal in order of seniority One can distinguish the following types of CDOs. Arbitrage cno s . In an arbitrage CDO, an issuer seeks to capture an arbi trage between the pricing/yield of high-yield sub-investment-grade securities that are acquired in the capital markets and yield on investment-grade bond assets that are sold to investors. This allows investors who could not otherwise invest in sub-investment-grade assets to participate in this market. There are two basic types of arbitrage CD Os, namely cash-flow CD Os and market value CDOs. While for the former, credit events short of default are not relevant for the performance, the collateral pool of the latter is marked to market regularly, and the asset manager is required to trade actively. Conventional cnos. A balance sheet CLO is typically created by a bank or financial institution wishing to securitize illiquid loan assets that they have originated. The loan assets may be fairly heterogenous, although most balance sheet CLOs have been done using investment-grade loans. 9.7.2 Review of Modelling Methods
To construct a model to evaluate CDOs, the following quantities have to be modelled: default probabilities for the asset in the pool, default dependence of the assets, loss in event of default, timing of default. We will now review several modelling approaches that have been proposed in the literature and are being used in practical applications. •
•
• •
9. Credit Risk
406
Moody's Binomial Expansion Technique. A simple approximation is given by Moody's Binomial expansion technique ( BET ) , which is based on a diversity score: represent the loss distribution of N bonds from the same industry by that of M ::; N independent identical bonds, i.e. construct a comparison portfolio. The parameter M is known as the diversity score ( DS ) . Table 9.7.2 reports the suggested diversity score for a standard application.
I Number of firms I Diversity score I 1 .0 1.5 2.0 2.3 2.6 3.0 3.2 3.5 3.7 4.0 case by case
1 2 3 4 5 6 7 8 9 10 >
10
Table 9 . 3 .
Moody's Diversity Score
Given the diversity score, Moody's technique is as follows. To calculate default probability under a diversity score of M, we can now use the Binomial distribution. Thus Z, the number of defaulting bonds, has distribution
qn
p
= JP(Z
= n
)=
(�)pn ( l _ p) M - n ,
with the default probability according to the rating of the tranche. Fur thermore, the expected loss is computed as
Ln
M LqnLn, n=O
where is the loss incurred when bonds default in the portfolio. The problematic aspects of this approach are that there is no probabilistic basis for diversity score. The idea is simply to match the mean and the standard deviation of the return distribution associated with the collateral pool. Infectious Defaults. This is a first application in this field of the important general phenomenon of market contagion. The idea has been put forward by Davis and Lo ( 2001a ) and Davis and Lo ( 2001b ) , to which we refer for details of the calculations involved. The basic idea is that a bond can either default directly or may be infected to default by the default of a different bond ( infectious default ) . n
9 . 7 Collateralized Debt Obligations (CDOs)
.. +Zn
407
So assume that we consider n bonds, and use indicator Zi 1 if bond i defaults, Zi 0 otherwise. Then N Zl + . is the number of defaulted bonds. For i = 1 , . . , n and j 1, . . . , n with i =f=. j let Xi , Yij be independent Bernoulli random variables with lP(Xi = 1) p and lP ( Yij = 1 ) = q . Then =
.
Zi = Xi
=
=
=
=
(
+ (1 - Xd 1 - Il( l - Xj }}i ) Hi
)
,
where the second term models infection. lP(N k) can be computed in closed form, also lE(N) = n ( 1 - ( 1 - p) ( l - pq ) n - l ) and WareN) = lE (N) + n(n l ) ,B�q (1E(N)) 2 . To see the effect of q, we keep the expected number of defaults constant while increasing q. We consider a total of n 50 bonds. So
-
=
=
q implied p std. dev. 0.5 3.54 a 6.05 0. 194 0.05 0.1 7.70 0. 1 16 0.2 10.32 0.064 Table 9 . 4 .
Effect of infection parameter
q can be used to model the volatility of the default (loss) distribution. To model the timing of defaults, the following extension of the model is used. Assume n bonds with exponentially distributed default time, then t p = 1 - e- A is probability of default within [0, t] . Let Nt be the number of defaults in [0, t] , then the default rate is proportional to the number of bonds alive: lP(default in [t, t + dtl l N ) lE(dNt I Nt ) = A(n - Nt ) dt. For met) = lE(Nd , we have t t
m et)
-
=
nAt
=
-J -
Am(s)ds
o
(as Mt = Nt J� A(n - Ns )ds is a martingale) . Thus met)
= n
(1
e - At ) .
Interaction is modelled by increasing the total hazard rate after a default by a factor a for an exponentially distributed time interval. It is possible to compute distribution of the number of defaults; see Davis and Lo (2001b) for details.
408
9. Credit Risk
Duffie and Garleanu (200 1 ) present an inten sity-based model. They model the default intensities of bonds as
Intensity-based Approach.
with X l ' . . . ' Xn , Xc independent affine processes ( say of CIR-type ) . The common intensity factor allows one to model dependency. Efficient simu lation methods are available for obtaining the distribution of the number of defaults. Ratings-based Modelling. One could also use a diffusion-driven Credit Metrics extension. Here every bond is modelled using the structural approach, i.e. we model the underlying value of the firm by a diffusion process. Default occurs if the value-of-the-firm process falls below a threshold. Dependencies within the pool can be captured by the cross-variation of the underlying Brownian Motions. See also Hamilton, James, and Webber (2001) for an extension of this approach.
A . Hilbert Space
Recall our use of n-dimensional Euclidean space JRn , the set of n-vectors or n-tuples x (X l , " " Xn) with each Xi E JR. Here one has the ordinary Euclidean length - the norm =
- and the inner product ( or dot product ) of two vectors x and y: (x, y) ,
or
X ·
y, :=
n
L XiYi . i= l
This setting is adequate for handling finite-dimensional situations, but not for infinite-dimensional ones. The simplest infinite-dimensional situation contain ing the above as a special case is the space £2 of square-summable sequences x = (Xl , X 2 , ' " ) with Because of the Cauchy-Schwarz inequality
if x , y E £2 , then is defined, and
(x , y ) :=
00
L XnYn
n= l
l ( x , y ) 1 :s II x li l l y ll
convergent, that is, L: :::'=l IXn l lYn l < the above ) . One may choose to work instead with complex sequences: all the above goes through, with the changes
( the series L::::'= l x nYn is absolutely - just replace Xn , Yn by IXn l , IYn l in
00
410
A . Hilbert Space 00
(x , y ) : = L X n Yn .
n= l A Hilbert space is a ( possibly - indeed, usually ) infinite-dimensional vector space endowed with such an inner product, and also complete ( in the sense of metric spaces; all Cauchy sequences converge - see Burkill and Burkill ( 1970) ) . For background, we must refer to the excellent textbook treatments of e.g. Young ( 1 988) and Bollobas ( 1 990) . Note that we already have a good supply of Hilbert spaces: finite-dimensional ones such as JRn and infinite dimensional ones such as £2 and L 2 . Hilbert spaces closely resemble ordinary Euclidean spaces in many re spects. In particular, one can take orthogonal complements. If M is a vector subspace of a Hilbert space H which is closed ( contains all its limit points ) , one can form its orthogonal complement MJ.. : = { y E H : (x, y ) = 0 \Ix E M } ;
then M J.. is also a closed vector subspace of H, and any expressible in the form z =x+y
with
z E H
is uniquely
x E M, y E MJ..
( and so (x, y ) = 0) . One then says that H is the direct sum of M and MJ.. , written H = M EB MJ.. . If z = x + y , II z l 1 2 = ( z , z) = (x + y , x + y ) = (x, x) + (x, y ) + ( y, x) + ( y , y ) = I I x l 1 2 + 2 (x, y ) + I I y l 1 2 .
In particular, if (x, y )
= 0 ( Le. x, y
are orthogona0 ,
This is the ( in general ) infinite-dimensional version of Pythagoras ' theorem. Hilbert spaces are the easiest of infinite-dimensional spaces to work with because they so closely resemble finite-dimensional, or Euclidean, ones. In particular, one can think geometrically in a Hilbert space, using diagrams as one would for ( say ) JR2 or JR3 ; see Appendix C. Functional analysis is the study of infinite-dimensional spaces, and so Hilbert-space theory forms an important part of it. Excellent textbook treat ments are available; we particularly recommend Young ( 1 988) for Hilbert space alone, Bollobas (1990) for the more general setting of functional anal ysis.
B. Projections and Conditional Expectations
Given a Hilbert space (or more generally, an inner product space) V, suppose V is the direct sum of a closed subspace M and its orthogonal complement
M.L :
In the direct-sum decomposition
P:z z P (pz) pz
of a vector E V into a sum of x E M and y E M.L , consider the map -+ x. This is called the orthogonal projection of V onto M. It is linear, and idempotent: since the direct-sum decomposition of x = is x = x + 0, = = x, or By Pythagoras' theorem,
pz
p2 = P.
I x l 1 2 I Pz l 1 2 :::; I l Pz 1 2 .
I PI P P I P I :::; R P) P, z,z pz); P ( P R P). pz z ( N.L z z P, N ( P) P. R( P) N ( P). The situation is symmetric between M and M.L : write Q : = 1 - P, with I the identity mapping. Then Q is linear, and as p2 = P, Q2 = (I - p)2 = I - 2P + p2 = I - 2P + P = I P = Q : Q2 = Q, and conversely, if Q2 = Q, then p2 = P. Thus also
= That is, application of decreases the In particular, norm of a vector: one says that has norm :::; 1 . Conversely, we quote that on an inner product space, these properties linear and idempotent with 1 characterise orthogonal projections. The range of that is, the set of x of the form x = for some is, as above, the set of vectors invariant under (that is, the set of with = is called the projection onto its range M = The orthogonal complement of the range - the set of with zero x-component - is the set of annihilated by the kernel (or nullspace) of Thus the direct-sum decomposition for the orthogonal projection P onto M is V = M 61 M .L = 61 -
-
-
412
B. Projections and Conditional Expectations v = M EI1 M -L
=
N(Q) EI1 R(Q) ,
Q :=
I
-
P.
In particular, when V is finite-dimensional, the content of the above re duces to linear algebra (there is no need to assume M closed, as closure is automatic, so analysis is not needed) . For a textbook treatment in this context, see Halmos ( 1958) . The above use of orthogonal projection, Pythagoras' theorem etc. under lies the theory of the familiar linear model of statistics: normally distributed errors, least squares, regression, analysis of variance etc. For such a geometric treatment of the linear model, see e.g. Rawlings ( 1 988) , Chapter 6. Conditional Expectations and Projections
We confine ourselves here to the L2 theory. Take the vector space L 2 L 2 (il, F , JP) of square-integrable random variables X on a probability space (il, F, JP) . This is a Hilbert space under the norm =
and inner product
(X, Y)
:=
JE(XY).
The space L2 is complete, by the Riesz-Fischer theorem (see e.g. Bollobas ( 1990) ) .
If M is a vector subspace of L 2 which is closed (equivalently: complete), given any X E L 2 one can 'drop a perpendicular' from X to M , obtaining Y E M with II X - Y I I = inf { II X W I I : W E M } and (X - Y, Z) 0 for all Z E M (see e.g. Williams ( 1 99 1 ) , §6 . 1 1 ) Then the map X � Y is the orthogonal projection onto M: it is linear, idempotent and of norm at most 1 . Suppose now that Q is a sub-a-field of F and M is the L 2 -space of Q measurable functions. Then -
=
.
X � JE(X IQ)
is the orthogonal projection of X onto M L 2 (il, Q , JP) . It gives the best predictor (in the least-squares sense) of X given Q (that is, it minimises the mean-square error of all predictors of X given the information represented by Q) : see e.g. Williams ( 1 99 1 ) , §9.4. The idempotence of this conditional expectation operator follows from the iterated conditional expectation oper ation of §2.5. This idempotence is also suggested by the above interpretation: forming our best estimate given available information should give the same result done once as done twice. =
B. Projections and Conditional Expectations
413
This picture of conditional expectation projection is powerful - partly because, in the L 2 -setting, it allows us to think and argue geometrically. The excellent text Neveu (1975) on martingales is based on this viewpoint. as
c . The Separating Hyperplane Theorem
In a vector space V , if x and y are vectors, the set of linear combinations ax + /3y, with scalars a, /3 2 0 with sum a + /3 = 1 , represents geometrically the line segment joining x to y. Each such linear combination .>..x + (1 - .>.. ) y, with 0 ::; .>.. ::; 1 , is called a convex combination of x and y . A set C in V is called convex if, for all pairs x and y of points in C, all convex combinations of x and y are also in C. If V has dimension n and U is a subspace of dimension n - 1, U is said to have codimension 1 . If U is a subspace, x+U
:=
{x
+
u : u
E U}
is called the translate of U by the vector x. A hyperplane in V is a translate of a subspace of codimension 1. Such a hyperplane is always representable in the form H = [f, a] : = {x : f (x) = a } ,
for some scalar a and linear functional f : that is, a map f : V -+ IR with f ( x + y) = f ( x)
+ f ( y)
( x, y E V) ,
f ( .>..x ) = .>.. f ( x) ( x
E V, .>.. E IR ) .
Such an f is of the form f ( x) = iI x! + . . . + fnXn i
then f ( iI , . . , fn ) defines a vector f in V, and the hyperplane H = [f, a] consists of those vectors x in V whose projections onto f have magnitude a. The hyperplane [j, a] bounds the set A c V if =
the hyperplane
·
f ( x) 2 a \Ix [f, a] separates
E V or
f ( x) ::; a \Ix E
Vi
the sets A, B c V if
f ( x) 2 a \Ix E
A and
f ( x) ::; a \Ix
E B,
or the same inequalities with A < B ( or 2 , ::; ) interchanged. The following result is crucial for many purposes, both in mathematics and in economics and finance.
416
C. The Separating Hyperplane Theorem
Theorem C.O.1 (Separating Hyperplane Theorem) . If A, B are two
non-empty disjoint convex sets in a vector space V, they can be separated by a hyperplane.
For proof and background, see e.g. Valentine ( 1 964) , Part II and Bott The restriction to finite dimension is not in fact necessary: the re sult is true as stated even if V has infinite dimension ( for proof, see e.g. BolloMs ( 1990) , Chapter 3) . In this form, the result is closely linked to the Hahn-Banach theorem, the cornerstone of functional analysis. Again, Bol lobas (1990) is a fine introduction. ( 1942) .
Remark. When using a book on functional analysis, it is usually a good idea to look out for the results whose proof depends on the Hahn-Banach theorem: these are generally the key results, and the hard ones. The same is true in mathematical economics or finance of the separating hyperplane theorem.
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of
Index
c5-admissible, 234 u-algebra, 31 - stopping time, 84 affine term structure, 340 algebra, 3 1 almost everywhere, 33 almost surely, 33 American put, 1 4 1 , 258 arbitrage, 1, 1 5 - free, 106 - opportunity, 19, 106, 232 - price, 1 1 5 - pricing technique, 8, 9, 328 - strategy, 106 arbitrageur, 6 ARCH (autoregressive conditional heteroscedasticity) , 3 1 7 Asian option - Geman-Yor method, 261 - Rogers-Shi method, 261 barrier option - Asian, 266 - down-and-out call, 264 - forward start, 266 - knockout discount, 265 - moving boundary, 266 - outside, 266 basket default swap, 400 Bayes formula, 225, 239 binomial model, 1 2 1 Black formula - caplets, 363 - swaption, 366 Black's futures option formula, 283 Black-Derman-Toy model, 341 Black-Scholes - complete, 248 - European call price, 133 - formula, 44, 251, 294 - hedging, 1 1 5, 252, 279
- martingale measure, 243 - model, 196, 243, 270 - partial differential equation, 253, 255 - risk-neutral valuation, 250 - stochastic calculus, 198 - volatility, 314 Borel u-algebra, 3 1 Brownian motion, 160 - geometric, 197 - martingale characterization, 1 7 1 , 2 1 5 - quadratic variation, 169 call - European - - convergence of CRR price, 133 - - Cox-Ross-Rubinstein price, 1 26 cap, 354 - Black's model, 355 caplet , 354, 362 central limit theorem - functional form, 223 Collateralized Debt Obligations, 404 complete market , 22 completeness theorem, 1 16, 238 conditional expectation - iteration, 50 conditional Jensen formula, 48 conditional mean formula, 48 conditional probability, 44 confluent hypergeometric function, 263 contingent claim, 2, 105, 1 16, 230, 248, 277 - attainable, 236 convergence - almost surely, 5 1 - i n pth mean, 5 2 - i n distribution, 52 - in probability, 52 - mean square, 52 - weak, 53, 221 convolution, 53 copula, 403
434
Index
cost process, 3 1 1 coupon, 328 coupon bonds, 328 Cox-Ross-Rubinstein model, 1 2 1 credit default swap, 399 credit migration, 376 currency, 5 currency option, 286 default correlation, 376 default probability, 376 derivative - Radon-Nikodym, 43 derivative securities, 2 diffusion, 159, 243 - constant elasticity of variance, 137 distribution - tn , 67 Bernoulli, 40 binomial, 41 - bivariate normal, 46 elliptically contoured, 66 generalized inverse Gaussian, 68 - hyperbolic, 42, 68 multinormal, 66 - normal, 4 1 , 56 Poisson, 42, 57 distribution function, 38 Doob Decomposition, 93 Doob-Meyer-decomposition, 1 70 dynamic completeness, 272 dynamic portfolio, 1 02 early-exercise - decomposition, 258 - premium, 260 elasticity coefficient , 255 equivalent martingale measure, 233 Esscher measure, 292 expectation, 39 expectation hypothesis, 346 , 348 expiry, 3 Follmer-Schweizer decomposition, 308 factor modelling, 401 factorization formula, 293 Feynman-Kac formula, 20 1 , 25 1 , 339 filtration, 75, 76, 153 - Brownian, 199, 243 financial market model, 101 , 229 finite market approximation, 271 finite-dimensional distributions, 153 first-passage time, 225, 264 first-to-default swap, 400
Flesaker-Hughston-model, 370 formula currency option, 286 - Geman-Yor, 263 - Levy-Khintchine, 64, 1 79 - risk-neutral pricing, 1 19, 1 20 - Stirling'S, 60 forward, 2 - contract , 3 - price, 3 forward LIBOR measure, 357 forward rate, 330 - instantaneous, 331 free lunch, 19 function - characteristic, 55 - finite variation, 37 indicator, 34 - Lebesgue-integrable, 35 - measurable, 34 - simple, 34 futures, 2, 282 gains process, 102, 230 GARCH (generalised autoregressive conditional heteroscedasticity) , 3 1 7 Gaussian process, 158 Girsanov pair, 1 99 Greeks, 254 - delta, 254 - gamma, 254 - rho, 254 - theta, 254 - vega, 254, 255 Heath-Jarrow-Morton - drift condition, 345 - model, 343 hedge - perfect, 1 1 9 hedgers, 6 hedging - mean-variance, 18, 307 - risk-minimizing, 18, 3 1 1 hedging strategy - CRR model, 127, 128 Hilbert space, 308, 410 hyperplane, 4 1 5 implied volatility, 3 1 4 independence, 40 index, 5 infinite divisible, 69
Index inner product, 409 integral - Lebesgue-Stieltjes , 36 - Legesgue, 34 - Riemann, 36 interest rate, 4 intrinsic value, 259 invariance principle, 223 Ito - calculus, 1 87 - lemma, 1 9 5 Ito formula - basic, 1 94 - for ItO process , 1 9 5 - for semi-martingales, 2 1 2 - multidimensional, 1 9 5 Ito process , 1 93 Levy process , 1 78, 294 Laplace transform, 263, 266 law of large numbers - weak, 58 Lebesgue measure, 3 1 LIBOR dynamics, 358, 3 6 1 LIBOR rate, 357 Lipschitz condition, 203, 274 local martingale, 209 localization, 192 lookback option - call, 267 - partial, 269 - put, 267 Levy exponent , 64 market - complete, 1 1 6 , 236 - incomplete, 289 , 295 , 3 1 5 market price of risk, 2 4 5 , 2 4 7 , 338 Markov chain, 78, 96 Markov process , 78, 1 5 8 - strong, 1 5 8 martingale, 78, 1 5 5 - local, 1 9 2 representation, 1 1 8 - square-integrable, 94 - transform, 80 martingale measure, 2 1 , 108 - forward risk-neutral, 24 1 , 346 - minimal, 3 1 6 - risk-neutral, 1 2 0 , 246 martingale modelling, 335 martingale representation, 308 martingale transform lemma, 8 1
435
maximal utility, 290 maximum likelihood, 257 mean reversion, 342 measurable space, 31 measure, 3 2 - absolutely continuous , 43 - equivalent , 43 measure space, 32 Merton model, 379 , 380 method of images , 265 Monte-Carlo, 266 multinomial models, 1 48 Newton-Raphson iteration, 256 no free lunch with vanishing risk, 235 norm, 409 Novikov condition, 1 99 numeraire, 2 1 , 1 0 1 , 230, 239 numeraire invariance theorem, 231 optimal capital structure, 389 optimal stopping problem, 91 option, 2 - American , 2 , 1 3 8 , 258 - Asian, 3 , 260 - barrier, 3, 263 - - discrete, 143 - binary, 269 - call, 2 - European, 2 - exotic, 5 - fair price (Davis) , 290 - futures call, 283 - lookback, 3, 266 - - discrete, 145 - put, 2 Ornstein-Uhlenbeck process, 205 orthogonal complement , 4 1 0 orthogonal projection, 4 1 1 partition, 3 7 point process, 1 7 5 Poisson process, 1 75 portfolio, 9 portfolio credit risk, 400 - asset-based, 401 - loss distribution, 402 predictable, 80, 1 9 2 , 209 previsible, 209 price - arbitrage, 1 1 9 pricing kernel , 368 probability - measure, 33
436
Index
- space, 33 probability space, 38 - filtered, 76 process - Bessel, 261 - Bessel-squared, 262 - maximum of BM, 264 - minimum of BM, 264 projection, 308 put-call parity, 143 quadratic covariation, 157 quadratic variation, 168, 2 1 1 random variable, 38 - expectation, 39 - variance, 39 reduced-form model, 391 - valuation, 395 reflection principle, 264 representation property, 238 Riccati equation, 340 Riesz decomposition, 259 risk - intrinsic, 295 - remaining, 295, 3 1 1 risk management, 273, 279 risk premium, 338 risk-neutral valuation, 1 15, 236, 250, 273, 335 sample space, 37 self-decomposability, 65 semi-martingale, 186, 209 - characteristics, 218 - good sequence of, 223 separating hyperplane Theorem, 19 Snell envelope, 89, 259 speculator, 6 spot LIBOR measure, 361 spot rate, 331 spreads, 382 state-price vector, 20, 276 stochastic basis, 76, 153 stochastic differential equations - strong solution, 204 - weak solution, 204 stochastic exponential, 197, 198, 216 stochastic integral - quadratic variation of, 192 stochastic process, 77, 1 53 - adapted, 77, 1 53 - cadlag, 154 - Poisson, 42
- progressively measurable, 243 - RCLL, 1 54 stochastic volatility, 219 stock, 4 stock price - jump, 136 stopping time, 82, 155 strategy - replicating, 1 1 2 strike price, 3 structural model - Black-Cox, 384 - Leland, 389 - Merton, 380 - stochastic interest rate, 388 structure-preserving property, 270 submartingale, 79, 155 supermartingale, 79, 155 swap, 2, 4 , 353, 363 swaption, 365 tail dependence, 403 term structure equation, 337, 339 theorem - central limit, 59 - Doob's Martingale convergence, 95 - Feynman-Kac, 202 - fundamental theorem of asset pricing, 1 19, 235 - Girsanov, 199 - local limit, 60 - monotone convergence, 35 - Optional Sampling, 85 - Poisson limit, 60, 61 - portmanteau, 222 - representation of Brownian martingales, 200 - Stopping time principle, 83 trading strategy, 18, 102 - admissible, 235 - mean-self-financing, 312 - replicating, 236 - self-financing, 230 - tame, 233 uniform integrability, 1 56 usual conditions ( for a stochastic basis) , 153 utility function, 290 value process, 102, 230 Vasicek model, 340 volatility, 196
Index - historic, 257 - implied, 256 - non-parametric estimation, 258 - stochastic, 257 volatility matrix, 243 volatility smile, 314
weak law of large numbers, 52 wealth process, 102 yield-to-maturity, 329 zero-coupon bond, 329, 330
437
