Handbook of Modeling High-Frequency Data in Finance (Wiley Handbooks in Financial Engineering and Econometrics)

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Handbook of

Modeling High-Frequency Data in Finance

Published Wiley Handbooks in Financial Engineering and Econometrics Viens, Mariani, and Florescu · Handbook of Modeling High-Frequency Data in Finance Forthcoming Wiley Handbooks in Financial Engineering and Econometrics Bali and Engle · Handbook of Asset Pricing Bauwens, Hafner, and Laurent · Handbook of Volatility Models and Their Applications Brandimarte · Handbook of Monte Carlo Simulation Chan and Wong · Handbook of Financial Risk Management Cruz, Peters, and Shevchenko · Handbook of Operational Risk Sarno, James, and Marsh · Handbook of Exchange Rates Szylar · Handbook of Market Risk

Handbook of

Modeling High-Frequency Data in Finance Edited by

Frederi G. Viens Maria C. Mariani Ionut¸ Florescu

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2012 John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Viens, Frederi G., 1969– Handbook of modeling high-frequency data in finance / Frederi G. Viens, Maria C. Mariani, Ionut¸ Florescu. — 1 p. cm. — (Wiley handbooks in financial engineering and econometrics ; 4) Includes index. ISBN 978-0-470-87688-6 (hardback) 1. Finance– Econometric models. I. Mariani, Maria C. II. Florescu, Ionut¸, 1973– III. Title. HG106.V54 2011 332.01 5193–dc23 2011038022

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents Preface Contributors

xi xiii

part One

Analysis of Empirical Data

1

1 Estimation of NIG and VG Models for High Frequency Financial Data

3

José E. Figueroa-López, Steven R. Lancette, Kiseop Lee, and Yanhui Mi 1.1 1.2 1.3 1.4 1.5 1.6

Introduction, 3 The Statistical Models, 6 Parametric Estimation Methods, 9 Finite-Sample Performance via Simulations, 14 Empirical Results, 18 Conclusion, 22 References, 24

2 A Study of Persistence of Price Movement using High Frequency Financial Data

27

Dragos Bozdog, Ionut¸ Florescu, Khaldoun Khashanah, and Jim Wang 2.1 Introduction, 27 2.2 Methodology, 29 2.3 Results, 35

v

vi

Contents

2.4 Rare Events Distribution, 41 2.5 Conclusions, 44 References, 45

3 Using Boosting for Financial Analysis and Trading

47

Germán Creamer 3.1 3.2 3.3 3.4 3.5

Introduction, 47 Methods, 48 Performance Evaluation, 53 Earnings Prediction and Algorithmic Trading, 60 Final Comments and Conclusions, 66 References, 69

4 Impact of Correlation Fluctuations on Securitized structures

75

Eric Hillebrand, Ambar N. Sengupta, and Junyue Xu 4.1 Introduction, 75 4.2 Description of the Products and Models, 77 4.3 Impact of Dynamics of Default Correlation on Low-Frequency Tranches, 79 4.4 Impact of Dynamics of Default Correlation on High-Frequency Tranches, 87 4.5 Conclusion, 92 References, 94

5 Construction of Volatility Indices Using A Multinomial Tree Approximation Method Dragos Bozdog, Ionut¸ Florescu, Khaldoun Khashanah, and Hongwei Qiu 5.1 5.2 5.3 5.4

Introduction, 97 New Methodology, 99 Results and Discussions, 101 Summary and Conclusion, 110 References, 115

97

vii

Contents

part Two

Long Range Dependence Models

117

6 Long Correlations Applied to the Study of Memory Effects in High Frequency (TICK) Data, the Dow Jones Index, and International Indices

119

Ernest Barany and Maria Pia Beccar Varela 6.1 6.2 6.3 6.4 6.5

Introduction, 119 Methods Used for Data Analysis, 122 Data, 128 Results and Discussions, 132 Conclusion, 150 References, 160

7 Risk Forecasting with GARCH, Skewed t

Distributions, and Multiple Timescales

163

Alec N. Kercheval and Yang Liu 7.1 7.2 7.3 7.4 7.5 7.6

Introduction, 163 The Skewed t Distributions, 165 Risk Forecasts on a Fixed Timescale, 176 Multiple Timescale Forecasts, 185 Backtesting, 188 Further Analysis: Long-Term GARCH and Comparisons using Simulated Data, 203 7.7 Conclusion, 216 References, 217

8 Parameter Estimation and Calibration for Long-Memory Stochastic Volatility Models Alexandra Chronopoulou 8.1 8.2 8.3 8.4

Introduction, 219 Statistical Inference Under the LMSV Model, 222 Simulation Results, 227 Application to the S&P Index, 228

219

viii

Contents

8.5 Conclusion, 229 References, 230

part Three

Analytical Results

233

9 A Market Microstructure Model of Ultra High Frequency Trading

235

Carlos A. Ulibarri and Peter C. Anselmo 9.1 9.2 9.3 9.4

Introduction, 235 Microstructural Model, 237 Static Comparisons, 239 Questions for Future Research, 241 References, 242

10 Multivariate Volatility Estimation with High Frequency Data Using Fourier Method

243

Maria Elvira Mancino and Simona Sanfelici 10.1 Introduction, 243 10.2 Fourier Estimator of Multivariate Spot Volatility, 246 10.3 Fourier Estimator of Integrated Volatility in the Presence of Microstructure Noise, 252 10.4 Fourier Estimator of Integrated Covariance in the Presence of Microstructure Noise, 263 10.5 Forecasting Properties of Fourier Estimator, 272 10.6 Application: Asset Allocation, 286 References, 290

11 The ‘‘Retirement’’ Problem Cristian Pasarica 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Introduction, 295 The Market Model, 296 Portfolio and Wealth Processes, 297 Utility Function, 299 The Optimization Problem in the Case π(τ ,T ] ≡ 0, 299 Duality Approach, 300 Infinite Horizon Case, 305 References, 324

295

ix

Contents

12 Stochastic Differential Equations and Levy Models with Applications to High Frequency Data

327

Ernest Barany and Maria Pia Beccar Varela 12.1 12.2 12.3 12.4

Solutions to Stochastic Differential Equations, 327 Stable Distributions, 334 The Levy Flight Models, 336 Numerical Simulations and Levy Models: Applications to Models Arising in Financial Indices and High Frequency Data, 340 12.5 Discussion and Conclusions, 345 References, 346

13 Solutions to Integro-Differential Parabolic Problem Arising on Financial Mathematics

347

Maria C. Mariani, Marc Salas, and Indranil SenGupta 13.1 13.2 13.3 13.4

Introduction, 347 Method of Upper and Lower Solutions, 351 Another Iterative Method, 364 Integro-Differential Equations in a Lévy Market, 375 References, 380

14 Existence of Solutions for Financial Models with Transaction Costs and Stochastic Volatility

383

Maria C. Mariani, Emmanuel K. Ncheuguim, and Indranil SenGupta 14.1 Model with Transaction Costs, 383 14.2 Review of Functional Analysis, 386 14.3 Solution of the Problem (14.2) and (14.3) in Sobolev Spaces, 391 14.4 Model with Transaction Costs and Stochastic Volatility, 400 14.5 The Analysis of the Resulting Partial Differential Equation, 408 References, 418

Index

421

Preface This handbook is a collection of articles that describe current empirical and analytical work on data sampled with high frequency in the financial industry. In today’s world, many fields are confronted with increasingly large amounts of data. Financial data sampled with high frequency is no exception. These staggering amounts of data pose special challenges to the world of finance, as traditional models and information technology tools can be poorly suited to grapple with their size and complexity. Probabilistic modeling and statistical data analysis attempt to discover order from apparent disorder; this volume may serve as a guide to various new systematic approaches on how to implement these quantitative activities with high-frequency financial data. The volume is split into three distinct parts. The first part is dedicated to empirical work with high frequency data. Starting the handbook this way is consistent with the first type of activity that is typically undertaken when faced with data: to look for its stylized features. The book’s second part is a transition between empirical and theoretical topics and focuses on properties of long memory, also known as long range dependence. Models for stock and index data with this type of dependence at the level of squared returns, for instance, are coming into the mainstream; in high frequency finance, the range of dependence can be exacerbated, making long memory an important subject of investigation. The third and last part of the volume presents new analytical and simulation results proposed to make rigorous sense of some of the difficult modeling questions posed by high frequency data in finance. Sophisticated mathematical tools are used, including stochastic calculus, control theory, Fourier analysis, jump processes, and integro-differential methods. The editors express their deepest gratitude to all the contributors for their talent and labor in bringing together this handbook, to the many anonymous referees who helped the contributors perfect their works, and to Wiley for making the publication a reality. Frederi Viens Maria C. Mariani Ionut¸ Florescu Washington, DC, El Paso, TX, and Hoboken, NJ April 1, 2011

xi

Contributors Peter C. Anselmo, New Mexico Institute of Mining and Technology, Socorro, NM Ernest Barany, Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM Maria Pia Beccar Varela, Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX Dragos Bozdog, Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ Alexandra Chronopoulou, INRIA, Nancy, France Germán Creamer, Howe School and School of Systems and Enterprises, Stevens Institute of Technology, Hoboken, NJ José E. Figueroa-Lòpez, Department of Statistics, Purdue University, West Lafayette, IN Ionut¸ Florescu, Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ Eric Hillebrand, Department of Economics, Louisiana State University, Baton Rouge, LA Alec N. Kercheval, Department of Mathematics, Florida State University, Tallahassee, FL Khaldoun Khashanah, Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ Steven R. Lancette, Department of Statistics, Purdue University, West Lafayette, IN Kiseop Lee, Department of Mathematics, University of Louisville, Louisville, KY; Graduate Department of Financial Engineering, Ajou University, Suwon, South Korea Yang Liu, Department of Mathematics, Florida State University, Tallahassee, FL xiii

xiv

Contributors

Maria Elvira Mancino, Department of Mathematics for Decisions, University of Firenze, Italy Maria C. Mariani, Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX Yanhui Mi, Department of Statistics, Purdue University, West Lafayette, IN Emmanuel K. Ncheuguim, Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM Hongwei Qiu, Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ Cristian Pasarica, Stevens Institute of Technology, Hoboken, NJ Marc Salas, New Mexico State University, Las Cruces, NM Simona Sanfelici, Department of Economics, University of Parma, Italy Ambar N. Sengupta, Department of Mathematics, Louisiana State University, Baton Rouge, LA Indranil Sengupta, Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX Carlos A. Ulibarri, New Mexico Institute of Mining and Technology, Socorro, NM Jim Wang, Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ Junyue Xu, Department of Economics, Louisiana State University, Baton Rouge, LA

Part One

Analysis of Empirical Data

Chapter

One

Estimation of NIG and VG Models for High Frequency Financial Data ´ PE Z a n d J O S E´ E . F I G U E ROA - L O STEVEN R. LANCETTE Department of Statistics, Purdue University, West Lafayette, IN

KISEOP LEE Department of Mathematics, University of Louisville, Louisville, KY; Graduate Department of Financial Engineering, Ajou University, Suwon, South Korea

YA N H U I M I Department of Statistics, Purdue University, West Lafayette, IN

1.1 Introduction Driven by the necessity to incorporate the observed stylized features of asset prices, continuous-time stochastic modeling has taken a predominant role in the financial literature over the past two decades. Most of the proposed models are particular cases of a stochastic volatility component driven by a Wiener process superposed with a pure-jump component accounting for the Handbook of Modeling High-Frequency Data in Finance, First Edition. Edited by Frederi G. Viens, Maria C. Mariani, and Ionut¸ Florescu. © 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

3

4

CHAPTER 1 Estimation of NIG and VG Models

discrete arrival of major influential information. Accurate approximation of the complex phenomenon of trading is certainly attained with such a general model. However, accuracy comes with a high cost in the form of hard estimation and implementation issues as well as overparameterized models. In practice, and certainly for the purpose motivating the task of modeling in the first place, a parsimonious model with relatively few parameters is desirable. With this motivation in mind, parametric exponential Lévy models (ELM) are one of the most tractable and successful alternatives to both stochastic volatility models and more general Itô semimartingale models with jumps. The literature of geometric Lévy models is quite extensive (see Cont & Tankov (2004) for a review). Owing to their appealing interpretation and tractability in this work, we concentrate on two of the most popular classes: the variance-gamma (VG) and normal inverse Gaussian (NIG) models proposed by Carr et al. (1998) and Barndorff-Nielsen (1998), respectively. In the ‘‘symmetric case’’ (which is a reasonable assumption for equity prices), both models require only one additional parameter, κ, compared to the two-parameter geometric Brownian motion (also called the Black–Scholes model). This additional parameter can be interpreted as the percentage excess kurtosis relative to the normal distribution and, hence, this parameter is mainly in charge of the tail thickness of the log return distribution. In other words, this parameter will determine the frequency of ‘‘excessively’’ large positive or negative returns. Both models are pure-jump models with infinite jump activity (i.e., a model with infinitely many jumps during any finite time interval [0, T ]). Nevertheless, one of the parameters, denoted by σ , controls the variability of the log returns and, thus, it can be interpreted as the volatility of the price process. Numerous empirical studies have shown that certain parametric ELM, including the VG and the NIG models, are able to fit daily returns extremely well using standard estimation methods such as maximum likelihood estimators (MLE) or method of moment estimators (MME) (c.f. Eberlein & Keller (1995); ¨ Eberlein & Ozkan (2003); Carr et al. (1998); Barndorff-Nielsen (1998); Kou & Wang (2004); Carr et al. (2002); Seneta (2004); Behr & Pötter (2009), Ramezani & Zeng (2007), and others). On the other hand, in spite of their current importance, very few papers have considered intraday data. One of our main motivations in this work is to analyze whether pure Lévy models can still work well to fit the statistical properties of log returns at the intraday level. As essentially any other model, a Lévy model will have limitations when working with very high frequency transaction data and, hence, the question is rather to determine the scales where a Lévy model is a good probabilistic approximation of the underlying (extremely complex and stochastic) trading process. We propose to assess the suitability of the Lévy model by analyzing the signature plots of the point estimates at different sampling frequencies. It is plausible that an apparent stability of the point estimates for certain ranges of sampling frequencies provides evidence of the adequacy of the Lévy model ¨ at those scales. An earlier work along these lines is Eberlein & Ozkan (2003), where this stability was empirically investigated using hyperbolic Lévy models and MLE (based on hourly data). Concretely, one of the main points therein was

1.1 Introduction

5

to estimate the model’s parameters from daily mid-day log returns1 and, then, measure the distance between the empirical density based on hourly returns and the 1-h density implied by the estimated parameters. It is found that this distance is approximately minimal among any other implied densities. In other words, if fδ (·; θd∗ ) denotes the implied density of Xδ when using the parameters θd∗ estimated from daily mid-day returns and if fh∗ (·) denotes the empirical density based on hourly returns, then the distance between fδ (·; θd∗ ) and fh∗ is minimal when δ is approximately 1 h. Such a property was termed the time consistency of Lévy processes. In this chapter, we further investigate the consistency of ELM for a wide rage of intraday frequencies using intraday data of the US equity market. Although natural differences due to sampling variation are to be expected, our empirical results under both models exhibit some very interesting common features across the different stocks we analyzed. We find that the estimator of the volatility parameter σ is quite stable for sampling frequencies as short as 20 min or less. For higher frequencies, the volatility estimates exhibit an abrupt tendency to increase (see Fig. 1.6 below), presumably due to microstructure effects. In contrast, the kurtosis estimator is more sensitive to microstructure effects and a certain degree of stability is achieved only for mid-range frequencies of 1 h and more (see Fig. 1.6 below). For higher frequencies, the kurtosis decreases abruptly. In fact, opposite to the smooth signature plot of σ at those scales, the kurtosis estimates consistently change by more than half when going from hourly to 30-min log returns. Again, this phenomenon is presumably due to microstructure effects since the effect of an unaccounted continuous component will be expected to diminish when the sampling frequency increases. One of the main motivations of Lévy models is that log returns follow ideal conditions for statistical inference in that case; namely, under a Lévy model the log returns at any frequency are independent with a common distribution. Owing to this fact, it is arguable that it might be preferable to use a parsimonious model for which efficient estimation is feasible, rather than a very accurate model for which estimation errors will be intrinsically large. This is similar to the so-called model selection problem of statistics where a model with a high number of parameters typically enjoys a small mis-specification error but suffers from a high estimation variance due to the large number of parameters to estimate. An intrinsic assumption discussed above is that standard estimation methods are indeed efficient in this high frequency data setting. This is, however, an overstatement (typically overlooked in the literature) since the population distribution of high frequency sample data coming from a true Lévy model depends on the sampling frequency itself and, in spite of having more data, high frequency data does not necessarily imply better estimation results. Hence, another motivation for this work is to analyze the performance of the two most common estimators, namely the method of moments estimators (MME) and the 1 These

returns are derived from prices recorded in the middle of the trading session. The idea behind the choice of these prices is to avoid the typically high volatility at the opening and closing of the trading session.

6


MLE, when dealing with high frequency data. As an additional contribution of this analysis, we also propose a simple novel numerical scheme for computing the MME. On the other hand, given the inaccessibility of closed forms for the MLE, we apply an unconstrained optimization scheme (Powell’s method) to find them numerically. By Monte Carlo simulations, we discover the surprising fact that neither high frequency sampling nor MLE reduces the estimation error of the volatility parameter in a significant way. In other words, estimating the volatility parameter based on, say, daily observations has similar performance to doing the same based on, say, 5-min observations. On the other hand, the estimation error of the parameter controlling the kurtosis of the model can be significantly reduced by using MLE or intraday data. Another conclusion is that the VG MLE is numerically unstable when working with ultra-high frequency data while both the VG MME and the NIG MLE work quite well for almost any frequency. The remainder of this chapter is organized as follows. In Section 1.2, we review the properties of the NIG and VG models. Section 1.3 introduces a simple and novel method to compute the moment estimators for the VG and the NIG distributions and also briefly describes the estimation method of maximum likelihood. Section 1.4 presents the finite-sample performance of the moment estimators and the MLE via simulations. In Section 1.5, we present our empirical results using high frequency transaction data from the US equity market. The data was obtained from the NYSE TAQ database of 2005 trades via Wharton’s WRDS system. For the sake of clarity and space, we only present the results for Intel and defer a full analysis of other stocks for a future publication. We finish with a section of conclusions and further recommendations.

1.2 The Statistical Models ´ 1.2.1 GENERALITIES OF EXPONENTIAL LEVY MODELS Before introducing the specific models we consider in this chapter, let us briefly motivate the application of Lévy processes in financial modeling. We refer the reader to the monographs of Cont & Tankov (2004) and Sato (1999) or the recent review papers Figueroa-López (2011) and Tankov (2011) for further information. Exponential (or Geometric) Lévy models are arguably the most natural generalization of the geometric Brownian motion intrinsic in the Black–Scholes option pricing model. A geometric Brownian motion (also called Black–Scholes model) postulates the following conditions about the price process (St )t≥0 of a risky asset: (1) The (log) return on the asset over a time period [t, t + h] of length h, that is, Rt,t+h := log

St+h St

is Gaussian with mean μh and variance σ 2 h (independent of t);

7

1.2 The Statistical Models

(2) Log returns on disjoint time periods are mutually independent; (3) The price path t → St is continuous; that is, P(Su → St , as u → t, ∀ t) = 1. The previous assumptions can equivalently be stated in terms of the so-called log return process (Xt )t , denoted henceforth as Xt := log

St . S0

Indeed, assumption (1) is equivalent to ask that the increment Xt+h − Xt of the process X over [t, t + h] is Gaussian with mean μh and variance σ 2 h. Assumption (2) simply means that the increments of X over disjoint periods of time are independent. Finally, the last condition is tantamount to asking that X has continuous paths. Note that we can represent a general geometric Brownian motion in the form St = S0 eσ Wt +μt , where (Wt )t≥0 is the Wiener process. In the context of the above Black–Scholes model, a Wiener process can be defined as the log return process of a price process satisfying the Black–Scholes conditions (1)–(3) with μ = 0 and σ 2 = 1. As it turns out, assumptions (1)–(3) above are all controversial and believed not to hold true especially at the intraday level (see Cont (2001) for a concise description of the most important features of financial data). The empirical distributions of log returns exhibit much heavier tails and higher kurtosis than a Gaussian distribution does and this phenomenon is accentuated when the frequency of returns increases. Independence is also questionable since, for example, absolute log returns typically exhibit slowly decaying serial correlation. In other words, high volatility events tend to cluster across time. Of course, continuity is just a convenient limiting abstraction to describe the high trading activity of liquid assets. In spite of its shortcomings, geometric Brownian motion could arguably be a suitable model to describe low frequency returns but not high frequency returns. An ELM attempts to relax the assumptions of the Black–Scholes model in a parsimonious manner. Indeed, a natural first step is to relax the Gaussian character of log returns by replacing it with an unspecified distribution as follows: (1 ) The (log) return on the asset over a time period of length h has distribution Fh , depending only on the time span h. This innocuous (still desirable) change turns out to be inconsistent with condition (3) above in the sense that (2) and (3) together with (1 ) imply (1). Hence, we ought to relax (3) as well if we want to keep (1 ). The following is a natural compromise: (3 ) The paths t → St exhibit only discontinuities of first kind (jump discontinuities).

8


Summarizing, an exponential Lévy model for the price process (St )t≥0 of a risky asset satisfies conditions (1 ), (2), and (3 ). In the following section, we concentrate on two important and popular types of exponential Lévy models.

1.2.2 VARIANCE-GAMMA AND NORMAL INVERSE GAUSSIAN MODELS The VG and NIG Lévy models were proposed in Carr et al. (1998) and BarndorffNielsen (1998), respectively, to describe the log return process Xt := log St /S0 of a financial asset. Both models can be seen as a Wiener process with drift that is time-deformed by an independent random clock. That is, (Xt ) has the representation Xt = σ W (τ (t)) + θτ (t) + bt,

(1.1)

where σ > 0, θ, b ∈ R are given constants, W is Wiener process, and τ is a suitable independent subordinator (nondecreasing Lévy process) such that Eτ (t) = t,

and

Var(τ (t)) = κt.

In the VG model, τ (t) is Gamma distributed with scale parameter β := κ and shape parameter α := t/κ, while in the NIG model τ (t) follows an inverse Gaussian distribution with mean μ = 1 and shape parameter λ = 1/(tκ). In the formulation (Eq. 1.1), τ plays the role of a random clock aimed at incorporating variations in business activity through time. The parameters of the model have the following interpretation (see Eqs. (1.6) and (1.17) below). 1. σ dictates the overall variability of the log returns of the asset. In the symmetric case (θ = 0), σ 2 is the variance of the log returns per unit time. 2. κ controls the kurtosis or tail heaviness of the log returns. In the symmetric case (θ = 0), κ is the percentage excess kurtosis of log returns relative to the normal distribution multiplied by the time span. 3. b is a drift component in calendar time. 4. θ is a drift component in business time and controls the skewness of log returns. The VG can be written as the difference of two Gamma Lévy processes Xt = Xt+ − Xt− + bt,

(1.2)

where X + and X − are independent Gamma Lévy processes with respective parameters √ 1 θ 2 κ 2 + 2σ 2 κ ± θκ + − α = α = , β± := . κ 2

9

1.3 Parametric Estimation Methods

One can see X + (respectively X − ) in Equation (1.2) as the upward (respectively downward) movements in the asset’s log return. Under both models, the marginal density of Xt (which translates into the density of a log return over a time span t) is known in closed form. In the VG model, the probability density of Xt is given by √

pt (x) =

⎛

θ(x−bt) σ2

⎞ κt − 21

⎛ ⎜ |x − K t −1 ⎝ κ 2

2e ⎜ |x − bt| ⎟ ⎠ √ t t ⎝ 2 2σ σ πκ κ ( κ ) + θ2 κ

2 bt| 2σκ σ2

⎞ +

θ2

⎟ ⎠, (1.3)

where K is the modified Bessel function of the second kind (c.f. Carr et al. (1998)). The NIG model has marginal densities of the form t

pt (x) =

te κ

+ θ(x−bt) 2

π ⎛

σ

(x − bt)2 + θ2 κσ 2

t2σ 2 κ

+ κ12

t σ 2 ⎜ (x − bt) + κ K1 ⎝ σ2

2 2

σ2 κ

− 21

⎞ + θ2 ⎟ ⎠.

(1.4)

Throughout the chapter, we assume that the log return process {Xt }t≥0 is sampled during a fixed time interval [0, T ] at evenly spaced times ti = iδn , i = 1, . . . , n, where δn = T /n. This sampling scheme is sometimes called calendar time sampling (Oomen, 2006). Under the assumption of independence and stationarity of the increments of X (conditions (1’) and (2) in Section 1.2.1), we have at our disposal a random sample ni := ni X := Xiδn − X(i−1)δn ,

i = 1, . . . , n,

(1.5)

of size n of the distribution fδn (·) := fδn (·; σ , θ, κ, b) of Xδn . Note that, in this context, a larger sample size n does not necessarily entail a greater amount of useful information about the parameters of the model. This is, in fact, one of the key questions in this chapter: Does the statistical performance of standard parametric methods improve under high frequency observations? We address this issue by simulation experiments in Section 1.4. For now, we introduce the statistical methods used in this chapter.

1.3 Parametric Estimation Methods In this part, we review the most used parametric estimation methods: the method of moments and maximum likelihood. We also present a new computational method to find the moment estimators of the considered models. It is worth

10


pointing out that both methods are known to be consistent under mild conditions if the number of observations at a fixed frequency (say, daily or hourly) are independent.

1.3.1 METHOD OF MOMENT ESTIMATORS In principle, the method of moments is a simple estimation method that can be applied to a wide range of parametric models. Also, the MME are commonly used as initial points of numerical schemes used to find MLE, which are typically considered to be more efficient. Another appealing property of moment estimators is that they are known to be robust against possible dependence between log returns since their consistency is only a consequence of stationarity and ergodicitity conditions of the log returns. In this section, we introduce a new method to compute the MME for the VG and NIG models. Let us start with the VG model. The mean and first three central moments of a VG model are given in closed form as follows (Cont & Tankov (2003), pp. 32 & 117): μ1 (Xδ ) := E(Xδ ) = (θ + b)δ, μ2 (Xδ ) := Var(Xδ ) = (σ 2 + θ 2 κ)δ,

(1.6)

μ3 (Xδ ) := E(Xδ − EXδ ) = (3σ θκ + 2θ κ )δ, 3

2

3 2

μ4 (Xδ ) := E(Xδ − EXδ )4 = (3σ 4 κ + 12σ 2 θ 2 κ 2 + 6θ 4 κ 3 )δ + 3μ2 (Xδ )2 . The MME is obtained by solving the system of equations resulting from substituting the central moments of Xδn in Equation 1.6 by their corresponding sample estimators:

k 1 n ¯ (n) , i − n i=1 n

ˆ k,n := μ

k ≥ 2,

(1.7)

¯ (n) := ni=1 ni /n. However, solving where ni is given as in Equation 1.5 and the system of equations that defines the MME is not straightforward and, in general, one will need to rely on a numerical solution of the system. We now describe a novel simple method for this purpose. The idea is to write the central moments in terms of the quantity E := θ 2 κ/σ 2 . Concretely, we have the equations μ2 (Xδ ) = δσ 2 (1 + E),

μ3 (Xδ ) = δσ 2 θκ(3 + 2E),

κ 1 + 4E + 2E 2 μ4 (Xδ ) . − 1 = δ (1 + E)2 3μ22 (Xδ ) From these equations, it follows that 3μ23 (Xδ ) E (3 + 2E)2 := f (E). = μ2 (Xδ ) μ4 (Xδ ) − 3μ22 (Xδ ) 1 + 4E + 2E 2 (1 + E)

(1.8)

11


In spite of appearances, the above function f (E) is a strictly increasing concave function from (−1 + 2−1/2 , ∞) to (−∞, 2) and, hence, the solution of the corresponding sample equation can be found efficiently using numerical methods. It remains to estimate the left-hand side of Equation 1.8. To this end, note that the left-hand side term can be written as 3Skw(Xδ )2 /Krt(Xδ ), where Skw and Krt represent the population skewness and kurtosis: Skw(Xδ ) :=

μ3 (Xδ ) μ4 (Xδ ) and Krt(Xδ ) := − 3. 3/2 μ2 (Xδ ) μ2 (Xδ )2

(1.9)

Finally, we just have to replace the population parameters by their empirical estimators: Varn :=

ˆ 3,n ˆ 1 n n := μ n := μ4,n − 3. ¯ n 2 , Skw i − , Krt 3/2 n − 1 i=1 ˆ 22,n μ ˆ 2,n μ n

(1.10)

Summarizing, the MME can be computed via the following numerical scheme: 1. Find (numerically) the solution Eˆn∗ of the equation 3 Skw n ; n Krt 2

f (E) =

(1.11)

2. Determine the MME using the following formulas:

(1 + Eˆn∗ )2 δn Krtn , κˆn := , 3 1 + Eˆn∗ 1 + 4Eˆn∗ + 2Eˆn∗ 2

ˆ 1 1 ¯n ˆ XT μ 3,n θˆn := , bˆ n := − θn = − θˆn . δn σˆ n2 κˆn 3 + 2Eˆn∗ δn T

σˆ n2

n Var := δn

1

(1.12)

(1.13)

We note that the above estimators will exist if and only if Equation 1.11 admits a solution Eˆ∗ ∈ (−1 + 2−1/2 , ∞), which is the case if and only if 3 Skw n < 2. n Krt 2

Furthermore, the MME estimator κn will be positive only if the sample kurtosis n is positive. It turns out that in simulations this condition is sometimes Krt violated for small-time horizons T and coarse sampling frequencies (say, daily or longer). For instance, using the parameter values (1) of Section 1.4.1 below and taking T = 125 days (half a year) and δn = 1 day, about 80 simulations out of ˆ while only 2 simulations result in invalid κˆ when δn = 1/2 1000 gave invalid κ, day.

12


Seneta (2004) proposes a simple approximation method built on the assumption that θ is typically small. In our context, Seneta’s method is obtained by making the simplifying approximation Eˆn∗ ≈ 0 in the Equations 1.12 and 1.13, resulting in the following estimators: σˆ n2 :=

Varn , δn

θˆn :=

ˆ 3,n μ = 3δn σˆ n2 κˆn

δn Krtn , 3 n ( Varn )1/2 Skw

κˆn :=

n δn Krt

(1.14) ,

XT bˆ n := − θˆn . T

(1.15)

Note that the estimators (Eq. 1.14) are, in fact, the actual MME in the restricted symmetric model θ = 0 and will indeed produce a good approximation of the MME estimators (Eqs. 1.12 and 1.13) whenever 3 Skw n := , Krtn 2

Q∗n

and, hence, Eˆn∗ is ‘‘very’’ small. This fact has been corroborated empirically by multiple studies using daily data as shown in Seneta (2004). The formulas (Eqs. 1.14 and 1.15) have appealing interpretations as noted already by Carr et al. (1998). Namely, the parameter κ determines the percentage excess kurtosis in the log return distribution (i.e., a measure of the tail fatness compared to the normal distribution), σ dictates the overall volatility of the process, and θ determines the skewness. Interestingly, the estimator σˆ n2 in Equation 1.14 can be written as n 1 XT 2 1 1 2 , Xiδn − X(i−1)δn − = RV n + O σˆ n = T − δn i=1 n T − δn n where RV n is the well-known realized variance defined by RV n :=

n

Xiδn − X(i−1)δn

2

.

(1.16)

i=1

Let us finish this section by considering the NIG model. In this setting, the mean and first three central moments are given by Cont & Tankov (2003) (p. 117): μ1 (Xδ ) := E(Xδ ) = (θ + b)δ, μ2 (Xδ ) := Var(Xδ ) = (σ 2 + θ 2 κ)δ,

(1.17)

μ3 (Xδ ) := E(Xδ − EXδ )3 = (3σ 2 θκ + 3θ 3 κ 2 )δ, μ4 (Xδ ) := E(Xδ − EXδ )4 = (3σ 4 κ + 18σ 2 θ 2 κ 2 + 15θ 4 κ 3 )δ + 3μ2 (Xδ )2 .

13


Hence, the Equation 1.8 takes the simpler form 3μ23 (Xδ ) 9E = := f (E), 2 5E +1 μ2 (Xδ ) μ4 (Xδ ) − 3μ2 (Xδ )

(1.18)

and the analogous equation (Eq. 1.11) can be solved in closed form as Eˆn∗ =

2 Skw n n − 5 Skw 3 Krt n 2

.

Then, the MME will be given by the following formulas:

Varn δn 1 1 + Eˆn∗ 2 Krtn , κˆ n := , σˆ n := δn 3 1 + Eˆn∗ 1 + 5Eˆn∗

ˆ 1 1 n ˆ X μ 3,n ¯ − θn = T − θˆn . θˆn := , bˆ n := δn σˆ n2 κˆ n 3 + 3Eˆn∗ δn T

(1.19)

(1.20)

(1.21)

1.3.2 MAXIMUM LIKELIHOOD ESTIMATION Maximum likelihood is one of the most widely used estimation methods, partly due to its theoretical efficiency when dealing with large samples. Given a random sample x = (x1 , . . . , xn ) from a population distribution with density f (·|θ) depending on a parameter θ = (θ1 , . . . , θp ), the method proposes to estimate θ with the value θˆ = θˆ (x) that maximizes the so-called likelihood function L(θ|x) :=

n

f (xi |θ).

i=1

When it exists, such a point estimate θˆ(x) is called the MLE of θ. In principle, under a Lévy model, the increments of the log return process X (which corresponds to the log returns of the price process S) are independent with common distribution, say fδ (·|θ), where δ represents the time span of the increments. As was pointed out earlier, independence is questionable for very high frequency log returns, but given that, for a large sample, likelihood estimation is expected to be robust against small dependences between returns, we can still apply likelihood estimation. The question is again to determine the scales where both the Lévy model is a good approximation of the underlying process and the MLE are meaningful. As indicated in the introduction, it is plausible that the MLE’s stability for certain range of sampling frequencies provides evidence of the adequacy of the Lévy model at those scales. Another important issue is that, in general, the probability density fδ is not known in a closed form or might be intractable. There are several approaches to deal with this issue such as numerically inverting the Fourier transform of fδ via

14


fast Fourier methods (Carr et al., 2002) or approximating fδ using small-time expansions (Figueroa-López & Houdré. 2009). In the present chapter, we do not explore these approaches since the probability densities of the VG and NIG models are known in closed forms. However, given the inaccessibility of closed expressions for the MLE, we apply an unconstrained optimization scheme to find them numerically (see below for more details).

1.4 Finite-Sample Performance via Simulations 1.4.1 PARAMETER VALUES We consider two sets of parameter values: √ κ = 0.422; θ = −1.5 × 10−4 ; 1. σ = 6.447 × 10−5 = 0.0080; b = 2.5750 × 10−4 ; 2. σ = 0.0127; κ = 0.2873; θ = 1.3 × 10−3 ; b = −1.7 × 10−3 ; The first set of parameters (1) is motivated by the empirical study reported in Seneta (2004) (pp. 182) using the approximated MME introduced in Section 3.1 and daily returns of the Standard and Poor’s 500 Index from 1977 to 1981. The second set of parameters (2) is motivated by our own empirical results below using MLE and daily returns of INTC during 2005. Throughout, the time unit is a day and, hence, for example, the estimated average rate of return per day of SP500 is S1 EX (1) = E log = θ + b = 1.0750 × 10−4 ≈ 0.1%, S0 or 0.00010750 × 365 = 3.9% per year.

1.4.2 RESULTS Below, we illustrate the finite-sample performance of the MME and MLE for both the VG and NIG models. The MME is computed using the algorithms described in Section 1.3.1. The MLE was computed using an unconstrained Powell’s method2 started at the exact MME. We use the closed form expressions for the density functions (Eqs. 1.3 and 1.4) in order to evaluate the likelihood function.

1.4.2.1 Variance Gamma. We compute the sample mean and sample standard deviation of the VG MME and the VG MLE for different sampling frequencies. Concretely, the time span δ between consecutive observations is taken to be 1/36,1/18,1/12,1/6,1/3,1/2,1 (in days), which will correspond to 10, 20, 30 min, 1, 2, 3 h, and 1 day (assuming a trading period of 6 h per day). 2 We

employ a MATLAB implementation due to Giovani Tonel obtained through MATLAB Central (http://www.mathworks.com/matlabcentral/fileexchange/).

15

1.4 Finite-Sample Performance via Simulations

Figure 1.1 plots the sampling mean σˆ¯ δ and the bands σˆ¯ δ ± std(σˆ δ ) against the different time spans δ as well as the corresponding graphs for κ, based on 100 simulations of the VG process on [0, 3 ∗ 252] (namely, three years) with the parameter values (1) above. Similarly, Fig. 1.2 shows the results corresponding to the parameter values (2) with a time horizon of T = 252 days and time spans δ = 10, 20, and 30 min, and also, 1/6, 1/4, 1/3, 1/2, and 1 days, assuming this time a trading period of 6 h and 30 min per day and taking 200 simulations. These are our conclusions: 1. The MME for σ performs as well as the computationally more expensive MLE for all the relevant frequencies. Even though increasing the sampling frequency slightly reduces the standard error, the net gain is actually very small even for very high frequencies and, hence, does not justify the use of high frequency data to estimate σ . 2. The estimation for κ is quite different: Using either high frequency data or maximum likelihood estimation results in significant reductions of the standard error (by more than 4 times when using both). 3. The computation of the MLE presents numerical issues (easy to detect) for very high sampling frequencies (say, δ < 1/6). 4. Disregarding the numerical issues and extrapolating the pattern of the graphs when δ → 0, we can conjecture that the MLE σˆ is not consistent when δ → 0 for a fixed time horizon T , while the MLE κˆ appears to be a consistent estimator for κ. Both of these points will be investigated in a future publication.

−3 11 x 10

MLE and MME for κ variance gamma model 0.7

Mean of the MLE Mean + Std of MLE Mean − Std of MLE Mean of the MME Mean + Std of MME Mean − Std of MME

10

Value κ = 0.42 Mean of the MLE Mean + Std of MLE Mean − Std of MLE Mean of the MME Mean + Std of MME Mean − Std of MME

0.65 0.6 0.55

9

Estimators for κ

Estimators for σ

MLE and MME for σ variance gamma model

8

7

0.5 0.45 0.4 0.35 0.3

6

0.25 5

0.2 0

0.2 0.4 0.6 0.8 δ = Time span between observations

1

0

0.2

0.4

0.6

0.8

1

δ = Time span between observations

FIGURE 1.1 Sampling mean and standard error of the MME and MLE for the param-

eters √σ and κ based on 100 simulations of the VG model with values T = 252 × 3, σ = 6.447 × 10−5 = 0.0080; κ = 0.422; θ = −1.5 × 10−4 ; b = 2.5750 × 10−4 .

16

CHAPTER 1 Estimation of NIG and VG Models MME and MLE for σ variance gamma model

MME and MLE for κ variance gamma model 0.6

0.015

Mean of MLE Mean + Std of MLE Mean − Std of MLE Mean of MME Mean + Std of MME Mean − Std of MME True value = 0.2873

0.55 0.014

0.5

0.013

Estimator for κ

Estimators for σ

0.45

0.012 Mean of MLE Mean + Std of MLE Mean − Std of MLE Mean of MME Mean + Std of MME Mean − Std of MME True value = 0.0127

0.011

0.01

0.4 0.35 0.3 0.25 0.2 0.15 0.1

0.009 0

0.2

0.4

0.6

0.8


1

0

0.2

0.4

0.6

0.8

1


FIGURE 1.2 Sampling mean and standard error of the MME and MLE for the parameters σ

and κ based on 200 simulations with values T = 252, σ = 0.0127; κ = 0.2873; θ = 1.3 × 10−3 ; b = −1.7 × 10−3 .

For completeness, we also illustrates in Fig. 1.3 the performance of the estimators for b and θ for the parameter values (2) based again on 200 simulations during [0, 252] with time spans of 10, 20, and 30 min, and 1/6, 1/4, 1/3, 1/2, and 1 days. There seems to be some gain in efficiency when using MLE and higher sampling frequencies in both cases but the respective standard errors level off for δ small, suggesting that neither estimator is consistent for fixed time horizon. One surprising feature is that the MLE estimators in both cases do not seem to exhibit any numerical issues for very small δ in spite of being based on ˆ the same simulations as those used to obtain σˆ and κ.

1.4.2.2 Normal Inverse Gaussian. We now show the estimation results for the NIG model. Here, we take sampling frequencies of 5, 10, 20, and 30 s, also 1, 5, 10, 20, and 30 min, as well as 1, 2, and 3 h, and finally 1 day (assuming a trading period of 6 h). Figure 1.4 plots the sampling mean σˆ¯ δ and bands σˆ¯ δ ± std(σˆ δ ) against the different time spans δ and the corresponding graphs for κ, based on 100 simulations of the NIG process on [0, 3 ∗ 252] with the parameter values (1) above. The results are similar to those of the VG model. In the case of σ , neither MLE nor high frequency data seem to do better than standard moment estimators and daily data. For κ, the estimation error can be reduced as much as 4 times when using high frequency data and maximum likelihood estimation. The most striking conclusion is that the MLE for the NIG model does not show any numerical issues when dealing with very high frequency. Indeed, we are able to obtain results for even 5-s time spans (although the computational time increases significantly in this case).

17

1.5 Finite-Sample Performance via Simulations

x 10

10

MME and MLE for θ variance gamma model

−3

x 10

4 Mean of MME Mean + Std of MME Mean − Std of MME Mean of MLE Mean + Std of MLE Mean − Std of MLE True value = 0.0013

2

0 Estimators for b

Estimators for θ

5

MME and MLE for b variance gamma model

−3

−2

−4

0

Mean of MME Mean + Std of MME Mean − Std of MME Mean of MLE Mean + Std of MLE Mean − Std of MLE True value = −0.0017

−6

−8

−5

−10 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1



FIGURE 1.3 Sampling mean and standard error of the MME and MLE for the parameters θ and b based on 200 simulations with values T = 252, σ = 0.0127; κ = 0.2873; θ = 1.3 × 10−3 ; b = −1.7 × 10−3 .

8.6 x 10

−3

MME and MLE for κ normal inverse gaussian

MLE and MME for σ normal inverse gaussian 0.8

Value = sqrt of .00006447 Mean of MME Mean + Std of MME Mean − Std of MME Mean of MLE Mean + Std of MLE Mean − Std of MLE

8.4

True value = 0.442 Mean of MME Mean + Std of MME Mean − Std of MME Mean of MLE Mean + Std of MLE Mean − Std of MLE

0.7

0.6 Estimators for κ

Estimators for σ

8.2

8

7.8

0.5

0.4

0.3 7.6

0.2

7.4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 δ = Time span between observations

1

0.1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ = Time span between observations

FIGURE 1.4 Sampling mean and standard error of the MME and MLE for the param-

eters √σ and κ based on 100 simulations of the NIG model with values T = 252 × 3, σ = 6.447 × 10−5 = 0.0080; κ = 0.422; θ = −1.5 × 10−4 ; b = 2.5750 × 10−4 .

18


1.5 Empirical Results 1.5.1 THE DATA AND DATA PREPROCESSING The data was obtained from the NYSE TAQ database of 2005 trades via Wharton’s WRDS system. For the sake of clarity and space, we focus on the analysis of only one stock, even though other stocks were also analyzed for this study. We pick Intel (INTC) stock due to its high liquidity (based on the number of trades or ticks). The raw data was preprocessed as follows. Records of trades were kept if the TAQ field CORR indicated that the trade was ‘‘regular’’ (namely, it was not corrected, changed, signaled as cancelled, or signaled as an error). In addition, the condition field was use as a filter. Trades were kept if they were regular way trades, that is, trades that had no stated conditions (COND=’’ or COND=‘*’). A secondary filter was subsequently applied to eliminate some of the remaining incorrect trades. First, for each trading day, the empirical distribution of the absolute value of the first difference of prices was determined. Next, the 99.9th percentile of these daily absolute differences was obtained. Finally, a trade was eliminated if, in magnitude, the difference of the price from the prior price was at least twice the 99.9th percentile of that day’s absolute differences and this difference was reversed on the following trade. Figure 1.5 illustrates the Intel stock prices before (a) and after processing (b).

1.5.2 MME AND MLE RESULTS The exact and approximated MMEs described in Section 1.3.1 were applied to the log returns of the stocks at different frequencies ranging from 10 s to 1 day.

(a) Raw Intel 5−second stock prices (January 2, 2005 − December 30, 2005)

30

29

29

28

28

27

27 Stock price

Stock price

30

26 25

(a) “Clean” Intel 5−second stock prices (January 2, 2005 − December 30, 2005)

26 25

24

24

23

23

22

22 21

21 0

50

100 150 Time in days

200

250

0

50

100 150 Time in days

200

FIGURE 1.5 Intel stock prices during 2005 before and after preprocessing.

250

19

1.5 Empirical Results

Subsequently, we apply the unconstrained Powell’s optimization method to find the MLE estimator. In each case, the starting point for the optimization routine was set equal to the exact MME. Tables 1.1–1.4 show the estimation results under both models together with the log likelihood values using a time horizon of one year. Figure 1.6 shows the graphs of the NIG MLE and approximated NIG MME against the sampling frequency δ based on observations during T = 1 year, T = 6 months, and T = 3 months, respectively.

1.5.3 DISCUSSION OF EMPIRICAL RESULTS In spite of certain natural differences due to sampling variation, the empirical results under both models exhibit some very interesting common features that we now summarize: 1. The estimation of σ is quite stable for ‘‘midrange’’ frequencies (δ ≥ 20 min), exhibiting a slight tendency to decrease when δ decreases from 1 day to 10 min, before showing a pronounce and clear tendency to increase for small time spans (δ = 10 min and less). This increasing tendency is presumably due to the influence of microstructure effects. 2. The point estimators for κ are less stable than those for σ but still their values are relatively ‘‘consistent’’ for mid-range frequencies of 1 h and more. This consistency of κˆ abruptly changes when δ moves from 1/6 of a day to 30 min, at which point a reduction of about half is experienced under both models. To illustrate how unlikely such a behavior is in our models, we consider the simulation experiment of Fig. 1.2 and find out that in only TABLE 1.1 INTC: VG MLE (Top), Exact VG MME (Middle), and Approximate VG MME (Bottom) δ

20 min

30 min

1/6

1/4

1/3

1/2

1

κˆ 0.0354 0.0542 0.1662 0.1724 0.2342 0.2098 0.2873 σˆ 0.0115 0.0117 0.0120 0.0121 0.0123 0.0125 0.0127 θˆ 0.0010 0.0023 0.0019 0.0011 0.0020 0.0020 0.0013 bˆ −0.0014 −0.0027 −0.0023 −0.0015 −0.0024 −0.0023 −0.0017 log L 2.2485e+4 1.4266e+4 6.0015e+3 3.7580e+3 2.6971e+3 1.6783e+3 745.8689 κˆ σˆ θˆ bˆ

0.0571 0.0834 0.1839 0.1804 0.2694 0.1579 0.1383 0.0116 0.0119 0.0120 0.0121 0.0123 0.0124 0.0125 0.0016 0.0010 0.0032 0.0019 0.0024 0.0028 0.0041 −0.0020 −0.0014 −0.0036 −0.0022 −0.0028 −0.0032 −0.0045 log L 2.2438e+4 1.4243e+4 5.9946e+3 3.7578e+3 2.6966e+3 1.6780e+3 745.5981 κˆ σˆ θˆ bˆ

0.0573 0.0835 0.1887 0.1819 0.2749 0.1603 0.1423 0.0116 0.0119 0.0121 0.0122 0.0124 0.0124 0.0126 0.0016 0.0010 0.0031 0.0018 0.0024 0.0027 0.0040 −0.0020 −0.0014 −0.0035 −0.0022 −0.0027 −0.0031 −0.0043 log L 2.2437e+4 1.4243e+4 5.9942e+3 3.7577e+3 2.6965e+3 1.6781e+3 745.6023

20


TABLE 1.2 INTC: VG MLE (Top), Exact VG MME (Middle), and Approximate VG MME (Bottom) δ κˆ σˆ θˆ bˆ log L κˆ σˆ θˆ bˆ log L κˆ σˆ θˆ bˆ log L

10 s

20 s

30 s

1 min

5 min

10 min

0.0128 0.0465 −0.0004 0.0000 5.2980e+6 0.0010 0.0169 −0.0001 −0.0003 4.3254e+6 0.0010 0.0169 −0.0001 −0.0003 4.3254e+6

0.0112 0.0300 −0.0004 −0.0000 2.4338e+6 0.0023 0.0152 0.0014 −0.0018 2.0063e+6 0.0023 0.0152 0.0014 −0.0018 2.0063e+6

0.0183 0.0303 −0.0004 0.0000 1.5115e+6 0.0052 0.0145 0.0025 −0.0029 1.2823e+6 0.0052 0.0145 0.0025 −0.0029 1.2823e+6

0.0354 0.0293 −0.0004 0.0000 6.6256e+5 0.0080 0.0138 −0.0040 0.0036 5.8998e+5 0.0081 0.0138 −0.0040 0.0036 5.8987e+5

0.0501 0.0173 −0.0004 0.0000 1.0540e+5 0.0153 0.0125 −0.0013 0.0009 1.0203e+5 0.0153 0.0125 −0.0013 0.0009 1.0203e+5

0.0191 0.0120 −0.0002 −0.0002 4.7949e+4 0.0282 0.0121 0.0011 −0.0015 4.7897e+4 0.0282 0.0121 0.0011 −0.0015 4.7897e+4

TABLE 1.3 INTC: NIG MLE (Top), Exact NIG MME (Middle), and Approx. NIG MME (Bottom) δ

20 min

κˆ 0.0557 σˆ 0.0116 θˆ 0.0019 bˆ −0.0022 log L 2.2498e+4 κˆ 0.0570 σˆ 0.0116 θˆ 0.0016 bˆ −0.0020 log L 2.2498e+4 κˆ 0.0573 σˆ 0.0116 θˆ 0.0016 bˆ −0.0020 log L 2.2498e+4

30 min

1/6

1/4

1/3

1/2

1

0.0874 0.0118 0.0017 −0.0021 1.4274e+4 0.0833 0.0119 0.0010 −0.0014 1.4274e+4 0.0835 0.0119 0.0010 −0.0014 1.4274e+4

0.2621 0.0121 0.0017 −0.0021 5.9988e+3 0.1791 0.0120 0.0033 −0.0037 5.9952e+3 0.1887 0.0121 0.0031 −0.0035 5.9957e+3

0.2494 0.0122 0.0012 −0.0016 3.7575e+3 0.1789 0.0121 0.0019 −0.0022 3.7564e+3 0.1819 0.0122 0.0018 −0.0022 3.7563e+3

0.3412 0.0124 0.0018 −0.0022 2.6969e+3 0.2640 0.0123 0.0025 −0.0028 2.6963e+3 0.2749 0.0124 0.0024 −0.0027 2.6964e+3

0.2024 0.0124 0.0019 −0.0023 1.6777e+3 0.1554 0.0124 0.0028 −0.0032 1.6775e+3 0.1603 0.0124 0.0027 −0.0031 1.6776e+3

0.2159 0.0126 0.0019 −0.0022 745.6436 0.1343 0.0125 0.0042 −0.0046 745.5409 0.1423 0.0126 0.0040 −0.0043 745.5465

1 out of the 200 simulations the exact MME estimator for κ increased by more than twice its value when δ goes from 30 min to 1/6 of a day (only 3 out 200 simulations showed an increment of more than 1.5). In none of the 200 simulation, the MLE estimator for κ increased more than 1.5 its value when δ goes from 30 min to 1/6 of a day. For the NIG model, using the

21

1.5 Empirical Results

TABLE 1.4 INTC: NIG MLE (Top), Exact NIG MME (Middle), and Approx. NIG MME (Bottom) δ

10 s

20 s

30 s

1 min

5 min

10 min

κˆ σˆ θˆ bˆ log L κˆ σˆ θˆ bˆ log L

0.1349 0.0341 −0.0002 −0.0000 3.8974e+6 0.0003 0.0194 0.0194 −0.0196 3.8863e+6

0.0061 0.0190 0.0007 −0.0009 1.8740e+6 0.0007 0.0161 0.0187 −0.0189 1.8718e+6

0.0012 0.0149 0.0086 −0.0088 1.2188e+6 0.0012 0.0148 0.0160 −0.0162 1.2135e+6

0.0024 0.0134 0.0095 −0.0097 5.8072e+5 0.0031 0.0134 0.0134 −0.0136 5.7856e+5

0.0125 0.0119 0.0042 −0.0044 1.0206e+5 0.0157 0.0119 0.0070 −0.0072 1.0204e+5

0.0220 0.0114 0.0037 −0.0038 4.7957e+4 0.0252 0.0114 0.0042 −0.0044 4.7955e+4

κˆ σˆ θˆ bˆ log L

0.0003 0.0194 0.0194 −0.0196 3.8863e+6

0.0007 0.0161 0.0187 −0.0188 1.8718e+6

0.0012 0.0148 0.0159 −0.0161 1.2135e+6

0.0031 0.0134 0.0132 −0.0134 5.7850e+5

0.0160 0.0120 0.0069 −0.0070 1.0204e+5

0.0255 0.0114 0.0042 −0.0044 4.7955e+4

Signature plots for σ estimators NIG model; INTC 2005

Signature plots for the κ estimators NIG model; INTC 2005

0.016

0.015

Estimators for σ

0.0145

MLE based on 1 year Approx. MME based on 1 year MLE based on 6 months Approx. MME based on 6 months MLE based on 3 months Approx. MME based on 3 months

0.5

Estimators for κ

0.0155

0.014 0.0135 0.013 0.0125 0.012

MLE based on 1 year Approx. MME based on 1 year MLE based on 6 months Approx. MLE based on 6 months MLE based on 3 months Approx. MME based on 3 months

0.4

0.3

0.2

0.1

0.0115 0.011

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sampling frequency δ (in days)

Sampling frequency δ (in days)

FIGURE 1.6 Signature plots for the MLE and MME for σ under a NIG model based on different time horizons.

simulations of Fig. 1.4, we found out that in only 3 out of 100 simulations the MME estimator for κ increased by more than 1.2 when δ goes from 30 min to 1/6 of a day (it never increased for more than 1.5). Such a jump in the empirical results could be interpreted as a consequence of microstructure effects.

22


3. According to our previous simulation analysis, the estimators for κ are more reliable when δ gets smaller. Hence, we recommend using the value of the estimator for δ as small as possible, but still in the range where we suspect that microstructure effects are relatively low. For instance, one can propose to take κˆ = 0.1662 under the VG model (respectively κˆ = 0.2621 under the NIG model), or alternatively, one could average the MLE estimators for δ > 1/2. 4. Under both models, the estimators for κ show a certain tendency to decrease as δ gets very small ( x) > 1 − α}.

(2.1)

The probability above is approximated using the constructed histogram of maximum price movements. We note that the rule above is different than the traditional quantile definition which uses nonstrict inequalities. The modification above is imposed by the specific nature of the tick data under study (i.e., discrete data).

31

0.25

0.25

0.20

0.20

Probability

Probability

2.2 Methodology

0.15 0.10 0.05

0.15 0.10 0.05

−4 −3 −2 −1 0 ΔP (a)

1

2

3

−4 −3 −2 −1 0 ΔP (b)

1

2

3

FIGURE 2.1 Two price change distributions. (a) Distribution with one interesting observation and (b) distribution lacking interesting observations.

For illustration consider the two distributions of the price change p(cents) in Fig. 2.1. Suppose we are interested in rare events that occur with probability α = 0.015. The rule in Equation 2.1 will select the observations corresponding to x = −4 for distribution in Fig. 2.1a and no observation in Fig. 2.1b. A traditional quantile rule for any level α ≤ 0.015 no matter how small will indeed select the observations corresponding to x = −4 for the distribution in Fig. 2.1a, however, for the distribution in Fig. 2.1b will select all the observations at x = −3 and x = 3. Therefore, using a traditional quantile rule would force us to analyze points from distributions which lack extreme observations. Note: Using rule (Eq. 2.1) with returns instead of change in price will be preferable in a trading environment. We use change in price (p) for clarity of exposition.

2.2.2.1 A Discussion About the Appropriateness of the Rule of Detecting Rare Events. Our rule is nonstandard and further discussion is necessary.

We first note that because of the way pn quantities are constructed they are not independent. Thus their histogram is only an approximation of the true probabilities of price movement. However, since we are only interested in extreme price movement, rule (Eq. 2.1) will identify candidate rare events which may or may not correspond to the true probability level α. We may have a better depiction of the true histogram of the price movement by considering nonoverlapping windows. There are two reasons why this is not feasible. First, by considering nonoverlapping windows, we may lose extreme price differences calculated using prices from these nonoverlapping windows. Second, in a previous study (Mariani et al., 2009), the authors have shown that returns calculated from tick data exhibit long memory behavior. Thus, even by considering nonoverlapping windows one cannot guarantee that the observations are independent. Furthermore, why do we use our rule and not a more traditional rule for detecting outliers such as 1.5*IQR rule or a parametric outlier test? A parametric detection rule does not make sense in our context since we do not want to hypothesize an underlying statistical model. The interquartile range (IQR) rule is useful for outlier detection not rare events. It is essentially equivalent with our

32

CHAPTER 2 A Study of Persistence of Price Movement

rule since it uses quantiles but it is a very rigid rule. In general, it does not find outliers very often for fat tailed distributions (such as the ones under study here).

2.2.3 RARE EVENT ANALYSIS—CHOOSING THE OPTIMAL LEVEL α After we obtain the rare event candidates, we need to develop a systematic methodology to evaluate them. According to our assumption the movement in price is abnormal and the equity should recover and reverse its momentum. We assume that a trade is placed at the time when a rare event is discovered. We consider a limited volume window (called the after-event window) and we analyze the price behavior (Fig. 2.2).

DEFINITION 2.1 We say that a favorable price movement occurs for a fixed rare event if either the price level within the after-event window decreases below the event price level for at least one trade if the event was generated by a negative value for rule (Eq. 2.1) the price level within the after-event window decreases below the event price level for at least one trade if the event was generated by a positive value for rule (Eq. 2.1)

ΔV < V0

ΔV < Vae

P[cents] 4480 Best return 4470

4460

Best dP dP

4450

4440 Price trade

Trade[#] 500

1000

1500 Worst dP Worst return

FIGURE 2.2 Visual depiction of the quantities used in the study.

2.2 Methodology

33

This definition allows to estimate the probability of a favorable price movement for a specific level α. Specifically, if n is the total number of rare events detected by rule (Eq. 2.1) and k is the number of favorable price movements among them then the probability desired is simply k/n. As we shall see this definition allows the optimal selection of the level α. As the level α increases the events will stop being rare and just plain events. Definition 2.1 does not allow the selection of the optimal volume window size V0 or the optimal after-event window size. To investigate this selection, we consider the return on a trade. To this end, we consider the following strategy: A trade is placed at every rare event, long or short, according to the sign of the quantile detected. An after-event window size is fixed at the moment of the trade. We close the position either during the after-event window if a favorable price movement takes place or at the last trade of the after-event window if a favorable price movement does not take place. The return of such a strategy depends on the price at which the position is closed during the after-event window. To determine the optimal window size and optimal α level, we use the following trading strategy.

DEFINITION 2.2 A position is opened at a point determined according to rule (1). The position is closed according to the following: If a favorable price movement takes place in the after-event window we close the position using the best return possible. If a favorable price movement does not take place within the after-event window we close the position using the worst return possible within the window. For a certain level α and an after-event window size Vae we calculate the expected return by averaging all the trade returns placed following the above strategy. We note that we shall use the trading rule in Definition 2.2 only for determining optimal level α and window size. In practice, using back-testing and strategy calibration will determine a satisfactory favorable price movement and the position will be closed as soon as that level is reached.

2.2.4 MULTISCALE VOLUME CLASSIFICATION Econometric analysis traditionally distinguishes between results obtained for highly traded stocks versus less frequently traded equities. Most of the studies

34


are focused on what are called large capitalization equities which are defined as having market capitalizations larger than a specified cutoff. This definition is often vague, varies over the years and, more importantly, does not necessarily have direct relevance to trading patterns. For example, an equity traditionally classified as a large-cap stock may have a small average daily volume (ADV) and since the later is essential for us, we use a different nomenclature based directly on ADV. The results obtained for a highly liquid equity do not necessarily hold true for less liquid stocks even if both belong to the same capitalization class. Herein, we analyze the change in price from the volume perspective; therefore, we recognize the need for classifying equities into classes based on the average daily traded volume. We refer to this classification as the multiscale volume classification. The histogram in Fig. 2.3 corresponds to the average daily trading volume (ADV) of the total universe of 5369 equities considered in this study. The distribution of the ADV among the stocks is skewed to the right and our selection criterion follows certain features. As a preliminary step in our analysis, we need to eliminate all equities with ADV below 30,000 shares. The 30,000 volume cutoff value is not arbitrary, but it is found to be the minimum level required to perform our analysis. These stocks are grouped in class index 1 and are not used in any of the further analysis. The highest ADV values are concentrated around major indexes and large capitalization equities with more than 10 million shares traded daily. The three intermediary classes contain large, medium, and small ADV stocks. The resulting five classes in our multiscale volume classification are summarized in Table 2.1.

2000

Equity count

1500

1000

500

0 Small-vol stocks

Mid-vol stocks

Large-vol stocks

FIGURE 2.3 Average daily volume distribution.

Super equity

35

2.3 Results

TABLE 2.1 Equities Partitioned into Five Classes

1 2 3 4 5

Class

Average Daily Volume (Shares)

Number Equities

— Small-vol stocks Mid-vol stocks Large-vol stocks Super equity

ADV ≤ 30, 000 30, 000 < ADV ≤ 100, 000 100, 000 < ADV ≤ 1, 000, 000 1, 000, 000 < ADV ≤ 10, 000, 000 10, 000, 000 < ADV

1305 1088 2117 799 60

2.3 Results The methodology described in Section 2.2 is applied to all the equity data within a class in a homogeneous way. For this purpose, we combine all the outlying events detected according to rule (Eq. 2.1) within each class. Table 2.2 presents the probabilities of a favorable price movement according to Definition 2.1. We note that to calculate the probability of favorable price movement as in Definition 2.1, we need to specify a level α for the detection rule, a volume level V0 as well as an after-event volume size (Vae ). To analyze the optimal choices of these parameters, Table 2.2 presents the results obtained for a discrete set of parameters. Specifically, we look at α ∈ {0.02, 0.015, 0.01, 0.005, 0.002, 0.0015, 0.001, 0.0005, 0.0002}, V0 ∈ {3000, 5000, 10, 000}, and Vae = k × V0 , where k ∈ {1, 2, 3}. For a better visualization and interpretation of these numbers we construct probability surfaces for each class and we plot them with respect to the α level and volume Vae in Fig. 2.4. According to the Definition 2.1, we expect the probabilities to increase as the α level becomes more selective, as well as the size of the after-event window volume to increase. Indeed, we observe this behavior in Fig. 2.4, but it is remarkable that the surfaces are parallel and smooth. This seems to indicate that the probability has a similar behavior for each class. Furthermore, by using a simple translation in α and Vae , we may be able to map each surface into another. This translation is very important because once we decide on a optimal level for one class it automatically translates into optimal levels for the other classes. To determine the optimal level for each class, we calculate the expected return of trades according to the Definition 2.2. Specifically, for fixed levels of α and Vae , we average all the returns within each class and present the results in Table 2.3. We also construct the corresponding surfaces in Fig. 2.5. Unlike the probability plots, the surfaces in Fig. 2.5 have different curvatures. For each class surface, we identify the α level which produces maximum return for each Vae . First, unlike the probability surfaces which were decreasing in α the return surfaces have a maximum for each Vae . Remarkably, within each class the maximum return is obtained for the same α level regardless

36


TABLE 2.2 Probability (%) of Favorable Price Movement for Equity Classes for All Days Class

α Level

V0 = 3000

V0 = 5000

V0 = 10,000

for Rule

Vae (shares)

Vae (shares)

Vae (shares)

(Eq. 2.1) 3000 6000 9000 5000 10,000 15,000 10,000 20,000 30,000 Small-vol stocks

Mid-vol Stocks

Large-vol Stocks

Super equity

0.02 0.015 0.01 0.005 0.002 0.0015 0.001 0.0005 0.0002 0.02 0.015 0.01 0.005 0.002 0.0015 0.001 0.0005 0.0002 0.02 0.015 0.01 0.005 0.002 0.0015 0.001 0.0005 0.0002 0.02 0.015 0.01 0.005 0.002 0.0015 0.001 0.0005 0.0002

84.13 85.30 86.68 89.52 92.68 93.72 94.52 n.a. n.a. 78.48 78.85 79.35 81.24 84.65 85.82 86.98 88.91 88.87 76.54 76.82 77.29 78.31 80.50 81.47 82.69 85.42 88.23 71.75 72.36 74.10 74.87 76.27 76.44 77.59 79.40 81.59

Abbreviation: n.a., Not available.

88.97 90.00 91.23 93.48 95.59 96.03 97.26 n.a. n.a. 84.82 85.09 85.43 86.95 89.32 90.42 91.25 92.78 92.23 83.14 83.36 83.72 84.46 85.98 86.72 87.74 89.62 92.01 79.76 80.43 81.90 82.73 83.25 83.25 84.50 86.06 87.91

91.05 91.78 92.66 94.59 96.63 96.86 98.63 n.a. n.a. 87.54 87.71 88.03 89.28 91.20 92.12 92.73 93.88 93.49 86.14 86.29 86.58 87.06 88.30 88.87 89.82 91.58 93.64 83.52 84.09 85.28 86.07 86.76 86.86 88.15 88.76 90.11

89.00 90.17 91.83 94.07 96.33 95.53 97.46 n.a. n.a. 83.39 83.70 84.58 86.37 89.70 90.96 91.71 93.21 94.28 80.55 80.99 81.46 82.40 84.05 84.94 86.20 89.28 92.67 77.36 77.46 78.00 78.72 80.53 80.96 82.60 84.36 84.97

92.68 93.43 94.62 96.08 98.16 97.65 99.15 n.a. n.a. 88.28 88.55 89.23 90.44 92.82 93.58 94.07 94.93 95.65 86.19 86.49 86.77 87.49 88.71 89.35 90.32 92.55 95.17 83.99 84.03 84.57 85.24 86.77 87.23 88.32 89.57 89.64

94.06 94.66 95.68 97.14 98.82 98.82 99.15 n.a. n.a. 90.35 90.54 91.17 92.31 94.34 94.96 95.36 96.11 97.25 88.73 88.99 89.23 89.78 90.76 91.25 92.17 94.09 96.27 87.05 87.12 87.68 88.12 89.50 90.05 90.60 91.37 91.97

93.80 95.73 96.36 94.68 96.46 97.00 95.02 96.83 97.30 96.81 97.75 98.05 98.22 98.57 98.75 98.77 99.08 99.08 98.63 100.00 100.00 n.a. n.a. n.a. n.a. n.a. n.a. 88.84 92.15 93.40 89.28 92.42 93.62 90.05 93.06 94.12 91.76 94.27 95.03 94.08 96.11 96.56 94.78 96.45 96.81 95.56 97.07 97.38 96.46 97.65 97.87 97.58 98.07 98.07 85.47 89.85 91.79 85.80 90.12 92.04 86.21 90.45 92.37 86.99 91.09 92.90 88.78 92.66 94.09 89.69 93.51 94.72 91.26 94.52 95.43 93.05 95.49 96.18 95.17 96.66 96.93 81.49 86.93 89.21 81.83 87.23 89.59 83.03 88.04 90.29 83.73 88.32 90.64 86.08 90.16 91.78 86.21 90.04 91.69 86.96 90.77 92.29 87.22 90.43 92.09 91.41 93.43 94.95

37

2.3 Results

Small-vol stocks

Mid-vol stocks

100

Large-vol stocks

Super equity

90

Probability(%) 80

0.020 0.015 0.010 a level

30,000 20,000 ΔV(shares)

10,000

0.005

FIGURE 2.4 Probability surfaces for equity classes. of the Vae value. The corresponding α level is thus construed as optimal. The following list presents these values. Class Small-vol stocks Mid-vol stocks Large-vol stocks Super equity

Optimal Level α 0.0025 0.0005 0.0001 L, lag = d)), and

(5.6)

Prob(ReturnS&P500 > 0 | (ReturnVol Index < L, lag = d)),

100

100

80

80

60 40 VIX cpVIX VIXprime cVIXb pVIXb

20 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Probability %

Probability %

where ReturnS&P500 is the return of S&P500, ReturnVol Index is the return of the particular VIX, L is a threshold for the index return, and d represents the same day for d = 0, and the previous day for d = 1. The graphs presented are grouped in two sections corresponding to the same day d = 0 or the forecast for the next day d = 1. On all of these graphs the x-axis plots the respective threshold that conditions the probabilities in Equation 5.6, while the y-axis plots the percent of days where the S&P500 moved in the predicted direction. The left image in a figure presents all the dataset, while the right image is restricted to the important financial events detailed in Table 5.3. We analyze separately the positive and negative movement in Figs. 5.6 and 5.7. It is pretty clear from these images that the CBOE VIX is the best indicator for the return/volatility evolution within the same day (d = 0). The profile of the probability curves for the major events selection follows a similar profile to the probability curves for all data analyzed. However, the fact that an increase in the VIX calculated from the calls indicates a drop in the S&P500 index was a surprise to us. For prediction purposes, we analyze the relationship between the previous day VIX and the return on the S&P500 the following day. The corresponding graphs once again split by the positive and negative thresholds are presented in Figs. 5.8 and 5.9, respectively. This time we observe that the CBOE VIX is one of the worst indicators for future day evolution of the S&P500. In some cases the probability of predicting a positive return correctly is well below 50%.


20 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Threshold for positive S&P500

Threshold for positive S&P500 events

(a)

(b)

FIGURE 5.6 Probability of a positive return on S&P500 when d = 0.

109

100

100

80

80


20 0 0.0

Probability %

Probability %

5.3 Results and Discussions

60 40 20 0

VIX cpVIX VIXprime cVIXb pVIXb

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Threshold for negative S&P500 events (b)

0.5 1.0 1.5 2.0 2.5 3.0 Threshold for negative S&P500 (a)

100

100

80

80


20 0 0.0

Probability %

Probability %

FIGURE 5.7 Probability of a negative return on S&P500 when d = 0. VIX cpVIX VIXprime cVIXb pVIXb

60 40 20 0

0.5 1.0 1.5 2.0 2.5 3.0 Threshold for positive S&P500 (a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Threshold for positive S&P500 events (b)

100

100

80

80


20 0 0.0

0.5 1.0 1.5 2.0 2.5 3.0 Threshold for negative S&P500 (a)

Probability %

Probability %

FIGURE 5.8 Probability of a positive return on S&P500 when d = 1.

60 40 20 0

VIX cpVIX VIXprime cVIXb pVIXb

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Threshold for negative S&P500 events (b)

FIGURE 5.9 Probability of a negative return on S&P500 when d = 1.

110

CHAPTER 5 Construction of Volatility Indices

In contrast, two of the indices that we calculated stand out. We note that a drop in the cVIX-b (calculated from call options) forecasts with the highest probability a positive return of the S&P500. Furthermore, an increase in the pVIX-b (calculated from the put options) has a clear advantage in forecasting second day negative S&P500 returns. The probabilities for the pVIX-b are in fact very high for all data sample and also for the major events in the market considered in this analysis. We hope that we convinced the reader that using different types of options in the VIX calculation has the potential to reveal more information about the market than the VIX.

5.4 Summary and Conclusion We propose a new methodology of calculating a value that represents the market volatility at a given moment in time by implementing a stochastic volatility technique. We believe this technique is a viable way to produce a market index. We propose several variants of such indices and we believe each of them is valuable as an indicator of a market movement. The index constructed from calls (cVIX-b) may be considered as an indicator of market’s positive movement while the index constructed from put options (pVIX-b) as an indicator of future negative movement in the market. The difference (spread) between the two indices may be indicative of future market movement. The average of these two indices (cpVIX) and the index constructed with all options (VIX ) both have value in determining when the VIX undervalues or overvalues the market volatility. Finally, we analyze the relations between all these types of indices. We believe all of them bring more information about the market and the methodology has the potential to produce market indicators each indicative of a certain aspect of the financial market.

Appendix A.1: Step-by-Step Explanation of the CBOE Procedure for Calculating VIX Index Using data obtained at the market close on September 8, 2009, we replicate the VIX value following the CBOE procedure. First, we need to do some calculation and rearrangement of the data. The procedure is described below. 1. Selection of Options Chains. VIX generally uses put and call options in the two nearest-term expiration months while there are more options chains trading in the market. Also, it should be pointed out that, with eight days left to expiration, the VIX ‘‘rolls’’ to the second and third contract months in order to minimize the pricing anomalies that might occur close to the maturity.

5.4 Summary and Conclusion

111

2. T , the time to expiration, is measured in minutes rather than in days. Specifically, the calculation of T is given by the following expression: T = {Mcurrentday + Msettlementday + Motherdays }/Myear ,

(5.7)

where Mcurrentday is the number of minutes remaining until midnight of the current day, Msettlementday is the number of minutes from midnight until 8:30 a.m. on SPX settlement day, and Motherdays is the number of minutes in all the days between current day and the settlement day, and Myear is the number of minutes in a year. 3. Calculating the at-the-money strike. This is done by finding the strike price at which the difference between the call and put prices is the smallest. 4. Calculation of F , the forward index level. This is based on the previously determined at-the-money option prices and the corresponding strike price: F = Strikeprice + erT (Call price − Put price)

(5.8)

Note that since two options chains are used in the calculation, two forward index level should be obtained, for the near term and next term options chains. 5. Selection of K0 , the strike price immediately below the forward index level, F . In the following, we demonstrate how to obtain the VIX value using CBOE procedure using data obtained when the S&500 options stopped trading at 3:15 p.m. (Middle time) on September 8, 2009. Step 1. Calculate the time to expiration T , forward index level F , K0 , the strike price immediately below F , and data arrangement: T1 = {Mcurrentday + Msettlementday + Motherdays }/Myear = (525 + 510 + 12,960)/525,600 = 0.026626712 T2 = {Mcurrentday + Msettlementday + Motherdays }/Myear = (525 + 510 + 53,280)/525,600 = 0.103339041 We should note that the total days in a year is 365 days. Since the smallest difference between call and put price is at strike $1025, the at-the-money strike is determined to be at $1025 for both near term options and next term options. Therefore, using federal funds effective rate at 0.15%, the forward

112


index level F1 for the near term options and forward index level F2 for the next term options are F1 = Strike price + erT1 (Call price − Put price) = 1025 + e0.0015×0.026626712 (13.25 − 13.9) = 1024.35 F2 = Strike price + erT2 (Call price − Put price) = 1025 + e0.0015×0.103339041 (29.15 − 30.35) = 1023.80 We also obtain K0 , the strike price immediately below F , which is $1020 for both expirations. Then we select call and put options that have strike prices greater and smaller, respectively, than K0 (it is 1020 here) and nonzero bid price. After encountering two consecutive options with a bid price of zero, do not select any other option. Note that the prices of the options are calculated using the mid-point of the bid-ask spread. At K0 , the average of call and put price is used. The data selected is summarized below: • Calculation of time to maturity: T1 = 0.026626712 T2 = 0.103339041 • At-the-money strike: $1025 for both near term options and next term options. • Federal funds effective rate: 0.15%. • The forward index level F1 for the near term options and forward index level F2 for the next term options are F1 = 1024.35 F2 = 1023.80 • K0 , the strike price immediately below F , is $1020 for both expirations. Step 2. Calculate the volatility for near term and next term options. We apply the following equations to calculate the VIX to the near term and next term options: 2 2 Ki rT1 1 F1 e Q(T , K ) − − 1 , 1 i T1 i Ki2 T1 K0 2 2 Ki rT2 1 F2 2 σ2 = e Q(T2 , Ki ) − −1 . T2 i Ki2 T2 K0 σ12 =

(5.9)

(5.10)

We need to pay attention to value of Ki . Generally, Ki is half the distance between the strike on either side of Ki , but at the upper and lower ends of any options chain, Ki is simply the distance between Ki and the adjacent strike price. We obtain: σ12 = 0.055576664 and σ22 = 0.066630428.

113

5.4 Summary and Conclusion

Step 3. Interpolate σ12 and σ22 to get a single value with a constant maturity of 30 days. VIX is 100 times of the square root of this value. σ = 2

where NT1 NT2 NT2 N365

NT T1 σ12 2 NT2

− N30 N30 − NT1 + T2 σ22 − NT1 NT2 − NT1

N365 , N30

(5.11)

is the maturity of the near term options in minutes (13,995), is the maturity of the next term options in minutes (54,315), is the maturity of a 30-day options in minutes (43,200), is the number of minutes in a year (525,600).

Therefore, the VIX = 100 × σ = 25.62, which is exactly the same value as the one provided by CBOE.

Appendix B.1: Explanation of the New Volatility Index Calculation The following gives details about the quadrinomial tree approximation and the new volatility estimation. In Fig. 5.10, we present a one-step construction. Assume that we are given an (empirical or theoretical) distribution for the stochastic volatility process at the time t when pricing is done. Sample from this volatility distribution to√ obtain the value ϕ. Given this value, we construct a grid of points of the form lϕ t with l taking integer values. No matter where the parent x is, it will fall at one such point or between two grid points. In this grid, let j be the integer that corresponds to the point right above x. Mathematically, j is the integer equal to the integer part of ϕ√xt + 1, as x3 < x < x2 . We will have

x1 = (j + 1)σ(Yi )√Δt

p1

p2 δ

x

x2 = jσ(Yi )√Δt

p3 x3 = (j – 1)σ(Yi )√Δt p4

x4 = (j – 2)σ(Yi )√Δt

FIGURE 5.10 Schematic of one step in the quadrinomial tree method.

114


√ two possible cases: either the point√ jϕ t on the grid corresponding to j (above) is closer to x, or the point (j − 1)ϕ t corresponding to j − 1 (below) is closer. We use δ to denote the distance from the parent x and the closest successor on the grid. We use q to denote the standardized value, that is, δ q := √ . ϕ t

(5.12)

There are two cases: first when x2 is closer to x and the second when x3 is closer to x. In the first case by considering the mean of the increment converging to the drift of the process Xt , the probabilities corresponding to each of the points on the grid can be calculated as 1 (1 + q + q2 ) − p 2 p2 = 3p − q2

p1 =

1 (1 − q + q2 ) − 3p 2 p4 = p

p3 =

(5.13)

In the second case, when x3 is closer to x the probabilities are p1 = p 1 (1 + q + q2 ) − 3p 2 p3 = 3p − q2 p2 =

p4 =

1 (1 − q + q2 ) − p 2

(5.14)

1 1 where p ∈ [ 12 , 6 ]. It is observed that when p is close to 16 , the option values obtained are stable even with few replications (Fig. 5.4 in Florescu and Viens (2008)). In this chapter, we set p = 0.135 throughout the algorithm.

APPENDIX B.2: STEP-BY-STEP EXPLANATION OF THE CONSTRUCTION OF VIX USING STOCHASTIC VOLATILITY QUADRINOMIAL TREE METHOD Here we use quadrinomial tree model to compute the price of a synthetic options with exact 30 days maturity using distribution of implied volatility obtained from S&P500 as input. Then by Black and Scholes (1973) formula, we obtain the implied volatility of this synthetic option. We want to study whether or not this implied volatility multiplied with 100 can better reflect the market volatility.

References

115

There are four steps in the construction of this VIX as follows: • Compute the implied volatilities of entire option chain on SP500 and construct an estimate for the distribution of current market volatility. The implied volatility is calculated by applying Black–Scholes formula. • Use this estimated distribution as input to the quadrinomial tree method. Obtain the price of an at-the-money synthetic option with exactly 30-day maturity. • Compute the implied volatility of the synthetic option based on Black–Scholes formula once the 30-day synthetic option is priced. • Obtain the estimated VIX by multiplying the implied volatility of the synthetic option by 100. Please note that the most important step in the estimation is the choice of proxy for the current stochastic volatility distribution.

REFERENCES Black F, Scholes M. The valuation of options and corporate liability. J Polit Econ 1973;81:637–654. Bollen N, Whaley R. Does net buying pressure affect the shape of implied volatility functions? J Finance 2004;59(2):711–753. CBOE. The new CBOE volatility index-vix. White papers, CBOE; 2003, http://www. cboe.com/micro/vix/vixwhite.pdf. Demeterfi K, Derman E, Kamal M, Zou J. More than you ever wanted to know about volatility swaps. Technical report, Goldman Sachs Quantitative Strategies Research Notes; 1999, http://www.ederman.com/new/docs/gs-volatility_swaps.pdf. Florescu I, Viens F. Stochastic volatility: option pricing using a multinomial recombining tree. Appl Math Finance J 2008;15:151–181. Jiang GJ, Tian YS. Extracting model-free volatility from option prices: an examination of the VIX index. J Deriv 2007;14(3):35–60.

Part Two

Long Range Dependence Models

Chapter

Six

Long Correlations Applied to the Study of Memory Effects in High Frequency (Tick) Data, the Dow Jones Index, and International Indices ERNEST BARANY Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM

M A R I A P I A B E CC A R VA R E L A Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX

6.1 Introduction In recent years, a growing concern is the presence of long-term memory effects in financial time series. The empirical characterization of stochastic processes usually requires the study of temporal correlations and the determination of asymptotic probability density function (pdfs). Major stock indices in developed countries have been previously analyzed in literature, see [1–11] and the references therein. Handbook of Modeling High-Frequency Data in Finance, First Edition. Edited by Frederi G. Viens, Maria C. Mariani, and Ionut¸ Florescu. © 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

119

120

CHAPTER 6 Long Correlations Applied to the Study of Memory

The estimated statistical properties of the temporal time series have been of a great importance comparing the financial markets. The first model that described the evolution of option prices was the Brownian motion. This model assumes that the increment of the logarithm of prices follows a diffusive process with Gaussian distribution [12]. However, the empirical study of temporal series of some of the most important indices shows that in short time intervals, the associated pdfs have greater kurtosis than a Gaussian distribution [5]. The first step to explain this behavior was done in 1963 by Mandelbrot [13]. He developed a model for the evolution of cotton prices by a stable stochastic non-Gaussian Levy process; these types of non-Gaussian processes were first introduced and studied by Levy [14]. The other major problem encountered in the analysis of the behavior of different time-series data is the existence of long-term or short-term correlations in the behavior of financial markets (established versus emerging markets [15], developed countries’ market indices [1–5], Bombay stock exchange index [16], Latin American indices [17], and the references therein). Studies that focus on particular country indices [16–18] generally show that a long-term memory effect exists. These problems may also be avoided considering a temporal evolution of financial markets described by a truncated Levy flight (TLF) [19] or by a standardized TLF, and this is the model used in this work. On the basis of our results we conclude that using the TLF model is an important and useful tool in the analysis of long-memory effects in time series. In many cases, TLF model fits the data very well. However, for a further clarification of the results the analysis should be complemented with the R/S and DFA (detrended fluctuation analysis) methods since in many cases these approaches bring new facts into the picture. Previous literature has concluded that the time series of financial indices are explained by the TLF model [17–19]. The Rescaled Range Analysis (R/S) and DFA methods are used to investigate long-range correlations. Previous work has shown that both methods are very powerful for characterizing fractional behavior [16–18, 20,21]. Since the exponents calculated can serve as verification and comparison of the results, both methods are used in this work. To display the versatility of these models, we apply them to distinct sets of data. We analyze stock indices in developed and developing countries using daily data, and we also present an analysis of high frequency data. One of the main interests in this work is to compare the international stock market indices with US market indices such as the S&P500. Specifically, this paper seeks to determine whether long-memory effects are present in well-diversified international market indices. Long correlation behavior for a financial index along with the rate of return of all companies within the index is analyzed as well. We detect long-range correlations in the rate of return of the companies and briefly discuss some features specific to the equity in comparison to the conclusions obtained for the financial indices. We also apply our methodology to high frequency data. Most of the previous studies that detected long-range correlations in financial indices concentrated on daily data. We wish to verify if the same conclusion applies to high frequency data. Following this line, we analyze a sample of 26 stocks of trade-by-trade (tick) data for a very typical day (April 10, 2007) devoid of any major events.

6.1 Introduction

121

We found that all unit-root tests performed rejected the existence of a unit-root type nonstationarity. The p-values of the tests were all under 0.01. We use Rescaled Range Analysis (R/S) and DFA methods to determine long-range correlations. Both methods characterize fractional behavior, but R/S analysis can yield more accurate results for small and stationary data sets and DFA analysis yields more accurate result for nonstationary data sets. The exponents calculated are complementary and could serve as verification and comparison of the results; therefore, both methods are used. We found evidence that even in an ordinary day without any notable information for about 75% of the market, the use of short-term memory models is inappropriate. Specifically, in only 23% of the studied cases one of the tests performed did not reject the Gaussian hypothesis (no memory or very short-term memory). There were no stocks for which both tests performed agreed that the data may be Gaussian. Finally, we study high frequency data corresponding to the Bear Stearns crash. On Friday, March 14, 2008, at about 9:14 a.m., JP Morgan Chase together with the Federal Reserve Bank of New York announced an emergency loan to Bear Stearns (about 29 billion, terms undisclosed) to prevent the firm from becoming insolvent. This bailout was declared to prevent the very likely crash of the market as a result of the fall of one of the biggest investment banks at the time. This measure proved to be insufficient for keeping the firm alive, and two days later, on Sunday March 16, 2008, Bear Stearns signed a merger agreement with JP Morgan Chase essentially selling the company for $2 a share (price revised on March 24 to $10/share). The same stock traded at $172 in January 2007 and $93 a share in February 2007. Today, this collapse is viewed as the first sign of the risk management meltdown of investment bank industry in September 2008 and the subsequent global financial crisis and recession. In this chapter we are not concerned with the details and causes that lead to the demise of Bear Stearns. Instead, we have two major objectives in mind. First, we would like to know how soon an investor who was lacking insider information but had at his/her disposal all the information contained in the equity prices could have discovered that a crash is imminent and take the necessary precautions. Second, although the crisis was restricted to the financial companies, we wish to find out if it was a market-wide phenomenon and if its effects may be observed in the evolution of price of technology companies or food producers for example (which technically should not be affected by the crisis in the financial sector). The premise we make in this study is a simple one. In the normal market conditions, all the participating agents have diverse views and accordingly the price process should not exhibit large long-term memory effects. Of course, even in normal market conditions, when working with high frequency data, the price process is far from the log-normal specification. On the other hand, when a crisis situation is anticipated all the agents start to behave in a similar way and accordingly the resulting price process starts to exhibit large memory effects. There is ample recent evidence for this fact, we only mention [8 and 9] and the references therein. We estimate the Hurst parameter (H ) as well as the Detrended Fluctuation parameter (α) and we compare with 0.5. The further these parameters are from

122

CHAPTER 6 Long Correlations Applied to the Study of Memory

0.5 the stronger the evidence of a crash event waiting to happen. The reason why we estimate both parameters is that the alpha parameter works better with nonstationary data than H . On the other hand, when working with stationary data H is much more relevant. The estimation methodology is described below. We conclude that stochastic volatility models, jump diffusion models, and general Levy processes seem to be needed for the modeling of high frequency data in any situation.

6.2 Methods Used for Data Analysis In this section we give details about the methodology used in data analysis.

6.2.1 THE TRUNCATED LEVY FLIGHT Levy [22] and Khintchine [23] solved the problem of the determination of the functional form that all the stable distributions must follow. They found that the most general representation is through the characteristic function ϕ(q), by the following equation: ⎧ ⎪ α 1 − iβ q tan π α ⎪ (α = 1) iμq − γ |q| ⎨ |q| 2 (6.1) ln(ϕ(q)) = q 2 ⎪ ⎪ ln |q| (α = 1) ⎩iμq − γ |q| 1 + iβ |q| π where 0 < α ≤ 2, γ is a positive scale factor, μ is a real number, and β is an asymmetry parameter that takes values in the interval [−1, 1]. The analytic form for a stable Levy distribution is known only in these cases: α = 1/2, β = 1, (Levy–Smirnov distribution), α = 1, β = 0 (Lorentz distribution), α = 2 (Gaussian distribution). In this work, symmetric distributions (β = 0) with zero mean value (μ = 0) are considered. In this case, the characteristic function takes the form: α

ϕ(q) = e−γ |q|

(6.2)

As the characteristic function of a distribution is its Fourier transform, the stable distribution of index α and scale factor γ is 1 PL (x) ≡ π

∞

α

e−γ |q| cos(qx) dq

(6.3)

0

The asymptotic behavior of the distribution for large values of the absolute value of x is given by

123

6.2 Methods Used for Data Analysis

PL (|x|) ≈

γ (1 + α) sin(πα/2) ≈ |x|−(1+α) π|x|1+α

(6.4)

and the value in zero PL (x = 0) by PL (x = 0) =

(1/α) παγ 1/α

(6.5)

The fact that the asymptotic behavior for huge values of x is a power law has as a consequence that the stable Levy processes have infinite variance. To avoid the problems arising in the infinite second moment, Mantegna and Stanley [19] considered a stochastic process with finite variance that follows scale relations called TLF . The TLF distribution is defined by ⎧ ⎪ x>l ⎨0 (6.6) P(x) = cPL (x) −l <x 0 if the corresponding wealth process X x,π ,C (·) is strictly positive on [0, ∞) a.s. An application of Itö’s rule to the product of the processes Z0 (·) and γ (·)X x,π ,C (·) leads to x,π ,C (t) + H (s)dC(s) H (t)X =x+

(0,t] t

H (s)X x,π ,C (s)(σ ∗ (s)π(s) − θ(s))∗ dW (s),

0 < t < ∞.

(11.42)

0

This shows, in particular, that for any pair (π, C) available at initial capital x > 0, the process H (·)X x,π ,C (·) + (0,·] H (s)dC(s) is a continuous, positive local martingale, hence a supermartingale, under P. Thus the limit limt→∞ H (t)X x,π ,C (t) + (0,t] H (s)dC(s) exists a.e. and the optional sampling theorem gives

x,π ,C E H (τ )X (τ ) + H (s)dC(s) ≤ x, τ ∈ S0 . (11.43) (0,τ ]

11.7.2 THE OPTIMIZATION PROBLEM The infinite horizon problem has not been treated by Karatzas and Wang (2000) with the convex duality methods as it presents the challenge that wealth can be accumulated at τ = ∞. For fixed stopping time τ ∈ S0 define

τ e−βt U1 (c(t)) + e−βτ U2 (ξ ) J∞ (x; π, C, τ ) E +

0 ∞

τ

e−βt U3 (c(t))dt · 1{τ NV is a Cauchy sequence in Wp2,1 (V ). Hence uN → u over V for the Wp2,1 -norm, and then u is a strong solution in V . It follows that u satisfies the equation on × (0, T ). Furthermore, it is clear that u(x, 0) = u0 (x). For M > N , we have that uM (x, t) = uN (x, t) = h(x, t) for x ∈ ∂ ∩ ∂N and t ∈ (0, T ). Thus, u satisfies the boundary condition u(x, t) = h(x, t) on ∂ × (0, T ).

13.2.3 A GENERAL INTEGRO-DIFFERENTIAL PROBLEM The generalized Black–Scholes model with stochastic volatility and jumps leads us to study a more general integro-differential parabolic problem than the one given by Equation 13.9. Mariani and Florescu of Ref. 7 proved the existence of a strong solution to a more general problem in an unbounded parabolic domain QT = × (0, T ). Here, is an open, unbounded subset of Rd , with a smooth boundary ∂. Once again, the fact that has a boundary reflects the idea that the asset price is unbounded, whereas the volatility is assumed to be bounded. One can use the same methodology as in the previous section to obtain a stronger existence result than the result given in Ref. 7. In this section, we state the general problem, the assumptions, and the two main results. We do not give the proofs because they are essentially the same as the proofs given in the previous section. First, we will consider the following initial-boundary value problem in the bounded parabolic domain VT = U × (0, T ), T > 0: ut − Lu = F(x, t, u)inVT , u(x, 0) = u0 (x) on ,

(13.24)

363

13.2 Method of Upper and Lower Solutions

u(x, t) = g(x, t) on ∂ × (0, T ). Then, we will try to extend our results to the corresponding initial-value problem in the unbounded domain QT = × (0, T ). As before, L = L(x, t) is a second-order elliptic operator in nondivergence form, namely, L(x, t) :=

d i,j=1

∂2 ∂ + bi (x, t) + c(x, t). ∂xi ∂xj ∂xi i=1 d

aij (x, t)

The integral operator is defined by F(x, t, u) = f (x, t, u(x, t), y) dy.

(13.25)

This operator will be continuous as the ones defined in Equations 13.7–13.9 modeling the jump. The case in which f is decreasing respect to u and all jumps are positive corresponds to the evolution of a call option near a crash. Furthermore, we will impose assumptions A(1) through A(5) which were given in the previous section. Here, we change A(6) and A(7) to: A(6) For each y ∈ U , f (x, t, z, y) belongs to the space C δ,δ/2,δ (V T × R) and f (x, t, 0, y) = 0. A(7) The operator F(x, t, z) is nondecreasing with respect to z. We now state an important result.

LEMMA 13.9 Suppose ϕ ∈ C 2+δ,1+δ/2 V T is defined as in Lemma 2.1. Then there exists a classical solution u ∈ C 2+δ,1+δ/2 (V T ) to the problem ut − Lu = F(x, t, u) in VT , u(x, 0) = u0 (x) on U , u(x, t) = ϕ(x, t) on ∂U × (0, T ),

(13.26)

and α = 0 is a lower solution. Moreover, if β is an upper solution, then 0 ≤ u(x, t) ≤ β(x, t), for all (x, t) ∈ VT .

Using this result and a standard diagonal argument as in the previous section, one can extend this result to the unbounded domain QT = × (0, T ):

364

CHAPTER 13 Solutions to Integro-Differential Parabolic Problem

THEOREM 13.10 There exists a strong solution u ∈ Wp2,1 (QT ) to the problem ut − Lu = F(x, t, u) in QT , on , u(x, 0) = u0 (x) u(x, t) = g(x, t) on ∂ × (0, T ),

(13.27)

and α = 0 is a lower solution. Moreover, if β is an upper solution, then 0 ≤ u(x, t) ≤ β(x, t), for all (x, t) ∈ QT .

13.3 Another Iterative Method In the last part of the previous section, we discussed a Black–Scholes model with jumps, where the volatility was assumed to be stochastic. Moreover, the asset price was modeled by a Poisson process. This gave rise to a parabolic integro-differential problem, where the integral operator depended on u(x, t). In this section, we consider another Black–Scholes model with jumps. However, the asset price will now be modeled by an exponential Lévy model. Furthermore, we assume the volatility is constant. This will lead us to a parabolic integro-differential problem, where the integral operator depends on u(x, t) and ux (x, t). We will then generalize the problem to an unbounded parabolic domain, just as we have done in the previous section. This will lead us to a integro-differential problem, where the integral operator depends on u(x, t) and ∇u(x, t). The presence of the gradient in the integrand prevents us from being able to use the method of upper and lower solutions to analyze this class of problems. Therefore, we must use some other method to study these problems. The iterative method we will use for this problem was developed by Chadam and Yin in Ref. 22 to study a similar partial integro-differential problem.

13.3.1 STATEMENT OF THE PROBLEM As pointed out in Ref. 17, when modeling high frequency data in applications, a Lévy-like stochastic process appears to be the best fit. When using these models, option prices are found by solving the resulting PIDE. For example, integrodifferential equations appear in exponential Lévy models, where the market price of an asset is represented as the exponential of a Lévy stochastic process. These models have been discussed in several published works such as Refs 17 and 23.

365

13.3 Another Iterative Method

In this section, we consider the following integro-differential model for a European call option ∂C σ 2S2 ∂ 2C ∂C (S, t) − rC(S, t) (S, t) + rS (S, t) + ∂t ∂S 2 ∂S 2 ∂C y y + ν(dy) C(Se , t) − C(S, t) − S(e − 1) (S, t) = 0, ∂S

(13.28)

where the market price of an asset is represented as the exponential of a Lévy stochastic process (see Chapter 12 of Ref. 17). Also, we assume the option has the final payoff C(S, T ) = max(ST − K , 0),

(13.29)

where K > 0 is the strike price. If we introduce the change of variables τ = T − t,

S x = ln + rτ , K erτ x−rτ ,T − τ , C Ke u(x, τ ) = K then Equation 13.28 becomes ∂u σ 2 ∂ 2u ∂u (x, τ ) − (x, τ ) + F(u, ux ), (x, τ ) = ∂τ 2 ∂x 2 ∂x

(13.30)

with the initial condition u(x, 0) = u0 (x) for all x ∈ R.

(13.31)

The term

∂u y F(u, ux ) = u(x + y, τ ) − u(x, τ ) − (e − 1) (x, τ ) ν(dy) ∂x

(13.32)

is an integro-differential operator modeling the jump. We shall derive Equations 13.30–13.32. First, notice that to convert back to the original variables, we use the equations t = T − τ, S = K ex−rτ , C(S, t) = C K ex−rτ , T − τ = K e−rτ u(x, τ ).

366


Next, we compute each partial derivative in Equation 13.28. We do this by using the chain and product rules repeatedly and the expression K e−rτ u(x, τ ) for C(S, t): ∂C ∂τ ∂C ∂u ∂u ∂C = =− = rK e−rτ u(x, τ ) − rK e−rτ − K e−rτ , ∂t ∂τ ∂t ∂τ ∂x ∂τ ∂C ∂C ∂x 1 ∂C 1 ∂u = = = K e−rτ , ∂S ∂x ∂S S ∂x S ∂x

∂ 1 −rτ ∂u 1 1 −rτ ∂ 2 u ∂x ∂ 2C −rτ ∂u = K e K e = − + Ke , ∂S 2 ∂S S ∂x S2 ∂x S ∂x 2 ∂S =−

2 1 1 −rτ ∂u −rτ ∂ u K e K e . + S2 ∂x S2 ∂x 2

Furthermore, notice that the first term in the integral operator of Equation 13.28 can be expressed as C(Sey , t) = C K ex+y−rτ , T − τ = K e−rτ u(x + y, τ ). If we substitute everything into Equation 13.28 and divide through by K e−rτ , we obtain Equations 13.30 and 13.33. It is clear that S > 0 implies x ∈ R and t = T implies τ = 0. Using these two facts, Equation 13.29 becomes C(S, T ) = Ku(x, 0) = u˜ 0 (x). This justifies the initial condition in Equation 13.31, where u0 (x) = u˜ 0 (x)/K . Once again, for the classical Black–Scholes model and for any other Black–Scholes model, such as models which take into account jumps, it follows that C(S, t) ∼ 0 when S ∼ 0 and C(S, t) ∼ S when S is very large. As in the previous section, this observation will justify why we will continuously be using the same boundary condition later in this section.

13.3.2 A GENERAL PARABOLIC INTEGRO-DIFFERENTIAL PROBLEM In a more general context, the previous discussion motivates us to consider more general integro-differential parabolic problems. First, we consider the following initial-boundary value problem in the bounded parabolic domain QT = × (0, T ), T > 0: ut − Lu = F(x, t, u, ∇u) in QT , on , u(x, 0) = u0 (x) u(x, t) = g(x, t) on ∂ × (0, T ).

(13.33)

367


Then, we try to extend our results to the corresponding initial-value problem in the unbounded domain RTd +1 = Rd × (0, T ): ut − Lu = F(x, t, u, ∇u) in RTd +1 , u(x, 0) = u0 (x) on Rd .

(13.34)

Here, L = L(x, t) is a second-order elliptic operator in nondivergence form, namely, L(x, t) :=

d i,j=1

∂2 ∂ + bi (x, t) + c(x, t). ∂xi ∂xj ∂xi i=1 d

aij (x, t)

The integro-differential operator is defined by F(x, t, u, ∇u) = f (x, t, y, u(x, t), ∇u(x, t)) dy.

(13.35)

This integro-differential operator will be a continuous integral operator as the ones defined in Equations 13.28 and 13.32 modeling the jump. The case in which f is decreasing respect to u and all jumps are positive corresponds to the evolution of a call option near a crash. Throughout this section, we impose the following assumptions: A(1) The coefficients aij (x, t), bi (x, t), c(x, t) belong to the Hölder space C δ,δ/2 (Q T ). A(2) For some 0 < λ < , aij (x, t) satisfies the inequality λ|v|2
0, f satisfies the estimate |f (x, t, y, z, p)| ≤ C0 (1 + |z| + |p|), for all (x, t, y, z, p) ∈ Q T × × Rd +1 , where C0 is independent of parameters of f . We shall prove the existence of a solution to Equation 13.33 using an iteration argument. We will do this by proving estimates based on a Green’s function. Afterward, we will use a standard argument to show that our solution can be extended to give us a solution to the initial-value problem in RTd +1 . In this section, QT = × (0, T ) always denotes a bounded parabolic domain, where ⊂ Rd is open and has smooth boundary ∂. Let us define the 2,1 (Q ). We function space C 1+1,0+1 (Q T ) to be the set of all u ∈ C 1,0 (Q T ) ∩ W∞ T 1+1,0+1 (Q T ) is a strong solution to the parabolic initial-boundary will say u ∈ C value problem (Eq. 13.33) provided that u satisfies the parabolic equation almost everywhere in QT and the initial-boundary conditions in the classical sense. Once again, the following lemma follows immediately from Theorem 10.4.1 in Ref. 4.

LEMMA 13.11 There exists a unique solution ϕ ∈ C 2+δ,1+δ/2 Q T to the problem ut − Lu = 0

in QT ,

u(x, 0) = u0 (x) on ,

(13.36)

u(x, t) = g(x, t) on ∂ × (0, T ).

As we have already mentioned in the previous section, Lemma 3.1 can be thought of as generalization of the classical Black–Scholes model where the stock price S satisfies ε < S < Smax . In all practicality, one should not assume that S is bounded away from 0. The problem in the next theorem can be regarded as a generalization of Equations 13.30 and 13.34, where the stock price S is bounded above and bounded below away from 0 as in Lemma 13.6. We take the same boundary condition as in Lemma 13.6, because of our earlier comment regarding the behavior of the option value when S is really small or really large for any Black–Scholes model.


369

THEOREM 13.12 Let ϕ be defined as in Lemma 13.11. Then there exists a strong solution u ∈ C 1+1,0+1 (Q T ) to the problem ut − Lu = F(x, t, u, ∇u) in QT , u(x, 0) = u0 (x) on , u(x, t) = ϕ(x, t) = g(x, t) on ∂ × (0, T ).

(13.37)

Proof . First, we introduce a change of variables to transform our problem into one with a zero boundary condition. If we let v(x, t) = u(x, t) − ϕ(x, t), v0 (x) = u0 (x) − ϕ(x, 0) = 0, then v will satisfy the initial-boundary value problem vt − Lv = F(x, t, v + ϕ, ∇(v + ϕ)) in QT , v(x, 0) = 0 on , v(x, t) = 0 on ∂ × (0, T ).

(13.38)

We further change variable τ = At , where A is a constant which will be chosen later. By abuse of notation we denote ALv by Lv and AF by F. Then if T ∗ = TA , Equation 13.38 becomes vτ − Lv = F(x, τ , v + ϕ, ∇(v + ϕ)) in QT ∗ , v(x, 0) = 0 on , v(x, τ ) = 0 on ∂ × (0, T ∗ ).

(13.39)

If problem (Eq. 13.39) has a strong solution, then Equation 13.37 will have a strong solution since u = v + ϕ. We use an iteration procedure to construct the solution to Equation 13.39. Consider the problem βτ − Lβ = F(x, τ , α + ϕ, ∇(α + ϕ)) in QT ∗ , β(x, 0) = 0 on , β(x, τ ) = 0 on ∂ × (0, T ∗ ),

(13.40)

where α ∈ C 2+δ,1+δ/2 (Q T ∗ ,U ) is arbitrary. Using the same argument as in Section 13.2, we can show that F(x, τ , α + ϕ, ∇(α + ϕ)) ∈ C δ,δ/2 (Q T ∗ ). By Theorem 10.4.1 in Ref. 4, there exists a unique solution β ∈ C 2+δ,1+δ/2 (Q T ∗ ) to problem (Eq. 13.40).

370


Using this result, we can now define vn ∈ C 2+δ,1+δ/2 (Q T ∗ ), n ≥ 1, to be the unique solution to the linearized problem ∂τ vn − Lvn = F(x, τ , vn−1 + ϕ, ∇(vn−1 + ϕ)) in QT ∗ , vn (x, 0) = 0 on , vn (x, τ ) = 0 on ∂ × (0, T ∗ ), (13.41) where v0 = v0 (x) = 0 ∈ C 2+δ,1+δ/2 (Q T ∗ ,U ). To prove the existence of a solution to problem (Eq. 13.39), we will show that this sequence converges. Since the operator ∂t∂ − L is parabolic, therefore from Ref. 24, there exists a Green’s function G(x, y, τ , τ ) for problem (Eq. 13.41). For n ≥ 1, the solution vn can be written as τ n v (x, τ ) = G(x, y, τ , τ ) F(y, τ , vn−1 + ϕ, ∇(vn−1 + ϕ)) dy dτ 0 G(x, y, τ , 0)v0 (y) dy + = 0

τ

G(x, y, τ , τ ) F(y, τ , vn−1 + ϕ, ∇(vn−1 + ϕ)) dy dτ ,

because v0 (y) = 0. Here, F(y, τ , vn−1 + ϕ, ∇(vn−1 + ϕ)) = f (y, τ , z, (vn−1 + ϕ)(y, τ ), ∇(vn−1 + ϕ)(y, τ )) dz.

For convenience, we will write F n−1 (y, τ ) = F(y, τ , vn−1 + ϕ, ∇(vn−1 + ϕ)) f (y, τ , z, (vn−1 + ϕ)(y, τ ), ∇(vn−1 + ϕ)(y, τ )) dz. =

Now we take the first and second derivatives of vn (x, τ ) with respect to x: τ n vxi (x, τ ) = Gxi (x, y, τ , τ ) F n−1 (y, τ ) dy dτ , vxni xj (x, τ ) =

0

0

τ

Gxi xj (x, y, τ , τ ) F n−1 (y, τ ) dy dτ.

For the parabolic operator, we again have from Chapter IV.16 in Ref. 24, the estimates

|x − y|2 − d2 , (13.42) |G(x, y, τ , τ )| ≤ c1 (τ − τ ) exp −C2 τ −τ

371


|Gxi (x, y, τ , τ )| ≤ c1 (τ − τ )

− d+1 2

|Gxi xj (x, y, τ , τ )| ≤ c1 (τ − τ )−

d+2 2

|x − y|2 , exp −C2 τ −τ

|x − y|2 , exp −C2 τ −τ

(13.43) (13.44)

where τ > τ and the constants c1 and C2 are independent of all parameters of G. If we combine everything, we get vn (·, τ )W∞2 () = vn (·, τ )L∞ () d

+

vxni (·, τ )L∞ ()

+

i=1

≤ +

d i=1

+

vxni xj (·, τ )L∞ ()

i,j=1

τ

0

d

G(·, y, τ , τ )L∞ () |F n−1 (y, τ )| dy dτ

τ

0

d i,j=1 0

Gxi (·, y, τ , τ )L∞ () |F n−1 (y, τ )| dy dτ

τ

Gxi xj (·, y, τ , τ )F n−1 (y, τ ) dy

L∞ ()

dτ.

Our goal is to show that vn (·, τ )W∞2 () is uniformly bounded on the interval [0, T ∗ ] so that we can use the Arzelà –Ascoli theorem and a weak compactness argument (Theorem 3 of Appendix D in Ref. 25). From A(6), we have |F n−1 | = F n−1 . We obtain the following estimates by using A(7): F n−1 (y, τ ) ≤ ≤

|f (y, τ , z, (vn−1 + ϕ)(y, τ ), ∇(vn−1 + ϕ)(y, τ ))| dz C0 (1 + |vn−1 (y, τ )|

+ |ϕ(y, τ )| + |∇vn−1 (y, τ )| + |∇ϕ(y, τ )|) dz d n−1 n−1 C0 v (·, τ )L∞ () + vyi (·, τ )L∞ () ≤

i=1

+ C0 1 + sup |ϕ(y, τ )| + QT ∗

d

sup |ϕyi (y, τ )|

dz

i=1 QT ∗

≤ C3 vn−1 (·, τ )W∞2 () + CT ∗ , where C3 is a constant independent of T ∗ , whereas CT ∗ is a constant that depends on T ∗ . By a direct calculation, we can easily see that (with |x − y|2 =

372


(x1 − y1 )2 + · · · + (xd − yd )2 ),

d |x − y|2 dy (τ − τ )− 2 exp −C2 τ −τ

d |x − y|2 dy ≤ (τ − τ )− 2 exp −C2 τ −τ Rd 2 − d2 = C2 e−σ dσ

=

Rd

π C2

d2

.

We can see this by computing the integral in one dimension:

∞ 1 (x1 − y1 )2 (τ − τ )− 2 exp −C2 dy1 τ −τ −∞ ∞ 1 τ − τ −ω2 = (τ − τ )− 2 e 1 dω1 C2 −∞ ∞ 2 −1 = C2 2 e−ω1 dω1

=

−∞

π C2

12

,

where we use

ω1 =

τ

C2 (x1 − y1 ). −τ

The integral in Rd is a product of these one-dimensional integrals. This gives us the desired result. The Green’s function estimate Gx x (·, y, τ , τ ) dy ≤ C4 (τ − τ )−γ , (13.45) i j L∞ ()

where C4 is a constant independent of T ∗ , 0 < γ < 1 and τ > τ can be found in Ref. 26. Using all of our previous estimates and Equation 13.46, we obtain

v (·, τ )W∞2 () = v (·, τ ) n

n

L∞ ()

+

d

vxni (·, τ )L∞ ()

i=1

+

d i,j=1

vxni xj (·, τ )L∞ ()

373


≤

τ

1

A + B(τ − τ )− 2 + D(τ − τ )−γ

0

× C3 vn−1 (·, τ )W∞2 () + CT ∗ dτ

τ 1−γ 1/2 +D = CT ∗ Aτ + 2Bτ 1−γ τ 1 + C3 A + B(τ − τ )− 2 + D(τ − τ )−γ vn−1 (·, τ ) 0

W∞2 () dτ ≤ C(T ∗ , γ ) + C

τ

1

A + B(τ − τ )− 2 + D(τ − τ )−γ

0

vn−1 (·, τ )W∞2 () dτ where the constants A, B, D, and C are independent of T ∗ . The constant C(T ∗ , γ ) depends only on T ∗ and γ . Therefore we have vn (·, τ )W∞2 () ≤ C(T ∗ , γ ) τ 1 +C A + B(τ − τ )− 2 + D(τ − τ )−γ 0

v

n−1

(·, τ )W∞2 () dτ.

(13.46)

Observe that there exist an upper bound of the integral

τ

1

A + B(τ − τ )− 2 + D(τ − τ )−γ

dτ ,

0

for τ ∈ [0, T ∗ ]. Choose A (where τ = At , as defined before) such that this upper bound is ε where |εC| < 1. This is possible as C does not depend on T ∗ . We observe from Equation 13.46 that v1 (·, τ )W∞2 () ≤ C(T ∗ , γ ), v2 (·, τ )W∞2 () τ 1 ≤ C(T ∗ , γ ) + C A + B(τ − τ )− 2 + D(τ − τ )−γ 0

v1 (·, τ )W∞2 () dτ ≤ C(T ∗ , γ ) + C(T ∗ , γ )Cε, v3 (·, τ )W∞2 () τ 1 ∗ ≤ C(T , γ ) + C A + B(τ − τ )− 2 + D(τ − τ )−γ 0

v (·, τ )W∞2 () dτ 2

374


≤ C(T ∗ , γ ) + C(C(T ∗ , γ ) + C(T ∗ , γ )Cε)ε = C(T ∗ , γ ) + C(T ∗ , γ )Cε + C(T ∗ , γ )C 2 ε2 . Proceeding this way vn (·, τ )W∞2 () ≤ C(T ∗ , γ ) 1 + Cε + · · · + C n−1 εn−1 . ∗

(T ,γ ) Since |εC| < 1, we obtain vn (·, τ )W∞2 () ≤ C1−εC , where n n = 0, 1, 2, . . .. Consequently v (·, τ )W∞2 () is uniformly bounded on the closed interval [0, T ∗ ]. Using this result along with Equation 3.11, we can easily show that vτn (·, τ )L∞ () is also uniformly bounded on [0, T ∗ ]. Since vn (·, τ )W∞2 () and vτn (·, τ )L∞ () are continuous functions of τ on the closed interval [0, T ∗ ], it follows that |vn |, |vxni |, |vxni xj | and |vtn | are uniformly bounded on Q T ∗ . Thus vn (·, τ ) is equicontinuous in C(Q T ∗ ). By the Arzelà –Ascoli theorem, there exists a subsequence {vnk }∞ k=0 such that as k → ∞,

vnk → v ∈ C(Q T ∗ ) and vxnik → vxi ∈ C(Q T ∗ ) , where the convergence is uniform. Furthermore, by Theorem 3 in Appendix D of Ref. 25, vxnikxj → vxi xj ∈ L∞ (Q T ∗ ) and vτ k → vτ ∈ L∞ (Q T ∗ ), n

as k → ∞. Here, the convergence is in the weak sense. Therefore, vnk converges uniformly on the compact set Q T ∗ to a function v ∈ C 1+1,0+1 (Q T ∗ ). By a standard argument [20], we have that v satisfies the parabolic equation in Equation 13.39 almost everywhere and the initial-boundary conditions in the classical sense. Hence, v is a strong solution to problem (Eq. 13.39). Consequently, u is a strong solution to Equation 13.37. Now, we show that we can extend this solution to give us a strong solution on the unbounded domain RTd +1 = Rd × (0, T ).

THEOREM 13.13 There exists a strong solution u ∈ Wp2,1 (RTd +1 ) to the problem ut − Lu = F(x, t, u, ∇u) in RTd +1 , u(x, 0) = u0 (x) on Rd such that the solution u(x, t) → g(x, t) as |x| → ∞.

(13.47)

375

13.4 Integro-Differential Equations in a Lévy Market

Proof . We approximate the domain Rd by a nondecreasing sequence {N }∞ N =1 of bounded smooth subdomains of . For simplicity, we will let N = B(0, N ) be the open ball in Rd centered at the origin with radius N . Also, we let VN = N × (0, T ). Using the previous theorem, we let uM ∈ C 1+1,0+1 (V M ) be a solution to the problem ut − Lu = F(x, t, u, ∇u) in VM , u(x, 0) = u0 (x) on M , u(x, t) = g(x, t) on ∂M × (0, T ).

(13.48)

Since M ≥ 1 is arbitrary, we can use a standard diagonal argument (Theorem 13.10) to extract a subsequence that converges to a strong solution u to the problem on the whole unbounded space RTd +1 . Clearly, u(x, 0) = u0 (x) and u(x, t) → g(x, t) as |x| → ∞.


The Black–Scholes models with jumps arise from the fact that the driving Brownian motion is a continuous process, and so there are difficulties fitting the financial data presenting large fluctuations. The necessity of taking into account the large market movements, and a great amount of information arriving suddenly (i.e., a jump) has led to the study of PIDE in which the integral term is modeling the jump [1,7]. In this chapter we model the jump in such a way that it can be transformed to a convolution integral. Si 2 We assume n assets S = (S1 , . . . , Sn ) and we assume ni=1 ln |E| ≤ R 2 , r −λp

for some constant R . Let us define the region by U. Define αi = − 12 ( σi 2 /2i − 1), for i = 1, . . . , n and = ni=1 αi − 1. We consider the constant volatility case with n independent assets. Then in Equation 13.8, ρij = 0 for i = j. We consider ri to be the riskless rate for Si . Then Equation 13.8 becomes (if we model the jump as the integral of the following equation) n n n ∂C 1 2 2 ∂ 2 C ∂C + (r − λp )S − ri C + σi Si i i i ∂t 2 ∂Si ∂Si2 i=1 i=1 i=1 −1 n α +1 i G(S, P)C(P, t) Pi dP = 0, + λ|E| U

(13.49)

i=1

for some random variable P = (P1 , . . . , Pn ) ∈ U, where λ is the jump intensity. We take G(S, P) = g(ln PS11 , . . . , ln PSnn ), where g is probability density function of its variables, pi = E(Pi − 1), where E is the expectation operator and the

376


random variable Pi − 1 measures the percentage change in the stock price for Si if jump occurs. We assume σi = σ , i = 1, . . . , n. So Equation 13.49 becomes 1 2 ∂ 2C ∂C ∂C Si + (r − λp )S − ri C + σ2 i i i ∂t 2 i=1 ∂Si2 ∂Si i=1 i=1 −1 n α +1 Pi i dP = 0. + λ|E| G(S, P)C(P, t) n

n

n

U

We set

Si = |E|exi ,

(13.50)

i=1

Pi = |E|eyi ,

and C(S1 , . . . , Sn , t) = |E| exp

n

t=T−

τ σ 2 /2

,

αi u(x1 , . . . , xn , τ ).

i=1

Then we get −

∂u + γ u + Δu + λ ∂τ

B(R )

g(x − Y )u(Y )dY = 0,

(13.51)

where B(R ) = {x = (x1 , . . . , xn ) ∈ Rn | ni=1 xi2 ≤ R 2 } and n n ri ri − λpi 2 γ = (αi + (ki − 1)αi ) − k , ki = , k = i=1 . 2 2 σ /2 σ /2 i=1 Clearly Equation 13.51 is the transformed version of Equation 13.49. We choose g(X ) =

1 Jν (c|X |) , NR (c|X |)ν

(13.52)

where Jν is the Bessel function for order ν, with ν = (n−2) and NR is a 2 normalizing constant such that B(R ) g(X )dX = 1. To solve the problem (Eq. 13.51) with g given by Equation 13.52, we need the following two theorems. Proofs of them may be found in Ref. 27.

THEOREM 13.14 Suppose x = (r, η) and y = (r , ξ ) are in R2 where η and ξ are angular parts of x and y, respectively. Then J0 (c|x − y|)eikξ dξ = 2πJk (cr)Jk (cr )eikη . S1

377


THEOREM 13.15 Suppose x = (r, η) and y = (r , ξ ) are in Rn , where where η and ξ are angular parts of x and y, respectively, and ν = (n−2) 2 . Then

23ν+1 Jν (c|x − y|) s S (ξ )dξ = Δ (ν, cr)Δn (ν, cr )Sks (η), ν k ν−1 n n−1 (c|x − y|) π S π ν where Δm (ν, r) = 2r Jν+m (r). We consider here the case n ≥ 3. The case n = 2 will be similar and simpler. We denote Hl as the space of degree l spherical harmonics on the n-sphere. We look for a solution of Equation 13.51 of the form u(x, τ ) =

∞ h(N ,p)

TNl (τ )RNl (r)SNl (η),

(13.53)

N =0 l=1

where x = (x1 , . . . , xn ) = (r, η). Then the integral term of Equation 13.51 becomes (with Y = (Y1 , . . . , Yn ) = (r , ξ )) with the use of Theorem 13.15 λ g(x − Y )u(Y )dY B(R ) λ Jν (c|x − Y |) u(Y )dY = NR B(R ) (c|x − Y |)ν ∞ h(N ,p) Jν (c|x − Y |) λ R p+1 r dr T (τ )RNl (r )SNl (ξ )dξ = ν Nl NR N =0 0 S n−1 (c|x − Y |) l=1

=

λ NR

∞ h(N ,p) R

N =0 l=1

0

r p+1 dr

23ν+1 ΔN (ν, cr)ΔN (ν, cr )TNl (τ )RNl (r )SNl (η) π ν−1

∞ h(N ,p) 23ν+1 λ ΔN (ν, cr)TNl (τ ) = π ν−1 NR N =0 l=1 R ΔN (ν, cr )RNl (r )r p+1 dr SNl (η). 0

Therefore, Equation 13.15 becomes −

∞ h(N ,p) N =0 l=1

TNl (τ )RNl (r)SNl (η)

+γ

∞ h(N ,p) N =0 l=1

TNl (τ )RNl (r)SNl (η)

378


+

∞ h(N ,p)

TNl (τ )r 1−n

N =0 l=1

−

∞ h(N ,p)

∂ ∂r

TNl (τ )RNl (r)

N =0 l=1

r n−1

∂RNl (r) l SN (η) ∂r

N (N + n − 2) l SN (η) r2

∞ h(N ,p) 23ν+1 λ ΔN (ν, cr)TNl (τ ) + π ν−1 NR N =0 l=1 R

× 0

ΔN (ν, cr )RNl (r )r p+1 dr SNl (η) = 0.

Since SNl (η) are linearly independent comparing the coefficients, we have the following equations for N = 0, 1, . . . and l = 1, . . . , h(N , p).

∂ ∂RNl (r) TNl (τ )RNl (r) = γ TNl (τ )RNl (r) + TNl (τ )r 1−n r n−1 ∂r ∂r − TNl (τ )RNl (r)

N (N + n − 2) r2

23ν+1 λ + ν−1 ΔN (ν, cr)TNl (τ ) π NR

R

ΔN (ν, cr )RNl (r )r p+1 dr .

0

Therefore, we have the following equations TNl (τ ) = TNl (τ ),

and γ RNl (r) + r 1−n

N (N + n − 2) ∂RNl (r) r n−1 − RNl (r) ∂r r2

∂ ∂r

+ ζ ΔN (ν, cr)I = RNl (r), where is a constant, ζ =

(13.54)

23ν+1 λ π ν−1 NR

I=

R

(13.55)

and

ΔN (ν, cr )RNl (r )r p+1 dr .

0

Initial values for TNl (τ ) and the boundary values of R(r) are obtained from the given problem. Solution of (13.54) is given by TNl (τ ) = TNl (0)eτ .


379

Solution of Equation 13.55 can be obtained by standard techniques such as homotopy perturbation method [28,29]. Here we give an outline of that. Observe that Equation 13.55 can be rewritten as

N (N + n − 2) ∂ 2 RNl (r) (n − 1) ∂RNl (r) + γ− + − RNl (r) ∂r 2 r ∂r r2 R + ζ ΔN (ν, cr) ΔN (ν, cr )RNl (r )r p+1 dr = 0. (13.56) 0

By homotopy perturbation technique, we construct a homotopy H (v, p) =

∂ 2 v(r) ∂ 2 y0 (r) ∂ 2 y0 (r) − + p ∂r 2 ∂r 2 ∂r 2 N (N + n − 2) (n − 1) ∂v(r) −p ( + − γ )v(r) − 2 r r ∂r R −ζ ΔN (ν, cr) (13.57) ΔN (ν, cr )v(r )r p+1 dr = 0, 0

where y0 (r) is the initial approximation. According to homotopy perturbation theory, we can first use the embedding parameter p as a small parameter, and assume that the solution of Equation 13.57 can be written as a power series in p. That is v(r) = v0 (r) + pv1 (r) + p2 v2 (r) + · · · .

(13.58)

Setting p = 1, we can get the solution for Equation 13.56 as RNl (r) = v0 (r) + v1 (r) + v2 (r) + · · · .

(13.59)

Substituting Equation 13.58 in Equation 13.57 and equating the coefficients of like powers of p, we obtain ∂ 2 v0 (r) ∂ 2 y0 (r) − = 0, (13.60) ∂r 2 ∂r 2 ∂ 2 v1 (r) ∂ 2 y0 (r) N (N + n − 2) (n − 1) ∂v0 (r) p1 : + − [( + − γ )v0 (r) − ∂r 2 ∂r 2 r2 r ∂r R − ζ ΔN (ν, cr) ΔN (ν, cr )v0 (r )r p+1 dr ] = 0, (13.61)

p0 :

0

∂ 2 vk (r) N (N + n − 2) (n − 1) ∂vk−1 (r) − [( + − γ )vk−1 (r) − pk : 2 2 ∂r r r ∂r R − ζ ΔN (ν, cr) ΔN (ν, cr )vk−1 (r )r p+1 dr ] = 0, k ≥ 2. (13.62) 0

380


Then starting with an initial approximation y0 (r) and solving successively the above equations, we can find vk (r) for k = 0, 1, 2, . . . . Therefore we can k (r) = get the k-th approximation of the exact solution (Eq. 13.59) as RNl v0 (r) + v1 (r) + · · · + vk−1 (r). Observe that according to homotopy perturbation k theory lim RNl (r) = RNl (r). k→∞

REFERENCES 1. Merton RC. Continuous-time finance. Wiley-Blackwell, UK; 1992. 2. Salas M. Parabolic Problems Arising in Financial Mathematics and Semiconductor Physics. PhD Dissertation, New Mexico State University, Las Cruces, NM, August 2010. 3. Adams RA, editor. Sobolev spaces. Academic Press, Netherlands; 1975. 4. Krylov NV. Lectures on elliptic and parabolic equations in Hölder spaces. Volume 12, Graduate studies in mathematics. American Mathematical Society, Providence, Rhode Island; 1996. 5. Wang C, Wu Z, Yin J. Elliptic and parabolic equations. World Scientific Publishing, Singapore; 2006. 6. Amster P, Averbuj C, De Napoli P, Mariani MC. A parabolic problem arising on financial mathematics. Nonlinear Anal R World Appl 2010;11:759–763. 7. Florescu I, Mariani MC. Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market. Electron J Differ Equat 2010;2010(62):1102010. 8. Black F, Scholes M. The valuation of options and corporate liability. J Polit Econ 1973;81:637–654. 9. Duffie D. Dynamic asset pricing theory. 3rd ed. Princeton University Press, Princeton, New Jersey; 2001. 10. Hull JC. Options, futures, and other derivatives. 7th ed. Prentice Hall, New Jersey; 2008. 11. Ikeda N. Stochastic differential equations and diffusion processes. 2nd revised ed. North Holland, Maryland Heights, Missouri; 1989. 12. Jarrow RA. Modelling fixed income securities and interest rate options. 2nd ed. Stanford Economics and Finance, California; 2002. 13. Heston SL. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 1993;6(2):327–343. 14. Avellaneda M, Zhu Y. Risk neutral stochastic volatility model. Int J Theor Appl Finance 1998;1(2):289–310. 15. Berestycki H, Busca J, Florent I. Computing the implied volatility in stochastic volatility models. Commu Pure and Appl Math 2004;57(10):1352 –1373. 16. Andersen L, Andreasen J. Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing. Rev Deriv Res 2000;4:231–262. 17. Cont R, Tankov P. Financial modelling with jumps processes. CRC Financial mathematics series. Chapman & Hall, Boca Raton, Florida; 2003.

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18. Florescu I. Stochastic volatility stock price: approximation and valuation using a recombining tree. sharp estimation of the almost sure lyapunov exponent estimation for the anderson model in continuous space. PhD thesis, Purdue University, West Lafayette, IN, December 2004. 19. Florescu I, Viens F. Stochastic volatility: option pricing using a multinomial recombining tree. Appl Math Finance 2008;15(2):151–181. 20. Friedman A. Partial differential equations of parabolic type. Prentice Hall, New Jersey; 1964. 21. Zheng S. Nonlinear evolution equations. Chapman & Hall, Boca Raton, Florida; 2004. 22. Chadam J, Yin HM. An iteration procedure for a class of integrodifferential equations of parabolic type. J Integr Equat 1989;2(1):31–47. 23. Geman H. Pure jump levy processes for asset pricing modeling. J Bank Finance 2002;26:1297–1316. 24. Ladyzenskaja OA, Solonikov VA, Ural’ceva NN. Linear and quasilinear equations of parabolic type. Am Math Soc 1964;23. 25. Evans LC. Partial differential equations. Volume 19, Graduate studies in mathematics. American Mathematical Society, Providence, Rhode Island; 1998. 26. Yin HM. A uniqueness theorem for a class of non-classical parabolic equations. Appl Anal 1989;34:67–78. 27. SenGupta I. Differential operator related to the generalized superradiance integral equation. J Math Anal Appl 2010;369:101–111. 28. Dehghan M, Shakeri F. Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method. Prog Electromagn Res 2008;78:361–376. 29. He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–262.

Chapter

Fourteen

Existence of Solutions for Financial Models with Transaction Costs and Stochastic Volatility MARIA C. MARIANI Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX

EMMANUEL K. NCHEUGUIM Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM

I N D R A N I L S E N G U P TA Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX

14.1 Model with Transaction Costs In a complete financial market without transaction costs, the celebrated Black–Scholes model [1] provides not only a rational option pricing formula, but also a hedging portfolio that replicates the contingent claim. In the Black–Scholes analysis, it is assumed that hedging takes place continuously, Handbook of Modeling High-Frequency Data in Finance, First Edition. Edited by Frederi G. Viens, Maria C. Mariani, and Ionut¸ Florescu. © 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

383

384

CHAPTER 14 Existence of Solutions for Financial Models

and therefore, in a market with proportional transaction costs, it tends to be infinitely expensive. So the requirement of replicating the value of the option continuously has to be relaxed. The first model in that direction was initiated by Leland (1985)[2]. He assumes that the portfolio is rebalanced at discrete time δt fixed and transaction costs are proportional to the value of the underlying; that is, the costs incurred at each step is κ|ν|S, where ν is the number of shares of the underlying bought (ν > 0) or sold (ν < 0) at price S and κ is a constant depending on individual investors. Leland derived an option price formula which is the same as the Black–Scholes formula for European calls and puts with an adjusted volatility σˆ = σ 1 +

2 κ √ π σ δt

1/2 .

Following Leland’s idea, Hoggard et al. [3] derive a nonlinear PDE (partial differential equation) for the option price value in the presence of transaction costs. We outline the steps used in the next section.

14.1.1 OPTION PRICE VALUATION IN THE GEOMETRIC BROWNIAN MOTION CASE WITH TRANSACTION COSTS Let C(S, t) be the value of the option and be the value of the hedge portfolio. The asset follows a geometric Brownian motion. Using discrete time, we assume the underlying asset follows the process √ δS = μSδt + σ S δt,

(14.1)

where is drawn from a standard normal distribution, μ is a measure of the average rate of growth of the asset price also known as the drift, σ is a measure of the fluctuation (risk) in the asset prices and corresponds to the diffusion coefficient. Then the change in the value of the portfolio over the timestep δt is given by δ = σ S

√ ∂C 1 2 2 ∂ 2C 2 ∂C ∂C φ + μS − δt + σ S + − μ S δt − κS|ν| ∂S 2 ∂S 2 ∂S ∂t

We consider the delta-hedging strategy. Specifically, let the quantity of asset held short at time t, Δ = ∂C ∂S (S, t). The timestep is assumed to be small, thus the number of assets traded after a time δt is ν=

∂C ∂ 2C ∂C ∂ 2C (S + δS, t + δt) − (S, t) = δS 2 + δt + ··· ∂S ∂S ∂S ∂t∂S

385

14.1 Model with Transaction Costs

√ Since δS = σ S δt + O(δt), keeping only he leading term yields ν

√ ∂ 2C σ S δt. 2 ∂S

Thus, the expected transaction cost over a timestep is 2 √ 2 2 ∂ C E[κS|ν|] = κσ S 2 δt, π ∂S √ where 2/π is the expected value of ||. Therefore, the expected change in the value of the portfolio is 2 ∂ C ∂C 1 2 2 ∂ 2 C 2 2 δt. E(δ) = − κσ S σ S ∂t 2 ∂S 2 πδt ∂S 2 If the portfolio is a hedging portfolio standard no arbitrage arguments imply that the portfolio will earn the riskfree interest rate r, and ∂C E(δ) = r C − S δt. ∂S Hence, Hoggard, Whalley, and Wilmott derive the model for option pricing with transaction costs as ∂C ∂ 2C 1 ∂C + σ 2 S 2 2 + rS − rC − κσ S 2 ∂t 2 ∂ S ∂S

2 π δt

2 ∂ C ∂S 2 = 0, (S, T ) ∈ (0, ∞) × (0, T ) (14.2)

with the terminal condition C(S, T ) = max(S − E, 0), S ∈ (0, ∞)

(14.3)

for European call options with strike price E, and a suitable terminal condition for European puts. We note that Equation 14.2 contains the usual Black–Scholes terms with an additional nonlinear term that models the presence of transaction costs in the model. A related work in quantum mechanics has been done in Refs. 4,5. We review some background in functional analysis useful for us in Section 14.2. In Section 14.3, we study the existence of solutions for problem (Eqs. 14.2 and 14.3) and finally in Sections 14.4 and 14.5, we extend it to the case of stochastic volatility. We prove the existence of classical solutions for the model which includes transaction cost and stochastic volatility. Our main results regarding the existence of classical solutions are given in Theorems 14.25 and 14.26.

386


14.2 Review of Functional Analysis 14.2.1 LP SPACES Let ∈ IRn be an open set and p ≥ 1 be a real number.

DEFINITION 14.1 The space LP ( ) represents the class of all measurable functions on such that 1/p p |u(x)| dx < ∞. ||u||LP ( ) =

Lp ( ) is a Banach space when endowed with the norm ||u||Lp ( ) .

DEFINITION 14.2 p

The space Lloc ( ) represents the class of all measurable functions on such that |u(x)|p dx < ∞ K

for all compact subset K of .

REMARK 14.3 Lp ( ) ⊂ L1loc ( ) for all 1 ≤ p < ∞.

14.2.2 WEAK DERIVATIVES AND SOBOLEV SPACES

DEFINITION 14.4 Suppose u and v ∈ L1loc ( ), and α is a multiindex. v is said to the αth-weak or distributional partial derivative of u, denoted Dα u = v, if uDα φdx = (−1)|α| vφdx

for any test function φ ∈ C ∞ ( )c . Here Cc∞ ( ) represents the space of infinitely differentiable functions on with compact support.

387

14.2 Review of Functional Analysis

Note that the weak partial derivative of a function u, when it exits, is unique up to a set of measure zero. Weak derivatives of functions are not always functions and Sobolev spaces are particular classes of Lp functions, whose derivatives are also Lp functions.

DEFINITION 14.5 The Sobolev space H m ( ) = {u ∈ L2 ( ) : Dα u ∈ L2 ( ), for any multiindex α with |α| ≤ m}, where the derivatives are taken in the weak sense. H m ( ) is a Hilbert space when endowed with the inner product (Dα u, Dα v)L2 ( ) . (u, v)H m ( ) = |α|≤m

The space H01 ( ) represents the closure of Cc∞ ( ) in H 1 ( ) and it is shown that H01 ( ) = {u ∈ H 1 such that u = 0 on ∂ }. The space H −1 ( ) denotes the topological dual of H01 ( ) and for f ∈ H −1 ( ), ||f ||H −1 ( ) = sup{< f , u > such that ||u||H 1 ( ) ≤ 1 for all 0 u ∈ H01 ( }. Here denotes the pairing between H −1 ( ) and H01 ( ).

REMARK 14.6 We have H01 ( ) ⊂ H 1 ( ) ⊂ L2 ( ) = H 0 ( ) ⊂ H −1 ( ) and the embedding of H01 ( ) in L2 ( ) is compact.

14.2.3 SPACES INVOLVING TIME Let X be a Banach space and T be a nonnegative integer. The space L2 (0, T ; X ) consists of all measurable functions u from (0, T ) to X with ||u||L2 (0,T ;X ) :=

0

T

1/2 ||u(t)||2X dt

< ∞.

L2 (0, T ; X ) is a Banach space when endowed with the norm ||u||L2 (0,T ;X ) .

388


The space C([0, T ]; X ) consists of all continuous functions u : [0, T ] −→ X with ||u||C ([0,T ];X ) := max ||u(t)||X < ∞. 0≤t≤T

C([0, T ]; X ) is a Banach space when endowed with the norm ||u||C ([0,T ];X ) .

THEOREM 14.7

(See Ref. 6 Theorem 3 in Section 5.9.2)

If u ∈ L2 (0, T ; H01 (BR )) and

∂u ∂t

∈ L2 (0, T ; H −1 (BR )), then

(i) u ∈ C([0, T ]; L2 (BR )), (ii) the mapping t → ||u(t)||2L2 (B ) is absolutely continuous with R

d ||u(t)||2L2 (B ) = 2 R dt

BR

∂u udt for a.e 0 ≤ t ≤ T . ∂t

(14.4)

REMARK 14.8 2 −1 For u ∈ L2 (0, T ; H01 (BR )) and ∂u ∂t ∈ L (0, T ; H (BR )), u(0) is under1 2 stood in the sense of the embedding L (0, T ; H0 (BR )) → C([0, T ]; X ).

¨ 14.2.4 HOLDER SPACES Next, we discuss spaces with classical derivatives, known as Hölder spaces. We will follow the notation and definitions given in the books [7] and [8]. We k define Cloc ( ) to be the set of all real-valued functions u = u(x) with continuous classical derivatives Dα u in , where 0 ≤ |α| ≤ k. Next, we set |u|0; = [u]0; = sup |u| ,

[u]k;

= max Dα u0; . |α|=k

If the seminorm |u(x) − u(y)| |x − y|δ x,y∈

[u]δ; = sup x=y

389

14.2 Review of Functional Analysis

is finite, then we say the real-valued function u is Hölder continuous in with exponent δ. For a k-times differentiable function, we will set

[u]k+δ; = max Dα u δ; . |α|=k

DEFINITION 14.9 k The space C k ( ) is the set of all functions u ∈ Cloc ( ) such that the norm k [u]j; |u|k; = j=0

is finite. With this norm, it can be shown that C k ( ) is a Banach space.

DEFINITION 14.10 The Hölder space C k+δ ( ) is the set of all functions u ∈ C k ( ) such that the norm |u|k+δ; = |u|k; + [u]k+δ; is finite. With this norm, it can be shown that C k+δ ( ) is a Banach space. For any two points P1 = (x1 , t1 ), P2 = (x2 , y2 ) ∈ QT = × (0, T ), we define the parabolic distance between them as

1/2 d(P1 , P2 ) = |x1 − x2 |2 + |t1 − t2 | . For a real-valued function u = u(x, t) on QT , let us define the seminorm [u]δ,δ/2;QT =

sup

P1 ,P2 ∈QT P1 =P2

|u(x1 , t1 ) − u(x2 , t2 )| . d δ (P1 , P2 )

If this seminorm is finite for some u, then we say u is Hölder continuous with exponent δ. The maximum norm of u is given by |u|0;QT = sup |u(x, t)|. (x,t)∈QT

390


DEFINITION 14.11 The space C δ,δ/2 Q T is the set of all functions u ∈ QT such that the norm |u|δ,δ/2;QT = |u|0;QT + [u]δ,δ/2;QT is finite. Furthermore, we define C 2k+δ,k+δ/2 Q T = {u : Dα ∂tρ u ∈ C δ,δ/2 Q T , 0 ≤ |α| + 2ρ ≤ 2k}. We define a seminorm on C 2k+δ,k+δ/2 Q T by [u]2k+δ,k+δ/2;QT =

[Dα ∂tρ u]δ,δ/2;QT ,

|α|+2ρ=2k

and a norm by

|u|2k+δ,k+δ/2;QT =

|Dα ∂tρ u|δ,δ/2;QT .

0≤|α|+2ρ≤2k

Using this norm, it can be shown that C 2k+δ,k+δ/2 Q T is a Banach space.

14.2.5 INEQUALITIES 14.2.5.1 Cauchy’s Inequality with ε. ab ≤ εa2 +

b2 4ε

(a, b > 0, ε > 0).

14.2.5.2 Gronwall’s Inequality (Integral Form). Let η(t) be a nonnegative, absolutely continuous function on [0, T ] which satisfies for a.e. t the differential inequality η (t) ≤ φ(t)η(t) + ψ(t), where φ(t) and ψ(t) are nonnegative, summable functions on [0, T ]. Then t t φ(s)ds η(t) ≤ e 0 η(0) + ψ(s)ds . 0

14.3 Solution of the Problem (14.2) and (14.3) in Sobolev Spaces

391

14.2.5.3 Hölder’s Inequality. Assume 1 ≤ p, q ≤ ∞, 1p + 1q = 1. Then if

u ∈ Lp ( ), v ∈ Lq ( ), we have |uv|dx ≤ ||u||Lp ( ) ||v||Lq ( ) .

14.2.5.4 Poincare’s inequality. Assume is a bounded open subset of IRn .

Suppose u ∈ H01 ( ). Then

||u||L2 ( ) ≤ C||Du||L2 ( ) where the constant C depends only on n and .

14.2.6 SCHAEFER’S FIXED POINT THEOREM X is a real Banach space.

DEFINITION 14.12 A nonlinear mapping A : X −→ X is said to be compact if and only if for ∞ each bounded sequence {uk }∞ k=1 , the sequence {A[uk ]}k=1 is precompact; ∞ that is there exists a subsequence {ukj }j=1 such that {A[ukj ]}∞ j=1 converges in X .

THEOREM 14.13

(Schaefer’s Fixed Point Theorem)

Suppose A : X −→ X is a continuous and compact mapping. Assume further that the set {u ∈ X such that u = λA[u] for some 0 ≤ λ ≤ 1} is bounded. Then A has a fixed point.

Schaefer’s fixed point Theorem will be useful to show the existence of solutions in a ball.

14.3 Solution of the Problem (14.2) and (14.3)

in Sobolev Spaces

14.3.1 SOLUTION OF THE PROBLEM POSED IN A BALL To begin the analysis of the problem, we set 1 x = log(S/E), t = T − τ/ σ 2 , and C = EV (X , τ ), 2

392


then Equation 14.2 becomes 2 ∂V ∂V ∂ 2V ∂V ∗ ∂ V , (x, τ ) ∈ IR × (0, T ∗ ) + (k − 1) + − kV = κ 2 − − ∂τ ∂x 2 ∂x ∂x ∂x (14.5) and the initial condition V (x, 0) = max(ex − 1, 0), x ∈ IR, where k = r/ 12 σ 2 , κ ∗ = κ π σ82 δt , and T ∗ = 12 σ 2 T . Next set V (x, τ ) = ex U (x, τ ), then Equation 14.5 yields 2 ∂U ∂U ∂ 2U ∂U ∗ ∂ U , (x, τ ) ∈ IR × (0, T ∗ ) + (k + 1) + =κ 2 + − ∂τ ∂x 2 ∂x ∂x ∂x (14.6) and the initial condition U (x, 0) = max(1 − e−x , 0). The previous discussion motivates us to consider the following problem that can accommodate cost structures that go beyond proportional transaction costs: ∂U ∂U ∂ 2U ∂U ∂ 2 U +α + = βF , , (x, t) ∈ IR × (0, T ) (14.7) − ∂t ∂x 2 ∂x ∂x ∂x 2 and U (x, 0) = U0 (x), x ∈ IR,

(14.8)

where α and β are nonnegative constants. The goal in this section is to show that the theoretical problem (Eqs. 14.7 and 14.8) has a strong solution where the derivatives are understood in the distribution sense. We assume that (H1) F : IR × IR −→ IR+ is continuous, (H2) F (p, q) ≤ |p| + |q|, ∂ 2 2 2 (IR), ∂x F (U , ∂U (H3) For U ∈ Hloc ∂x ) ∈ L (0, T ; Lloc (IR)). Moreover, with BR = {x ∈ R : |x| < R} if Uk → U in L2 (0, T ; H01 (BR )), then ∂Uk ∂ ∂ ∂U 2 2 ∂x F (U , ∂x ) → ∂x F (U , ∂x ) in L (0, T ; L (BR )) and Uk0 → U0 in L2 (BR ),


393

1 (H4) U0 ∈ Hloc (IR), (H5) β < 1.

Let BR = {x ∈ IR : |x| < R} be the open ball centered at the origin with radius R. Assume that U0 is suitable cut into bounded functions defined on BR and such that (H1)–(H5) are satisfied in BR × [0, T ]. Set w = ∂U ∂x and consider an analogous problem in BR × [0, T ] with zero Dirichlet condition on the lateral boundary. ∂w ∂ 2 w ∂ ∂w ∂w − + 2 +α = β F w, (14.9) (x, t) ∈ BR × (0, T ), ∂t ∂x ∂x ∂x ∂x w(x, 0) = w0 (x) w(x, t) = 0,

x ∈ BR ,

(14.10)

(x, t) ∈ ∂BR × [0, T ].

(14.11)

DEFINITION 14.14 w is said to be a weak solution of Equations 14.9–14.11 if ∂w w ∈ L2 (0, T ; H01 (BR )), ∈ L2 (0, T ; H −1 (BR )) ∂t ∂w ∂φ ∂φ ∂w φ+ + αw dx ∂t ∂x ∂x ∂x BR ∂w ∂φ F w, dx = −β ∂x ∂x BR

(14.12)

for all φ ∈ H01 (BR ).

THEOREM 14.15

(A-Priori Estimates)

If w is a weak solution of Equations 14.9–14.11, then there exits a positive constant C independent of w such that ∂w max w(t)||L2 (BR ) + w(t)L2 (0,T ;H 1 (BR )) + ∂t 2 0 0≤t≤T L (0,T ;H −1 (BR )) ≤ C||w0 ||L2 (BR ) .

(14.13)

Proof . Choose w(t) ∈ H01 (BR ) as the test function in Equation 14.12, obtain ∂w ∂w ∂w ∂w ∂w ∂w dx = −β F w, w+ + αw dx. ∂t ∂x ∂x ∂x ∂x ∂x BR BR

394


Then by Equation 14.4 2 ∂w 1 d ∂ 2 1 ∂w ∂w + α F w, ||w(t)||2L2 (B ) + (w )dx = −β dx R 2 dt ∂x L2 (BR ) 2 BR ∂x ∂x ∂x BR

From Equation 14.11 2 ∂w 1 d F w, ∂w ∂w dx. ≤β ||w(t)||2L2 (BR ) + 2 dt ∂x L2 (BR ) ∂x ∂x BR Using (H 2), obtain 2 2 ∂w ∂w ∂w 1 d 2 ||w(t)||L2 (BR ) + |w| + dx. ≤β 2 dt ∂x L2 (BR ) ∂x ∂x BR By Cauchy inequality with ε > 0, we have 2 ∂w 2 ∂w 1 d 2 dx ||w(t)||L2 (BR ) + ≤β 2 dt ∂x L2 (BR ) BR ∂x 2 ∂w 1 2 dx + |w| dx . +ε 4ε BR BR ∂x Since β < 1 and choosing ε 1 yields d ||w(t)||2L2 (B ) + C1 ||w||2H 1 (B ) ≤ C2 w2L2 (B ) R R 0 R dt

(14.14)

for a.e. 0 ≤ t ≤ T , and appropriate positive constants C1 and C2 . Next write η(t) := w(t)2L2 (B ) , then by Equation 14.14 R

η (t) ≤ C2 η(t), for a.e. 0 ≤ t ≤ T . The differential form of Gronwall inequality implies η(t) ≤ eC2 t η(0)for a.e. 0 ≤ t ≤ T . Since η(0) = w(0)2L2 (B ) = w0 2L2 (B ) , then R

R

w(t)2L2 (B ) ≤ eC2 t w0 2L2 (B ) . R

R

Hence max w(t)L2 (BR ) ≤ Cw0 2L2 (BR ) .

0≤t≤T

(14.15)

395


To bound the second term, we consider Equation 3.10, and integrate from 0 to T to obtain T T ||w||2H 1 (B ) dt ≤ C2 ||w||2L2 (B ) dt. C1 0

0

R

R

0

Use inequality (Eq. 14.15) to obtain wL2 (0,T ;H 1 (BR )) ≤ Cw0 L2 (BR ) . 0

(14.16)

Finally, to obtain a bound for the third term, fix v ∈ H01 (BR ) with vH 1 (BR ) ≤ 1. 0 By Equation 14.12, we have BR

∂w vdx + ∂t

BR

∂w ∂v ∂w ∂v ∂v dx = −β F w, + αw dx. ∂x ∂x ∂x ∂x ∂x BR

Thus, ∂w ∂v ∂v ∂w ∂v ∂w vdx ≤ + αw dx . F w, dx + β ∂x ∂x ∂x ∂x ∂x BR ∂t BR BR By Holder inequality, we have

BR

2 1/2 2 1/2 ∂w ∂v ∂w dx dx vdx ≤ ∂t BR ∂x BR ∂x 1/2 2 1/2 ∂v dx |w|2 dx +α BR BR ∂x 2 1/2 2 1/2 ∂v ∂w dx F w, dx + . ∂x ∂x BR

BR

Since vH 1 (BR ) ≤ 1, use (H 2) and Poincarre inequality to deduce 0

BR

∂w vdx ≤ Cw(t)H 1 (BR ) . 0 ∂t

So ∂w (t) ≤ Cw(t)H 1 (BR ) . ∂t −1 0 H (BR )

396


Therefore 0

T

T ∂w 2 (t) dt ≤ C w(t)2H 1 (B ) dt ∂t −1 0 R 0 H (BR ) = Cw2L2 (0,T ;H 1 (B )) . 0

Then Equation 14.16 implies ∂w ≤ Cw0 2L2 (B ) . ∂t 2 R −1 L (0,T ;H (BR ))

R

(14.17)

Before we prove the existence theorem in a ball, we state the following Lemma from the linear theory of parabolic PDEs. The Lemma follows directly from Ref. 6, Theorem 2, page 354.

LEMMA 14.16

(Energy Estimates)

Consider the problem ⎧ ∂u ⎪ ⎪ ⎨ − L(v) = f (x, t) in BR × (0, T ) ∂t u(x, 0) = u0 (x) on BR × {0} ⎪ ⎪ ⎩ u(x, t) = 0 on ∂BR × [0, T ]

(14.18)

with f ∈ L2 (0, T ; L2 (BR )) and u0 ∈ L2 (BR ). Then there exists a unique u ∈ L2 (0, T ; H01 (BR )) ∩ C([0, T ]; L2 (BR )) solution of Equation 14.18 that satisfies ∂u max u(t)||L2 (BR ) + uL2 (0,T ;H 1 (BR )) + ∂t 2 0 0≤t≤T L (0,T ;H −1 (BR ))

≤ C ||f ||L2 (0,T ;L2 (BR )) + ||u0 ||L2 (BR ) , (14.19) where C is a positive constant depending only on BR , T and the coefficients of the operator L.

We need another Lemma. This follows directly from Ref. 6, Theorem 5, page 360.

397


(Improved Regularity)

LEMMA 14.17

Consider the problem ⎧ ∂u ⎪ ⎪ ⎨ − L(u) = f (x, t) in BR × (0, T ) ∂t u(x, 0) = u0 (x) on BR × {0} ⎪ ⎪ ⎩ u(x, t) = 0 on ∂BR × [0, T ] with f ∈ L2 (0, T ; L2 (BR )) and u0 ∈ L2 (BR ). Then there exists a unique weak solution of the problem u ∈ L2 (0, T ; H01 (BR )) ∩ C([0, T ]; L2 (BR )), 2 −1 with ∂u ∂t ∈ L (0, T ; H (BR )). Moreover u ∈ L2 (0, T ; H 2 (BR )) ∩ L∞ (0, T ; H01 (BR )),

We also have the estimate ess sup0≤t≤T u(t)||H 1 (BR ) 0

∂u ∈ L2 (0, T ; L2 (BR )). ∂t

∂u + uL2 (0,T ;H 2 (BR )) + ∂t 2 L (0,T ;L2 (BR ))

≤ C ||f ||L2 (0,T ;L2 (BR )) + ||u0 ||L2 (BR )

(14.20)

where C is a positive constant depending only on BR , T , and the coefficients of the operator L.

THEOREM 14.18

(Existence Based on Schaefer’s

Fixed Point Theorem)

If (H1)–(H5) are satisfied, then system (Eq. 14.9–14.11) has a weak solution w ∈ L2 (0, T ; H01 (BR )) ∩ C([0, T ]; L2 (BR )).

∂ Proof . Given w ∈ L2 (0, T ; H01 (BR )), set fw (x, t) := β ∂x F w, ∂w ∂x . By (H 3), fw ∈ L2 (0, T ; L2 (BR )). By Lemma (14.16), there exists a unique v ∈ L2 (0, T ; H01 (BR )) ∩ C([0, T ]; L2 (BR )) solution of ⎧ ∂v ∂v ∂ 2 v ⎪ ⎪ = fw (x, t) in BR × (0, T ) ⎨ − 2 −α ∂t ∂x ∂x v(x, 0) = v0 (x) on BR × {0} ⎪ ⎪ ⎩ v(x, t) = 0 on ∂BR × [0, T ]

(14.21)

398


Define the mapping A : L2 (0, T ; H01 (BR )) −→ L2 (0, T ; H01 (BR )) w −→ A(w) = v where v is derived from w via Equation 14.21. Let us show that the mapping A is continuous and compact. ⇒ Continuity Let {wk }k ⊂ L2 (0, T ; H01 (BR )) be a sequence such that wk −→ w in L2 (0, T ; H01 (BR )).

(14.22)

By the improved regularity (Eq. 14.20), there exists a constant C , independent of {wk }k such that

sup ||vk ||L2 (0,T ;H 2 (BR )) ≤ C ||fwk ||L2 (0,T ;L2 (BR )) + ||wk0 ||L2 (BR ) k

for vk = A[wk ], k = 1, 2, . . .. But by (H3) as wk → w in L2 (0, T ; H01 (BR )), we must have fwk (x, t) → fw (x, t) in L2 (0, T ; L2 (BR )), wk0 → w0 in L2 (BR ), and thus, ||fwk (x, t)||L2 (0,T ;L2 (BR ) → ||fw (x, t)||L2 (0,T ;L2 (BR ) , ||wk0 ||L2 (BR ) → ||w0 ||L2 (BR ) . Therefore the sequences {fwk (x, t)||L2 (0,T ;L2 (BR ) }k and {||wk0 ||L2 (BR ) }k are bounded. Thus, the sequence {vk }k is bounded uniformly in L2 (0, T ; H 2 (BR )). Thus by Rellich’s theorem (see Ref. 9), there exists a subsequence {vkj }j and a function v ∈ L2 (0, T ; H01 (BR )) with vkj −→ v in L2 (0, T ; H01 (BR )) Therefore,

∂vkj

BR

∂vkj ∂φ ∂φ φ+ + αvkj ∂t ∂x ∂x ∂x

(14.23)

dx = BR

fwkj (x, t)φdx

for each φ ∈ H01 (BR ). Then using Equations 14.22 and 14.23, we see that BR

∂v ∂φ ∂φ ∂v φ+ + αv ∂t ∂x ∂x ∂x

dx =

fw (x, t)φdx BR

Thus, v = A[w]. Therefore, A[wk ] −→ A[w]in L2 (0, T ; H01 (BR )). ⇒ The compactness result follows from similar arguments. Finally, to apply Schaefer’s fixed point Theorem with X = L2 (0, T ; H01 (BR )), we need to show that the set {w ∈ L2 (0, T ; H01 (BR )) : w = λA[w] for some


399

0 ≤ λ ≤ 1} is bounded. This follows directly from the A-priori estimates (Theorem 14.15) with λ = 1.

REMARK 14.19 1 2 Theorem(14.18) shows that w = ∂U ∂x ∈ L (0, T ; H0 (BR )) solves problem 1 2 (Eqs. 14.9–14.11). Hence U ∈ L (0, T ; H0 (BR ) ∩ H 2 (BR )) and is a strong solution of problem (Eqs. 14.7–14.8) in the bounded domain BR × [0, T ] with zero Dirichlet condition on the lateral boundary of the domain.

14.3.2 CONSTRUCTION OF THE SOLUTION IN THE WHOLE REAL LINE The next step is to construct a solution of problem (Eqs. 14.9–14.11) in the whole real line. To do that, we approximate the real line by IR = ∪N ∈IN BN = limN −→∞ BN , where BN = {x ∈ IR : |x| < N } and w0 by a sequence of bounded function wN 0 defined in BN such that |wN 0 | ≤ |w0 | and wN 0 −→ w0 in L2loc (IR). For N ∈ IN, there exists wN ∈ L2 (0, T ; H01 (BN )) ∩ C([0, T ]; L2 (BN )) with ∂w∂tN ∈ L2 (0, T ; H −1 (BN )), weak solution of ∂w ∂ 2 w ∂w ∂ ∂w − + 2 +α = β F w, ∂t ∂x ∂x ∂x ∂x w(x, 0) = wN 0 (x) w(x, t) = 0,

(x, t) ∈ BN × (0, T )

x ∈ BN (x, t) ∈ ∂BN × [0, T ].

(14.24) (14.25) (14.26)

For any given ρ > 0, the following sequences are bounded uniformly in N > 2ρ:

{wN }N in L2 (0, T ; H01 (Bρ )) ∂wN in L2 (0, T ; H −1 (Bρ )) ∂t N

Since these spaces are compactly embedded in L2 (Bρ × (0, T )) therefore the sequence {wN }N is relatively compact in L2 (Bρ × (0, T )). Suppose we take ρ ∈ N. Then using the compactness just described, we can construct a sequence consisting of diagonal elements of the converging subsequences with each ρ ∈ N. Denoted this sequence also by {wN }N . Then there exists w ∈ L2 (0, T ; L2loc (IR)) ∩ 1 L2 (0, T ; Hloc (IR)) with w(x, 0) = w(x), so that wN −→ w a.e and in L2 (0, T ; L2loc (R))

400


and also 1 (R)). wN −→ w weakly in L2 (0, T ; Hloc

Since F is continuous, passing to the limit in Equation 14.10 yields that w is a weak solution of the problem (Eqs. 14.7–14.8) in R.

14.4 Model with Transaction Costs

and Stochastic Volatility

In the standard Black–Scholes model, a basic assumption is that the volatility is constant. Several models proposed in recent years, however, allowed the volatility to be nonconstant or a stochastic variable. For instance, in [10] a model with stochastic volatility is proposed. In this model the underlying security S follows, as in the Black–Scholes model, a stochastic process dS = μS dt + σ S dX1 , where X1 is a standard Brownian motion. Unlike the classical model, the variance v(t) = σ 2 (t) also follows a stochastic process given by dv = κ(θ − v(t)) dt + γ

√ v dX2 ,

where X2 is another standard Brownian motion. The correlation coefficient between X1 and X2 is denoted by ρ: Cov(dX1 , dX2 ) = ρ dt. This leads to a generalized Black–Scholes equation. A similar model has been considered in Refs 11 and 12. The process used is a modified Hull–White process [13,14], to contain a mean reverting term in the volatility process. The process above may also be viewed as a generalization of the SABR process used in practice [15]. For the present chapter, we follow the model used in Ref. 16. As observed in Ref. 16, when the volatility is stochastic we may consider the process dS = μSdt + σ SdX1 ,

(14.27)

dσ = ασ dt + βσ dX2 ,

(14.28)

where the two Brownian motions X1 and X2 are correlated with correlation coefficient ρ: E[dX1 dX2 ] = ρdt.

(14.29)

14.4 Model with Transaction Costs and Stochastic Volatility

401

When working with stochastic volatility models the option prices are not uniquely determined by the asset price. Indeed, restated in a simple language the fundamental theorem of asset pricing [17,18] says: Let M = the number of underlying traded assets in the model and let R = the number of random sources in the model. Then we have The model is arbitrage free if and only if M ≤ R. The model is complete if and only if M ≥ R. The model is complete and arbitrage free if and only if M = R. We avoided using the equivalent martingale measure in the previous statement, for an exact statement we refer the reader to the original work [17,18]. In the case of stochastic volatility models M = 1 and R = 2 thus the market is arbitrage free but not complete. This means that the derivative prices (options) are not uniquely determined by the traded asset price. The same conclusion has been reached when approximating the stochastic volatility process with a Markov chain [19]. The solution in the above citation was to fix the price of a certain derivative as a given asset and express all the other derivative prices in terms of the price of the given derivative. Next, we exemplify the choice by deriving the same nonlinear PDE in two different cases. One, when we pick the price of an option as given or two, when the volatility is a traded asset (the case of S&P500).

14.4.1 THE PDE DERIVATION WHEN WE TAKE A PARTICULAR OPTION AS A GIVEN ASSET As we have mentioned, with the two sources of randomness, we need two contracts to hedge the option: one being the underlying assets S as usual, and the second is a particular option V1 written on the asset S. We consider a portfolio that contains the option we are trying to price with value V (S, σ , t), a quantity Δ of the asset S and a quantity −Δ1 of the given option V1 (S, σ , t). We have = V − ΔS − Δ1 V1 .

(14.30)

We apply Itô formula to get the dynamics of V and V1 , then we substitute to obtain the change in value of the portfolio as 2 1 2 2 ∂ 2V 1 2 2 ∂ 2V ∂V 2 ∂ V dt + β σ + ρβσ S + σ S d = ∂t 2 ∂S 2 2 ∂σ 2 ∂S∂σ 2 1 2 2 ∂ 2 V1 1 2 2 ∂ 2 V1 ∂V1 2 ∂ V1 − Δ1 dt + β σ + ρβσ S + σ S ∂t 2 ∂S 2 2 ∂σ 2 ∂S∂σ ∂V ∂V1 ∂V1 ∂V dσ + − Δ1 − Δ dS + − Δ1 ∂S ∂S ∂σ ∂σ − κS|ν|

402


where κS|ν| represents the transaction cost for buying or selling quantity ν of the main asset S during the time step dt. We explain in the next subsection why the portfolio rebalancing is done only through the asset S and not the option V1 as well. To eliminate all randomness from the portfolio (terms containing dX1 and dX2 ), we must choose ∂V1 ∂V − Δ1 − Δ = 0, ∂S ∂S and

∂V1 ∂V − Δ1 ∂σ ∂σ

= 0.

This also eliminates the drift terms (containing μ and a) and the portfolio dynamics become: 2 1 2 2 ∂ 2V 1 2 2 ∂ 2V ∂V 2 ∂ V + β σ + ρβσ S + σ S dt d = ∂t 2 ∂S 2 2 ∂σ 2 ∂S∂σ 2 1 2 2 ∂ 2 V1 1 2 2 ∂ 2 V1 ∂V1 2 ∂ V1 + σ S + β σ + ρβσ S dt − Δ1 ∂t 2 ∂S 2 2 ∂σ 2 ∂S∂σ (14.31)

− κS|ν|

14.4.1.1 What is the Cost of Transaction. Note that we have two quantities that need to be rebalanced in our portfolio Δ and Δ1 . Changes in Δ represent quantities of the stock that need to be bought or sold and changes in Δ1 represent adjustments in the option. Let us denote Δ1 with λ and note we have ∂V ∂V1 −1 V (σ + δσ ) − V (σ ) ≈ λ = Δ1 = . (14.32) ∂σ ∂σ V1 (σ + δσ ) − V1 (σ )

ASSUMPTION 14.20 We assume that λ is constant in time. In other words, the variation of V with respect to σ (the vega) is at all times proportional to the variation of V1 with respect to σ and the constant of proportionality is λ. The proportionality constant does not change in time.

If both V and V1 represent values of two options which belong to the same option chain, the Assumption 14.20 is very realistic. This is the normal behavior


403

of the option chain observed in practice. When the perceived instantaneous volatility changes this updates the entire option chain values not only one option. In fact the option traders quote the price of options in volatility not in dollars. We also note an important consequence. Since Δ1 = λ is a constant in time, as a result we only balance through buying and selling the underlying asset S. This is a very desirable feature since transaction costs are much more tractable and low then when only trading the asset. Thus we only need to consider the costs associated with trading the asset. If the number of assets held short at time t is Δt =

∂V1 (S, σ , t) ∂V (S, σ , t) − λ (S, σ , t), ∂S ∂S

(14.33)

after a timestep δt and rehedging, the number of assets we hold short is Δt+δt =

∂V1 (S, σ , t) ∂V (S + δS, σ + δσ , t + δt) − λ (S + δS, σ + δσ , t + δt). ∂S ∂S

Since the timestep δt is assumed small, the changes in asset and the volatility are also small, and applying the Taylor’s formula to expand Δt+δt yields Δt+δt + δσ

∂V ∂V1 ∂ 2V ∂ 2V (S, σ , t) − λ (S, σ , t) + δt (S, σ , t) + δS 2 (S, σ , t) ∂S ∂S ∂t∂S ∂S ∂ 2V ∂ 2 V1 ∂ 2 V1 ∂ 2 V1 (S, σ , t) − λδσ (S, σ , t) − λδt (S, σ , t) − λδS (S, σ , t) + · · · ∂σ ∂S ∂t∂S ∂S 2 ∂σ ∂S

Since δS = σ SδX1 + 0(δt) and δσ = βσ δX2 + 0(δt), 2 ∂V ∂ 2 V1 ∂V1 ∂ V −λ 2 Δt+δt −λ + σ SδX1 ∂S ∂S ∂S 2 ∂S 2 2 ∂ V1 ∂ V + βσ δX2 −λ ∂σ ∂S ∂σ ∂S

(14.34)

Subtracting Equation 14.33 from Equation 14.34, we find that the number of assets traded during a timestep is therefore 2 2 ∂ 2 V1 ∂ 2 V1 ∂ V ∂ V ν = σ SδX1 −λ − λ 2 + βσ δX2 . (14.35) ∂S 2 ∂S ∂σ ∂S ∂σ ∂S We do not know beforehand how many shares will be traded. However, we can compute the expected number of this variable and hence the expected transaction cost. Since X1 and X2 are correlated Brownian motions, we consider Z1 and Z2 two independent normal variables with mean 0 and variance 1 and thus we may write the distribution of X1 , X2 as √ δX1 = Z1 δt √ √ δX2 = ρZ1 δt + 1 − ρ 2 Z2 δt.

404


Substituting these expressions in ν and denoting: Vˆ = V − λV1 √ ∂ 2 Vˆ √ ∂ 2 Vˆ ϕ = σ S δt + βσρ δt ∂S 2 ∂σ ∂S √ ∂ 2 Vˆ ψ = βσ 1 − ρ 2 δt ∂σ ∂S

(14.36)

We write the change in the number of shares over a time step δt as ν = ϕZ1 + ψZ2 and thus the expected transaction cost is E[κS|ν| | S] = κS E |ϕZ1 + ψZ2 | . Since Z1 and Z2 are independent normals we can calculate the expected value of the expression easily as 2 2 E[κS|ν| | S] = κS ϕ + ψ 2 π Finally using Equation 14.31 and the notations in Equation 14.36, we write 2V ˆ ∂ Vˆ 1 2 2 ∂ 2 Vˆ 1 2 2 ∂ 2 Vˆ ∂ E[δ | S, σ ] = + σ S δt + β σ + ρβσ 2 S ∂t 2 ∂S 2 2 ∂σ 2 ∂S∂σ 2 2 (14.37) − κS ϕ + ψ 2 π Recall that we chose Δ and Δ1 values to make the portfolio riskfree, thus following standard no arbitrage arguments, over the small interval δt the portfolio has expected value ˆ ∂ V δt E[δ | S, σ ] = rδt = r (V − ΔS − Δ1 V1 ) δt = r Vˆ − rS ∂S We equate the two expressions and we thus find an equation in Vˆ = V − λV1 1 ∂ Vˆ ∂ Vˆ ∂ 2 Vˆ ∂ 2 Vˆ 1 ∂ 2 Vˆ + σ 2 S 2 2 + β 2 σ 2 2 + ρβσ 2 S + rS − r Vˆ ∂t 2 ∂S 2 ∂σ ∂S∂σ ∂S 2 2 2V 2V 2V ˆ ˆ ∂ 2 Vˆ ˆ ∂ ∂ 2 ∂ − κS + β 2σ 2 σ 2S2 + 2ρβσ 2 S 2 = 0. π δt ∂S 2 ∂S ∂S∂σ ∂S∂σ (14.38)

405


14.4.2 THE PDE DERIVATION WHEN THE VOLATILITY IS A TRADED ASSET This case is only applicable when there exists a proxy for the stochastic volatility. Today in the financial markets there exists only one example of such a case: the Standard and Poor index (in fact, the exchange traded fund that replicates it SPX or SPY) and the volatility index VIX that is supposed to represent the associated implied volatility of an option with strike price exactly at the money and with maturity exactly one month from the current date. The VIX is calculated using an interpolating formula from the (out-of-money) options available and traded on the market, nevertheless it may be viewed as a proxy for the stochastic volatility in the model we propose here. In the future, it is possible that more volatility indices will be traded on the market, and we denote in what follows S as the equity price and with σ the matching VIX. We are also taking σ a perfect proxy for the stochastic volatility, an in depth analysis about the appropriateness of the choice is beyond the scope of the current chapter. We consider a portfolio that contains one option, with value V (S, σ , t), and quantities Δ and Δ1 of S and σ , respectively. That is, = V − ΔS − Δ1 σ.

(14.39)

Similarly with the previous case, we apply Itô’s formula to get the dynamics of V , then we substitute to obtain the change in value of the portfolio as 1 2 2 ∂ 2V 1 2 2 ∂ 2V ∂ 2V ∂V 2 + σ S + β σ + ρσ βS d = dt ∂t 2 ∂S 2 2 ∂σ 2 ∂S∂σ ∂V ∂V − Δ dS + − Δ1 dσ − κS|ν| + ∂S ∂σ where κS|ν| represents the transaction cost for buying or selling quantity ν of the main asset S during the time step dt. To eliminate all randomness from the portfolio (terms containing dX1 and dX2 ), we must choose ∂V − Δ = 0, ∂S and

∂V − Δ1 ∂σ

= 0.

This also eliminates the drift terms (containing μ and α) and the portfolio dynamics is 1 2 2 ∂ 2V ∂ 2V 1 2 2 ∂ 2V ∂V 2 dt − κS|ν|. + β σ + ρσ βS d = + σ S ∂t 2 ∂S 2 2 ∂σ 2 ∂S∂σ (14.40)

406


14.4.2.1 What is the Cost of Transaction. We use a simplified assumption here.

ASSUMPTION 14.21 The price of the option is a linear function in σ and the coefficient of σ in this function does not depend on time t.

In this case changes in Δ represent changes in quantities of the stock owned and changes in Δ1 represent changes in VIX owned. We have Δ1 =

∂V V (σ + δσ ) − V (σ ) ≈ . ∂σ δσ

(14.41)

An important consequence of the above assumption is Δ1 is constant in time. Thus, we only need to consider the costs associated with trading the asset. If the number of assets held short at time t is Δt =

∂V (S, σ , t), ∂S

(14.42)

after a timestep δt and rehedging, the number of assets we hold short is Δt+δt =

∂V (S + δS, σ + δσ , t + δt). ∂S

Since the timestep δt is assumed small, the changes in asset and the volatility are also small, and applying the Taylor’s formula to expand Δt+δt yields Δt+δt

∂V ∂ 2V ∂ 2V (S, σ , t) + δt (S, σ , t) + δS 2 (S, σ , t) ∂S ∂t∂S ∂S 2 ∂ V + δσ (S, σ , t) + · · · ∂σ ∂S

Since δS = σ SδX1 + 0(δt) and δσ = βσ δX2 + 0(δt), Δt+δt

∂V ∂ 2V ∂ 2V + σ SδX1 2 + βσ δX2 . ∂S ∂S ∂σ ∂S

(14.43)

Subtracting Equation 14.42 from 14.43, we found that the number of assets traded during a timestep is therefore ν = σ SδX1

∂ 2V ∂ 2V + βσ δX . 2 ∂S 2 ∂σ ∂S

(14.44)


407

We do not know beforehand how many shares will be traded, but we can compute the expected number and hence the expected transaction cost. Since X1 and X2 are correlated Brownian motions, we consider Z1 and Z2 two independent normal variables with mean 0 and variance 1 and thus, we may write the distribution of X1 , X2 as √ δX1 = Z1 δt √ √ δX2 = ρZ1 δt + 1 − ρ 2 Z2 δt. Substituting these expressions in ν and denoting: √ ∂ 2V √ ∂ 2V α1 = σ S δt + βσρ δt 2 ∂S ∂σ ∂S √ ∂ 2V β1 = βσ 1 − ρ 2 δt ∂σ ∂S

(14.45)

We write the change in the number of shares over a time step δt as ν = α1 Z1 + β1 Z2 and thus the expected transaction cost is E[κS|ν| | S] = κS E |α1 Z1 + β1 Z2 | . Since Z1 and Z2 are independent normals, we can calculate the expected value of the expression easily as 2 κS α12 + β12 . E[κS|ν| | S] = π

14.4.2.2 The PDE Under Transaction Costs and Stochastic Volatility. Following Equation 14.40 and using the notations in Equations 14.45, we write 1 2 2 ∂ 2V ∂ 2V 1 2 2 ∂ 2V ∂V 2 + β σ + ρσ βS E[δ | S, σ ] = + σ S δt ∂t 2 ∂S 2 2 ∂σ 2 ∂S∂σ 2 κS α12 + β12 . (14.46) − π Recall that we chose Δ and Δ1 values to make the portfolio riskfree, thus following standard no arbitrage arguments, over the small interval δt the portfolio has expected value ∂V δt. E[δ | S, σ ] = rδt = r (V − ΔS − Δ1 σ ) δt = rV − rS ∂S (14.47)

408


We equate Equations 14.46 and 14.47 and we thus find an equation in V . 1 ∂V 1 ∂ 2V ∂ 2V ∂ 2V ∂V + σ 2 S 2 2 + β 2 σ 2 2 + ρσ 2 βS + rS − rV ∂t 2 ∂S 2 ∂σ ∂S∂σ ∂S 2 2 2 2 2 ∂ V ∂ V ∂ 2V ∂ 2V 2 2 2 2 2 − κS σ S + 2ρβσ S 2 = 0. +β σ 2 πδt ∂S ∂S ∂S∂σ ∂S∂σ (14.48)

14.5 The Analysis of the Resulting Partial

Differential Equation

We observe that the Equations 14.38 and 14.48 are identical (with the unknown functions as Vˆ and V , respectively). Therefore it is sufficient to consider any one of them for a solution procedure. We next analyze the nonlinear PDE presented in Equation 14.48. We use the following change of variables S = ex ,

σ = ey ,

t = T − τ,

V (S, σ , t) = Ev(x, y, τ ).

Since S, σ ∈ [0, ∞) this gives x, y ∈ (−∞, ∞). Then Equation 14.48 is transformed to ∂v ∂v 1 2 ∂ 2 v ∂v ∂ 2v 1 2y ∂ 2 v ∂v − − − + e β + ρey β +r − rv 2 2 ∂τ 2 ∂x ∂x 2 ∂y ∂y ∂x∂y ∂x 2 2 2 2 ∂ 2v ∂ v ∂ v ∂v 2 ∂v ∂ 2 v y 2 e2y − + 2ρβe − = 0. + β −κ 2 2 π δt ∂x ∂x ∂x ∂x ∂x∂y ∂x∂y

that is, −

1 ∂ 2v 1 ∂ 2v 1 ∂v ∂ 2v ∂v + e2y 2 + β 2 2 + ρey β + (r − e2y ) ∂τ 2 ∂x 2 ∂σ ∂x∂σ 2 ∂x ∂v ∂ 2 v ∂ 2 v 1 2 ∂v , − rv = F y, , 2 , − β 2 ∂y ∂x ∂x ∂x∂y

(14.49)

where ∂v ∂ 2 v ∂ 2 v F y, , 2 , ∂x ∂x ∂x∂y 2 2 2 2 ∂ 2 v ∂v 2 ∂ v ∂v ∂ 2 v ∂ v y 2 + β e2y − + 2ρβe − . =κ 2 2 πδt ∂x ∂x ∂x ∂x ∂x∂y ∂x∂y

14.5 The Analysis of the Resulting Partial Differential Equation

409

LEMMA 14.22 There exists a constant C > 0, independent of variables in F such that 2 2 2 2 F y, ∂v , ∂ v , ∂ v ≤ Cey ∂v + ∂ v + ∂ v . ∂x ∂x 2 ∂x∂y ∂x ∂x 2 ∂x∂y

Proof . Clearly 2 2 F y, ∂v , ∂ v , ∂ v ∂x ∂x 2 ∂x∂y 2 2 2 2 2 2v ∂ ∂ ∂ 2 v v v ∂v ∂ ∂v = κ + β2 e2y − + 2ey ρβ − 2 2 π δt ∂x ∂x ∂x ∂x ∂x∂y ∂x∂y ∂ 2v 2 ∂ 2 v 2 ∂v ∂ 2 v 2 ≤ κ 1 − ρ ey 2 − + ρβ + β π δt ∂x ∂x ∂x∂σ ∂x∂σ 2 ∂v ∂ v 2 y ∂ 2 v . ≤ κ e 2 + ey + (|ρβ| + |β 1 − ρ 2 |) π δt ∂x ∂x ∂x∂σ Therefore ∃C > 0 such that 2 2 2 v ∂ 2 v ∂v ∂ y F y, , ≤ Ce ∂v + ∂ v + ∂ v . , 2 2 ∂x ∂x ∂x∂y ∂x ∂x ∂x∂y

LEMMA 14.23 Suppose |ρ| < 1. Then the Equation 14.49 is of parabolic type.

Proof . For (vi , vj ) ∈ R2 and θ > 0, we have (σ 2 − θ)vi vi + (β 2 − θ)vj vj + 2ρσβvi vj 2 ! 2 2 ρσβ ρ σ vj + vj2 β 2 (1 − 2 = σ 2 − θvi + √ )−θ σ −θ σ2 − θ

410


Therefore ρ2σ 2 2 ) − θ = β 2 (1 − ρ 2 ). lim β (1 − 2 θ →0 σ −θ Since |ρ| < 1 and β = 0, we have ρ2σ 2 lim β (1 − 2 ) − θ > 0. θ →0 σ −θ

2

Thus, ∃θ1 > 0 in the neighborhood of 0 such that β 2 (1 −

ρ2σ 2 ) − θ 1 > 0. σ 2 − θ1

Therefore with this θ1 , ∀(vi , vj ) ∈ R2 ,

1 2 2 σ σ vi vi + β 2 vj vj + 2ρσβvi vj > θ1 (|vi |2 + |vj |2 ). 2 Thus Equation 14.49 is parabolic.

14.5.1 SOLUTION OF EQUATION 14.22 In this section, we prove the existence of a classical solution for Equation 14.49. Let us denote Lu =

∂ 2u ∂u 1 ∂u 1 2y ∂ 2 u 1 2 ∂ 2 u 1 y + + ρe β e β + (r − e2y ) − β 2 − ru. 2 2 2 ∂x 2 ∂σ ∂x∂σ 2 ∂x 2 ∂y

We first consider the following initial-boundary value problem in a bounded parabolic domain QT = × (0, T ), T > 0, and is a bounded domain in R2 . ∂ 2u ∂ 2u −uτ + Lu = F y, ∂u , , in QT , ∂x ∂x 2 ∂x∂y (14.50) on , u(x, y, 0) = u0 (x, y) u(x, y, τ ) = g(x, y, τ ) on ∂ × (0, T ). Then, we try to extend our results to the corresponding initial-value problem in = R2 × (0, T ) the unbounded domain R2+1 T ∂2u ∂2u −uτ + Lu = F y, ∂u , , in R2+1 T , ∂x ∂x 2 ∂x∂y u(x, y, 0) = u0 (x, y) on R2 . Throughout this section, we impose the following assumptions:

(14.51)

411


A(1) The coefficients of L belong to the Hölder space C δ,δ/2 (Q T ); A(2) The value of |ρ| < 1; A(3) u0 (x, y) and g(x, y, t) belong to the Hölder spaces C 2+δ (R2 ) and C 2+δ,1+δ/2 (Q T ), respectively. A(4) The two consistency conditions g(x, y, 0) = u0 (x, y), gτ (x, y, 0) − Lu0 (x, y) = 0, are satisfied for all x ∈ ∂ . We shall prove the existence of a solution to Equation 14.49 using an iteration argument. We will do this by proving estimates based on a Green’s function. Afterwards, we will use a standard argument to show that our solution can be extended to give us a solution to the initial-value problem in R2+1 T . Let us define the function space C 1+1,0+1 (Q T ) to be the set of all u ∈ 2,1 C 1,0 (Q T ) ∩ W∞ (Q T ). We will say u ∈ C 1+1,0+1 (Q T ) is a strong solution to the parabolic initial-boundary value problem (Eq. 5.1) provided that u satisfies the parabolic equation almost everywhere in QT and the initial-boundary conditions in the classical sense. The following lemma follows immediately from Theorem 10.4.1 in Ref. 7.

LEMMA 14.24 There exists a unique solution ϕ ∈ C 2+δ,1+δ/2 Q T to the problem −uτ + Lu = 0 in QT , u(x, y, 0) = u0 (x, y) on , u(x, y, τ ) = g(x, y, τ ) on ∂ × (0, T ).

(14.52)

We state and prove our main theorem.

THEOREM 14.25 Let ϕ be defined as in Lemma 14.24. Then there exists a strong solution u ∈ C 1+1,0+1 (Q T ) to the problem ∂u ∂ 2 u ∂ 2 u in QT , −uτ + Lu = F y, , 2 , ∂x ∂x ∂x∂y on , u(x, y, 0) = u0 (x, y) u(x, y, τ ) = ϕ(x, y, τ ) = g(x, y, τ ) on ∂ × (0, T ).

(14.53)

412


Proof . First, we introduce a change of variables to transform our problem into one with a zero boundary condition. If we let v(x, y, τ ) = u(x, y, τ ) − ϕ(x, y, τ ), v0 (x, y) = u0 (x, y) − ϕ(x, y, 0) = 0, then v will satisfy the initial-boundary value problem ∂(v + ϕ) ∂ 2 (v + ϕ) ∂ 2 (v + ϕ) in QT , , −vτ + Lv = F y, , ∂x ∂x 2 ∂x∂y v(x, y, 0) = 0 on , (14.54) v(x, y, τ ) = 0 on ∂ × (0, T ). We further change variable τ = Aτ , where A is a constant which will be chosen later. By abuse of notation we denote ALv by Lv and AF by F. Then if T ∗ = TA , Equation 14.54 becomes ∂(v + ϕ) ∂ 2 (v + ϕ) ∂ 2 (v + ϕ) , −v + Lv = F y, , in QT ∗ , ∂x ∂x 2 ∂x∂y v(x, y, 0) = 0 on , (14.55) on ∂ × (0, T ∗ ). v(x, y, τ ) = 0 τ

If problem (Eq. 14.55) has a strong solution, then Equation 14.53 will have a strong solution since u = v + ϕ. We use an iteration procedure to construct the solution to Equation 14.55. Consider the problem ∂(α + ϕ) ∂ 2 (α + ϕ) ∂ 2 (α + ϕ) , , in QT ∗ , −βτ + Lβ = F y, ∂x ∂x 2 ∂x∂y β(x, y, 0) = 0 on , (14.56) on ∂ × (0, T ∗ ), β(x, y, τ ) = 0 where α ∈ C 2+δ,1+δ/2 (Q T ∗ ,U ) is arbitrary. We can show that ∂(α + ϕ) ∂ 2 (α + ϕ) ∂ 2 (α + ϕ) , , F y, ∈ C δ,δ/2 (Q T ∗ ). ∂x ∂x 2 ∂x∂y By Theorem 10.4.1 in Ref. 7, there exists a unique solution β ∈ C 2+δ,1+δ/2 (Q T ∗ ) to problem (Eq. 14.56). Using this result, we can now define vn ∈ C 2+δ,1+δ/2 (Q T ∗ ), n ≥ 1, to be the unique solution to the linearized problem n−1 2 n−1 2 n−1 −∂τ vn + Lvn = F y, ∂(v ∂x +ϕ) , ∂ (v∂x2 +ϕ) , ∂ (v∂x∂y+ϕ) in QT ∗ , vn (x, 0) = 0 on , vn (x, τ ) = 0 on ∂ × (0, T ∗ ), (14.57)


413

where v0 = v0 (x) = 0 ∈ C 2+δ,1+δ/2 (Q T ∗ ,U ). To prove the existence of a solution to problem (Eq. 14.55), we will show that this sequence converges. From Ref. 20, there exists a Green’s function G(x, y, τ , τ ) for problem (Eq. 14.57). For n ≥ 1, the solution vn can be written as vn (x, y, τ ) τ ∂(vn−1 + ϕ) ∂ 2 (vn−1 + ϕ) ∂ 2 (vn−1 + ϕ) G(x, y, z, w, τ , τ ) F w, , = , dzdw dτ ∂z ∂z 2 ∂z∂w 0

+ G(x, y, z, w, τ , 0)v0 (z, w) dzdw

τ

= 0

∂(vn−1 + ϕ) ∂ 2 (vn−1 + ϕ) ∂ 2 (vn−1 + ϕ) , dzdw dτ , G(x, y, z, w, τ , τ ) F w, , ∂z ∂z 2 ∂z∂w

because v0 (z, w) = 0. For convenience, we will write F n−1 (z, w, τ ) = 2 n−1 ∂(vn−1 + ϕ) + ϕ) ∂ 2 (vn−1 + ϕ) ∂ (v F w, (z, w, τ ), (z, w, τ (z, w, τ ), ) . ∂z ∂z 2 ∂z∂w

Now we take the first and second derivatives of vn (x, y, τ ) with respect to x and y. Foe convenience denote x1 = x and x2 = y. vxni (x, y, τ ) = vxni xj (x, y, τ ) =

τ

τ

0

0

Gxi (x, y, z, w, τ , τ ) F n−1 (z, w, τ ) dzdw dτ , Gxi xj (x, y, z, w, τ , τ ) F n−1 (z, w, τ ) dzdw dτ .

From Chapter IV.16 in Ref. 20, we have the estimates (x − z)2 + (y − w)2 , |G(x, y, z, w, τ , τ )| ≤ c1 (τ − τ )−1 exp −C2 τ − τ (14.58) 3 (x − z)2 + (y − w)2 |Gxi (x, y, z, w, τ , τ )| ≤ c1 (τ − τ )− 2 exp −C2 , τ − τ (14.59) (x − z)2 + (y − w)2 −2 |Gxi xj (x, y, z, w, τ , τ )| ≤ c1 (τ − τ ) exp −C2 , τ − τ (14.60)

414


where τ > τ and the constants c1 and C2 are independent of all parameters of G. If we combine everything together, we get vn (·, ·, τ )W∞2 ( ) = vn (·, ·, τ )L∞ ( ) +

2

vxni (·, ·, τ )L∞ ( ) +

i=1

≤

τ

0

2 i=1

0

i,j=1 0

vxni xj (·, ·, τ )L∞ ( )

i,j=1

G(·, ·, z, w, τ , τ )L∞ ( ) |F n−1 (z, w, τ )| dzdw dτ +

τ

2

2

τ

Gxi (·, ·, z, w, τ , τ )L∞ ( ) |F n−1 (z, w, τ )| dzdw dτ +

Gx x (·, ·, z, w, τ , τ )F n−1 (z, w, τ ) dzdw i j

L∞ ( )

dτ .

Our goal is to show that vn (·, ·, τ )W∞2 ( ) is uniformly bounded on the interval [0, T ∗ ], so that we can use the Arzelà –Ascoli theorem and a weak compactness argument (Theorem 3 of Appendix D in Ref. 6). We obtain the following estimates by using Lemma 14.22 n−1 + ϕ) ∂ 2 (vn−1 + ϕ) n−1 w ∂(v |F (z, w, τ )| ≤ Ce + ∂x ∂x 2 2 n−1 ∂ (v + ϕ) n−1 + , ≤ C3 v (·, ·, τ )W∞2 ( ) + CT ∗ , ∂x∂y where C3 is a constant dependent on space variable (boundedness of is crucial) and independent of T ∗ , whereas CT ∗ is a constant which depends on T ∗ (the constant comes from the upper estimate of ϕ in [0, T ∗ ]). By a direct calculation, we can easily see that (x − z)2 + (y − w)2 dzdw (τ − τ ) exp −C2 τ − τ

(x − z)2 + (y − w)2 −1 dzdw (τ − τ ) exp −C2 ≤ τ − τ R2 π = . C2

−1

We can see this by computing the integral in one dimension

∞

−∞

(τ −

1 τ )− 2

(x1 − y1 )2 exp −C2 τ − τ

dy1

415


=

−∞ − 21

= C2 =

∞

(τ − τ )

∞

τ − τ −ω2 e 1 dω1 , C2

2

−∞

π C2

− 21

e−ω1 dω1 ,

12

,

where we use

ω1 =

C2 (x1 − y1 ). τ − τ

The integral in R2 is a product of these one-dimensional integrals. This gives us the desired result. The Green’s function estimate −γ Gx x (·, ·, z, w, τ , τ ) dzdw (14.61) i j ∞ ≤ C4 (τ − τ ) ,

L ( )

where C4 is a constant independent of T ∗ , 0 < γ < 1 and τ > τ can be found in Lemma 2.1 of Ref. 21. Using all of our previous estimates and Equation 14.61, we obtain vn (·, ·, τ )W 2 ( ) ∞

= vn (·, ·, τ )L∞ ( ) +

2

vxni (·, ·, τ )L∞ ( ) +

i=1

τ

≤ 0

1

τ

+ C3 0

1/2

vxni xj (·, ·, τ )L∞ ( )

i,j=1

A + B(τ − τ )− 2 + D(τ − τ )−γ

= CT ∗ Aτ + 2Bτ

2

+D

τ 1−γ 1−γ

C3 vn−1 (·, ·, τ )W 2 ( ) + CT ∗ dτ ∞

1 A + B(τ − τ )− 2 + D(τ − τ )−γ vn−1 (·, ·, τ )W 2 ( ) dτ ∞

≤ C (T ∗ , γ ) + C

τ 0

1 A + B(τ − τ )− 2 + D(τ − τ )−γ vn−1 (·, ·, τ )W 2 ( ) dτ , ∞

where the constants A, B, D, and C are independent of T ∗ . The constant C(T ∗ , γ ) depends only on T ∗ and γ . Therefore, we have vn (·, ·, τ )W 2 ( ) ∞

≤ C (T ∗ , γ ) + C

τ 0

1 A + B(τ − τ )− 2 + D(τ − τ )−γ vn−1 (·, ·, τ )W 2 ( ) dτ . ∞

(14.62)

416


Observe that there exist an upper bound of the integral

1 A + B(τ − τ )− 2 + D(τ − τ )−γ dτ ,

τ

0

for τ ∈ [0, T ∗ ]. Choose A (where τ = At , as defined before) such that this upper bound is ε where |εC| < 1. This is possible as C does not depend on T ∗ . We observe from Equation 14.62 that v1 (·, ·, τ )W∞2 ( ) ≤ C(T ∗ , γ ), v2 (·, ·, τ )W∞2 ( ) ≤ C (T ∗ , γ ) + C

τ

0

1 A + B(τ − τ )− 2 + D(τ − τ )−γ v1 (·, ·, τ )W∞2 ( ) dτ

≤ C (T ∗ , γ ) + C (T ∗ , γ )C ε,

v3 (·, ·, τ )W∞2 ( ) ∗

τ

≤ C (T , γ ) + C 0

1 A + B(τ − τ )− 2 + D(τ − τ )−γ v2 (·, ·, τ )W∞2 ( ) dτ

≤ C (T ∗ , γ ) + C (C (T ∗ , γ ) + C (T ∗ , γ )C ε)ε = C (T ∗ , γ ) + C (T ∗ , γ )C ε + C (T ∗ , γ )C 2 ε2 .

Proceeding this way

vn (·, ·, τ )W∞2 ( ) ≤ C(T ∗ , γ ) 1 + Cε + · · · + C n−1 εn−1 . ∗

(T ,γ ) , where n = 0, 1, 2, . . .. Since |εC| < 1, we obtain vn (·, ·, τ )W∞2 ( ) ≤ C1−εC n Consequently v (·, ·, τ )W∞2 ( ) is uniformly bounded on the closed interval [0, T ∗ ]. Using this result along with Equation 5.6, we can easily show that vτn (·, ·, τ )L∞ ( ) is also uniformly bounded on [0, T ∗ ]. Since vn (·, ·, τ )W∞2 ( ) and vτn (·, ·, τ )L∞ ( ) are continuous functions of τ on the closed interval [0, T ∗ ], it follows that |vn |, |vxni |, |vxni xj | and |vtn | are uniformly bounded on Q T ∗ . Thus vn (·, ·, τ ) is equicontinuous in C(Q T ∗ ). By the Arzelà –Ascoli theorem, there exists a subsequence {vnk }∞ k=0 such that as k → ∞,

vnk → v ∈ C(Q T ∗ ) and vxnik → vxi ∈ C(Q T ∗ ) ,

417


where the convergence is uniform. Furthermore, by Theorem 3 in Appendix D of [6], vxnikxj → vxi xj ∈ L∞ (Q T ∗ ) and vτ k → vτ ∈ L∞ (Q T ∗ ), n

as k → ∞. Here, the convergence is in the weak sense. Therefore, vnk converges uniformly on the compact set Q T ∗ to a function v ∈ C 1+1,0+1 (Q T ∗ ). By a standard argument Ref. 22, we have that v satisfies the parabolic equation in Equation 14.55 almost everywhere and the initial-boundary conditions in the classical sense. Hence, v is a strong solution to problem (Eq. 14.55). Consequently, u is a strong solution to Equation 14.53. Now, we show that we can extend this solution to give us a classical solution on the unbounded domain RTd +1 = Rd × (0, T ).

THEOREM 14.26 There exists a classical solution u ∈ C 2,1 (R2+1 T ) to the problem ∂u ∂ 2 u ∂ 2 u −uτ + Lu = F y, , 2 , ∂x ∂x ∂x∂y u(x, y, 0) = u0 (x, y) such that the solution u(x, y, t) → g(x, y, t) as

in

R2+1 T

on R2

(14.63)

x 2 + y2 → ∞.

Proof . We approximate the domain R2 by a nondecreasing sequence { N }∞ N =1 of bounded smooth subdomains of . For simplicity, we will let N = B(0, N ) be the open ball in R2 centered at the origin with radius N . Also, we let VN = N × (0, T ). Using the previous theorem, we let uM ∈ C 2,1 (V M ) be a solution to the problem ∂ 2u ∂ 2u −uτ + Lu = F y, ∂u , , in VM , ∂x ∂x 2 ∂x∂y (14.64) u(x, y, 0) = u0 (x, y) on M , u(x, y, t) = g(x, y, t) on ∂ M × (0, T ). Since M ≥ 1 is arbitrary, we can use a standard diagonal argument to extract a subsequence that converges to a solution u to the problem on the whole unbounded space R2+1 T . Clearly, u(x, y, 0) = u0 (x, y) and u(x, y, t) → g(x, y, t) 2 2 as x + y → ∞.

418


REFERENCES 1. Black F, Scholes M. The pricing of options and corporate liabilities. J Polit Econ 1973;81:637–659. 2. Leland HE. Option pricing and replication with transaction costs. J Finance 1985;40:1283–301. 3. Hoggard T, Whalley AE, Wilmott P. Hedging option portfolios in the presence of transaction costs. Adv Fut Opt Res 1994;7: 21. 4. SenGupta I. Spectral analysis for a three-dimensional superradiance problem. J Math Anal Appl 2011;375:762–776. 5. SenGupta I. Differential operator related to the generalized superradiance integral equation. J Math Anal Appl 2010;369:101–111. 6. Evans LC. Partial differential equations. Graduate studies in Mathematics, American Mathematical Society 1998. 7. Krylov NV. Lectures on elliptic and parabolic equations in Hölder spaces. Volume 12 of Graduate studies in mathematics. American Mathematical Society, Providence, Rhode Island; 1996. 8. Wang C, Wu Z, Yin J. Elliptic and parabolic equations. World Scientific Publishing, Singapore; 2006. 9. Folland GB. Introduction to partial differential equations. 2nd ed. Princeton University Press, Princeton, New Jersey; 1995. 10. Heston SL. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 1993;6:327–344. 11. Mariani MC, Florescu I, Beccar Varela MP, Ncheuguim E. Long correlations and Levy models applied to the study of memory effects in high frequency (tick) data. Physica A 2009;388(8): 1659–1664. 12. Mariani MC, Florescu I, Beccar Varela MP, Ncheuguim E. Study of memory effects in international market indices. Physica A 2010;389(8):1653–1664. 13. Hull JC, White AD. The pricing of options on assets with stochastic volatilities. J Finance 1987;42(2):281–300. 14. Wiggins, J. B., Option values under stochastic volatility: Theory and empirical estimates, J Financ Econ 19(2),351–372, December 1987. 15. Hagan P, Kumar D, Lesniewski A., Woodward D. Managing smile risk. Wilmott Magazine, 2002. 16. Florescu I, Mariani MC. Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market. Electron J Differ Equat 2010;2010(62):1–10. 17. Harrison JM, Pliska SR. Martingales and stochastic integrals in the theory of continuous trading. Stoch Proc Appl 1981;11(3):215–260. 18. Delbaen F, Schachermayer W. A general version of the fundamental theorem of asset pricing. Math Ann 1994;300:463–520. 19. Florescu I, Viens F. Stochastic volatility: option pricing using a multinomial recombining tree. Appl Math Finance 2008;15(2):151–181.

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20. Ladyzenskaja OA, Solonikov VA, Ural’ceva NN. Linear and quasilinear equations of parabolic type. Volume 23. American Mathematical Society; 1964. 21. Yin HM. A uniqueness theorem for a class of non-classical parabolic equations. Appl Anal 1989;34:67–78. 22. Friedman A. Partial differential equations of parabolic type. Prentice Hall, New Jersey; 1964.

Index Abnormal price movements, 45 Abundant data, high-low frequency vs. fixed frequency with, 208–212 Acceptable band, of likelihood ratio test, 202 Accounting ratios, 58 Accounting variables, 54–55 Activity-monitoring task, 64 Adaboost, 48–49, 51, 69 Adaptive reinforcement learning, 65 After-event window, 32 size of, 33, 40 Agent-based models, 63–64 Algorithmic modeling, 48, 67 Algorithmic trading, 42, 63–66 Algorithm speed, 199–202 All-overlapping (AO) estimator, 267, 272, 280, 282 α levels, 35–37 window size and, 40 Alpha parameter, 121–122 α-stable Lévy processes, 125 Alternating decision trees (ADTs), 49–51 structure of, 50 Alternative backtest, 196 Alternative backtest result tables, 196–199 Analysts’ earnings forecast, 62 Analytical/simulation results, new, xi ANOVA, 37 Anselmo, Peter C., xiii, 235 Antipersistent activity, 148 Approximation method, for MMEs, 12

A-priori estimates, 393 AR(1) model, 281 Arbitrary trading rule, 44–45 ARMA models, 287 ARMA process, with GARCH errors, 181–182 AR(p) process, 128 Artificial intelligence approaches, 63 Arzelà –Ascoli theorem, 371, 374, 414, 416 Asset allocation, 286–290 Asset behavior, establishing, 135 Asset-price models, 347–348 Asset price process, approximating, 99–100 Asset pricing, fundamental theorem of, 401 Asset trading costs, 403 Asymptotically normal estimator, 224 Asymptotically unbiased Fourier estimator, 265, 266 Asymptotic distribution, of the likelihood ratio test statistic, 191 Asymptotic theory, 267 Asynchronous trading model, 265 Asynchronous trading, regular, 264 At-the-money SPX, 98. See also Standard and Poor Index (SPX) At-the-money SPX put options, 105 At-the-money strike, 112 calculating, 111 Augmented Dickey–Fuller (ADF) test, 128–129

Handbook of Modeling High-Frequency Data in Finance, First Edition. Edited by Frederi G. Viens, Maria C. Mariani, and Ionut¸ Florescu. © 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

421

422 Augmented log likelihood, 172 Autocorrelation, of GARCH filtering, 202 Autocorrelation function (ACF), 177, 221 for minute data, 202–203 Automated trading platforms, 235 Automated trading systems, 63–64, 68 Autoregressive conditional duration (ACD) model, 27–28 Autoregressive conditionally heteroskedastic (ARCH) models, 272 Average daily volume (ADV), 34 classification of equity based on, 45 Average estimator, 279 BAC data series, DFA and Hurst methods applied to, 155 Backtest, evaluating results of, 192 Backtest algorithm, 189 Backtest failure ratio, 192 Backtesting, 188–203 Backtest null hypothesis, 202 Backtest results, using GARCH, 204–205 Backtest result tables, 192–195, 199–200 Backtest variant, 195–196 Balanced capital structure, 59 Balanced scorecards (BSCs), 48, 52–53, 69. See also Board balanced scorecards (BSCs); BSC entries; Enterprise BSC; Executive BSC Ball solution, 391–399 Banach spaces, 349, 350, 351, 386, 387–388, 389 Bandwidth choices, 269 Barany, Ernest, xiii, 119, 327 Bartlett-type kernels, 261, 263 Base learner, 48 Bear Stearns crash, high-frequency data corresponding to, 121, 131–132 Bear Stearns crash week, high-frequency data from, 148–160 Beccar Varela, Maria Pia, xiii, 119, 327 Bernoulli LRT, 191. See also Likelihood ratio test (LRT) Bernoulli MLE, 190. See also Maximum likelihood estimation (MLE)

Index Bernoulli(p) distribution, 190 Bessel function, 9, 376 Bessel function of the third kind, modified, 166 Best practices, 51 Bias, 253–254 estimated, 258, 259 of the Fourier covariance estimator, 264–266 Bias-corrected estimator, 261 Bid/ask orders, 29 Bid-ask price behavior, 236 Bid-ask spreads, 228, 229, 236, 238–239, 240 Big values, asymptotic behavior for, 338 Binary prediction problems, 48 Black–Litterman model, 68 Black–Scholes analysis, 383–384 Black–Scholes equation, 352, 400 Black–Scholes formula, 114, 115 Black–Scholes model(s), 4, 6–7, 334 boundary condition for, 354–355 in financial mathematics, 352 with jumps, 375 option prices under, 219 volatility and, 400 Black–Scholes PDE, 348. See also Partial differential equation (PDE) methods Board balanced scorecards (BSCs), 51–52, 59. See also Balanced scorecards (BSCs) designing, 59 Board performance, quantifying, 52 Board strategy map, 59–60 Boosting, 47–74 adapting to finance problems, 68 applications of, 68–69 combining with decision tree learning, 49 as an interpretive tool, 67 Boundary value problem, 319, 320 Bounded parabolic domain, 352, 368 Bozdog, Dragos, xiii, 27, 97 Brownian motion, 78, 120, 220 BSC indicators, 52, 53. See also Balanced scorecards (BSCs) BSC management system, 51–52

Index Calendar time sampling, 9 Call options chains, constructed VIX using, 105–106 Cantor diagonal argument, 361–362. See also Standard diagonal argument Carrying capacity, 328 Cauchy sequence, 362 Cauchy’s inequality, 390, 394 Cauchy-stable distribution, 337 CBOE index calculation procedure, 110–113. See also Chicago Board Options Exchange (CBOE) Market Volatility Index (VIX) CBOE procedure, vs. quadrinomial tree method, 100–101 CBOE VIX indicator, 108 CBOE white paper, 98 CDO tranches, 76 Central limit theorem, 123, 124, 187 Central moments, 10, 12 CEO compensation, 53, 59, 60–62 Chadam–Yin method, 364 Characteristic function, 122, 123, 169, 338 Characteristic parameter, 337 Chicago Board Options Exchange (CBOE) Market Volatility Index (VIX), 97. See also CBOE entries calculation of, 98–99 Chronopoulou, Alexandra, xiii, 219 ‘‘Circuit breakers’’, 241 Citi data series, DFA and Hurst methods applied to, 155 City Group, Lévy flight parameter for, 341 Classical risk forecast, 163 Classical time series analysis, 177 Combined Stochastic and Dirichlet problem, 317 Comparative analysis, 239–241 Compensation committees, 53 Compensation policy, 59 Complex models, 23 Compustat North America dataset, 54 Conditional density function, 173 Conditional distribution, 29, 30 Conditional expected returns, 181 Conditional normal distribution, density of, 173 Conditional VaR, 188–189, 207. See also Value at risk (VaR)

423 Conditional variances, 203, 206, 208 of the GARCH(1,1) process, 180 Confidence intervals, for forecasts, 187–188 Consecutive trades, 129 Consensus indicators, 62 Constant coefficient case, 311 Constant default correlation, 79–81 Constant default correlation model, 76 Constant rebalanced portfolio technical analysis (CRP-TA) trading algorithm, 65–66 Constant variance, 181 Constant volatility, 353 Constructed indices, comparison of, 106–107 Constructed volatility index (VIX). See also Volatility index (VIX) comparing, 105–106 convergence of, 105 Contaminated returns, variance and covariance of, 257 Continuous integral operator, 367 Continuous semimartingales, 246, 253 Continuous-time long-memory stochastic volatility (LMSV) model, 220 Continuous-time stochastic modeling, 3 Continuous-time vintage, 78 Convergence-of-interests hypothesis, 54 Convex duality method, 296 Copula models, 77 Copulas, 75–76 CorpInterlock, 62, 63 Corporate governance, 53–54 of S&P500 companies, 54–60 Corporate governance best practices, 59 Corporate governance scorecards, 51–52 Corporate governance variables, 69 interpreting S&P500 representative ADTs with, 58–59 Corporate performance, predicting, 69 Correlation coefficient, 400 Correlation fluctuations impact on securitized structures, 75–95 products and models related to, 77–79 Cost structures, 392

424 Covariance(s) estimating, 244 forecasting, 280–285 Covariance function, 252 Covariance matrix, 170 Covariance stationarity, 177, 179, 181 Covariation-realized covariance estimator, 266 Covolatility function, 249 Covolatility measurement/forecasting, as a key issue in finance, 243 Cox, Ingersoll, Ross (CIR) square-root model, 257 cpVIX, 103 Crash imminence, precautions against, 121 Creamer, Germán, xiii, 47 Crisis detection, 131 Crisis-related equity behavior, 150 Cubic-type kernels, 261, 263 Cumulative abnormal return, 62, 63 Cumulative consumption process, 297, 305–306 Cumulative distribution curve, 346 Cumulative distribution function, 176 Current market volatility distribution, estimating, 115 Current weighting, 49 Customer perspective, 51 Cutting frequency, 258, 259 cVIX-1, 101, 102. See also Volatility index (VIX) cVIX-2, 101, 102 cVIX-b, 102, 103, 105 forecasting, 110 Daily-based forecasts, 210 Daily GARCH process, 215–216. See also Generalized autoregressive conditionally heteroskedastic (GARCH) methodology Daily portfolio, determination of, 286–287 Daily returns, 4, 14 Daily returns scenario, 215–216 Daily return/volatility, 211–212 Daily sampled indices, analysis of, 132–141 Daily VaR forecast, backtesting, 199–200. See also Value at risk (VaR)

Index Data for NIG and VG model estimation, 18 statistical behavior of, 345 Data analysis methods, 122–128 truncated Lévy flight, 122–125 Data-generating processes, 275 Data manipulations, avoiding, 257 Data-modeling approach, 48 Data preprocessing, for NIG and VG model estimation, 18 Datasets, stationary and nonstationary, 127 Data synchronization, 244 Dayanik–Karatzas theory, 312 d -dimensional hyperbolic distribution, 171 Default behavior, modeling, 77 Default correlation, 75–76. See also Constant default correlation copula models for, 77 high-frequency tranche price sensitivity to, 88–89 logistic transitional, 84–87 regime-switching, 81–84 across vintages, 93 Default correlation dynamics impact on high-frequency tranches, 87–92 impact on low-frequency tranches, 79–87 Default rates, 79 Default risk, 93 Defiltering, 182 Delta-hedging strategy, 384 Density, of the skewed t distribution, 170 Density function, Laplace transform of, 169 Density of GH distributions, 167–169 Derivative of a product, 328 Derivative security pricing, 348 Deterministic equation, 331 Detrended fluctuation analysis (DFA) method, 120, 121, 127, 130–131, 132, 138, 140. See also DFA entries results of, 141–145 Detrended fluctuation parameter, 121–122 DFA estimates, 150. See also Detrended fluctuation analysis (DFA) method DFA exponent values, 138

Index DFA parameters, 135 DFA regression plots, 137 Differential equations, 331–334 Diffusion coefficients, 384 measuring, 243–244 pathwise computing of, 251 Director compensation, 59. See also CEO compensation Direct reinforcement learning, 65 Dirichlet (DIR) kernel, 255, 261 Discrete time model, 220 DIS data series, DFA and Hurst methods applied to, 153 Disjoint union, 314, 317 Distributional partial derivative, 386 Distribution distortions, 93 Distribution family, choice of, 164 Diversification opportunities, indicating, 137–138 Dominated convergence theorem, 360 Dot-com bubble, 136 Double-auction market, 237 Double-auction prices, 238 Dow Jones data, analysis of, 141–147 Dow Jones Index components of, 145 Lévy flight parameter for, 342 memory effects pattern in, 145–147 Dow Jones Index data series DFA method applied to, 146 R/S analysis applied to, 149 Dow Jones Index data series components DFA method applied to, 146 R/S analysis applied to, 149 Dow Jones industrial average (DJIA), 128–129 Drift, 384 Drift (μ) parameter, 225 Drift terms, 402, 405 Duality approach, 300–305, 308–311 Duality gap, 318, 319 Dual problem value, 322 Dual value function, 303, 318, 321 quantities associated with, 309 Dynamic default correlation, 76 high-frequency tranche price sensitivity to, 89–92 Dynamics, in default correlation, 76 Dynkin’s formula, 314

425 EAFE index, 138. See also MSCI EAFE stock index Early market activity, 42 Earnings game, 60 Earnings prediction, 60–63 Earnings surprises, 62, 63 Econometric analysis, 33–34 Econometric models, quantitative evaluation of, 47 EEM index, 137–138, 139 analysis results for, 142 EEM index exponents, value range of, 132–135. See also Emerging Markets Index (EEM) EFA index, 137, 139. See also iShares MSCI EAFE Index (EFA) EFA index exponents, value range of, 132–135 Efficiency ratio, 58 Efficient price, noise dependent on, 282 Eigenfunction stochastic volatility (ESV) models, 273–274 Elasticity degree assumption, 28–29 Electronic financial markets, development of, 67 Electronic trading, 28 Elliptic operator, 355, 363, 367 Emerging markets, diversifying into, 141 Emerging Markets Index (EEM), 138. See also EEM entries Empirical CDF, quantile–quantile plots of, 136 Empirical distribution, 164 of losses, 187 Empirical distribution function, 129 Energy estimates, 396 Enterprise BSC, 55. See also Balanced scorecards (BSCs) Entrenchment hypothesis, 54 Environment, carrying capacity of, 328 Epps effect, 244, 264, 269 Equities, classifying, 34 Equity behavior, 45 crisis-related, 150 Equity classes expected return for, 38 optimal after-event window size for, 40–41 probability of favorable price movement for, 36 rare-events distributions and, 42

426 Equity data high-frequency tick data for, 147–148 typical behavior of, 129 Equity price, rare events and, 44 Equity tranche, 79, 82 Equity tranche prices, 83, 86 Equivalent martingale measure, 298, 306, 401 ‘‘Erroneous trade-break rules,’’241 E-step, of an iterated two-step process, 172–173 Estimated bias, 258, 259 Estimated DFA parameter, 134 Estimated Hurst parameter, 133 Estimation error, 24 Euler discretization scheme, 227 Euler Monte Carlo discretization, 268, 275 European call option integro-differential model for, 365 pricing, 228 European option prices, 219 European options, 348, 353, 385 ExecuComp dataset, 54 Executive BSC, 55. See also Balanced scorecards (BSCs) Executive compensation, 53 Executive compensation variables, 55 Executive stock options, 53 Exit time, 313, 314 Expectation-maximization (EM) algorithm, 164, 171–175, 183 dependence on sample size, 183 for skewed t distributions, 175 Expected discounted utility, 300 maximizing, 307 Expected return for equity classes, 38 of trades, 35 Expected return surfaces, 39 Expected shortfall (ES), 163 Expected transaction cost, 407 Expected utility problems, 295 Expected value, 172 Expected variance, 99 Expert weighting algorithm, 66 Exponential Lévy models, 6–8, 364

Index Exponential martingale process, 297 Extreme price movement, 31 Fair value, of future variance, 98, 99 Favorable price movement defined, 32 probability of, 35–36 Federal funds effective rate, 112 Fejer kernel, 261, 263 Feller’s condition, 281 Feynman–Kac lemma, 353 Figueroa-López, José E., xiii, 3 Finance, volatility and covolatility measurement/forecasting in, 243 Finance problems, methods used for, 68 Financial Accounting Standards Board (FASB), 53 Financial analysis, using boosting for, 47–74 Financial asset returns, computing covariance of, 263–264 Financial data, 176 behavior of, 202 GH distributions for describing, 165 Financial databases, 62 Financial events observations centered on, 107 probability curves for, 108 Financial market behavior, correlations in, 120 Financial mathematics model, 348 Financial mathematics, Black–Scholes model in, 352 Financial models, with transaction costs and stochastic volatility, 383–419 Financial perspective, 51, 55 Financial returns, 164, 216 Financial sector estimates, 150 Financial time series, 176 long-term memory effects in, 119 Finite-sample performance, via simulations, 14–17 Finite value function, 315, 322 Finite variance, 123 Fitted Gaussian distributions, 22 Fixed frequency, vs. high-low frequency, 208–212 Fixed-frequency approach, drawback of, 183–185 Fixed-frequency density, 210

Index Fixed-frequency method, 200 Fixed-point theorem, 391 applying, 398–399 existence based on, 397 Fixed portfolio/consumption processes, 308 Fixed rare event, favorable price movement for, 32 Fixed stopping time, 307, 308 Fixed time interval, 9 Fixed timescale, risk forecasts on, 176–185 ‘‘Flash-crash’’ of 2010, 236 Flat-top realized kernels, 261 Florescu, Ionu, xiii, 27, 97 Fluctuating memory effect, 145 Forecast horizon, monthly, 196–199 Forecasting of covariance, 280–285 of Fourier estimator properties, 272–285 of volatility, 273–275 Forecast pdfs, 209–210 Forecasts, confidence intervals for, 187–188 Foreign stocks index, 128 Forward index level, calculating, 111–112 Fourier coefficients, 247, 251–252 Fourier covariance estimator, finite sample properties of, 264 Fourier cutting frequency, 274 Fourier estimator(s), 244–245 asymptotic properties of, 248–250 cutting frequency and, 259–260 forecasting performance of, 245 forecasting properties of, 272–285 gains offered by, 245, 286 of integrated covariance, 263–272 of integrated volatility, 254, 252–263 microstructure noise and, 260–261, 274 of multivariate spot volatility, 246–252 of multivariate volatility, 266 performance of, 273 results of, 276–279 robustness of, 252–253 of volatility of variance and leverage, 250–252

427 Fourier estimator MSE (MSEF ), microstructure noise and, 256. See also Mean squared error (MSE) Fourier estimator performance, ranking, 279 Fourier–Fejer summation method, 247, 251, 252 Fourier method high-frequency data using, 243–294 gains yielded by, 290 Fourier transform(s), 122, 246, 335 numerically inverting, 13–14 Fractional Brownian motion (FBM), 125, 220, 221 FRE data series, DFA and Hurst methods applied to, 154 Frequency range, identifying, 22 Frequency sampling, 5 Functional analysis, review of, 386 Functions, weak derivatives of, 387 Function space, 368 Fundamental solutions, 312 Fundamental theorem of asset pricing, 401 Future earnings announcement, 62 Future integrated volatility, forecasting, 276 Future variance, fair value of, 98, 99 Gamma distribution, 171 Gamma Lévy processes, 8 GARCH(1, 1) process, 179–181, 185, 186, 202. See also Generalized autoregressive conditionally heteroskedastic (GARCH) methodology GARCH(2, 2), 202, 203 GARCH(3, 3), 202, 203 GARCH(p2 ,q2 ) errors, 181 GARCH(p, q) process, 178–179, 207 GARCH calibration, dependence on sample size, 185 GARCH errors, ARMA process with, 181–182 GARCH filter, 164, 165, 177–182 GARCH filtering, 217 autocorrelation of, 202 GARCH forecasts, 203 GARCH method, 176 GARCH model, 268, 275

428 GARCH process, stationary distribution of, 181 GARCH sum, simulation of, 186–187 Gaussian copula methods, 75 Gaussian copula models, 76, 91, 93 Gaussian default modeling, 75–76 Gaussian distribution, 120, 122, 337 Gaussian random variable, 336 Gauss–Whittle contrast function, 225 General integro-differential problem, 362–364 Generalized autoregressive conditionally heteroskedastic (GARCH) methodology, 165. See also Daily GARCH process; GARCH entries; Higher-order GARCH models; Long-term GARCH; Low-order GARCH models long-term behavior of, 203–208 roles in high-low frequency approach, 188 weekly return process and, 212–215 Generalized hyperbolic (GH) distributions, 164, 165, 167–169, 217 linear transformations of, 169–170 subfamilies of, 171 Generalized inverse Gaussian (GIG) distribution, 166–167, 169, 170 Generalized tree process, 354 General semilinear parabolic problem, 355–362 General utility functions, 311 Genetic algorithms, 63, 64 Geometric Brownian motion, 4, 6–7 Geometric Brownian motion case, transaction costs and option price valuation in, 384–386 Geometric Lévy models. See Exponential Lévy models German Society of Financial Analysts, 51 Girsanov theorem, 307 Goodness of fit, 22 Goodness of fit p-values, 139 Google, Lévy flight parameter for, 343 Google data series, DFA and Hurst methods applied to, 148 Governance index, 51 GPH estimator, 221, 222–223, 227 asymptotic behavior of, 222 computing, 223

Index Green’s function, 312, 317, 370, 411, 413 estimates based on, 368, 372, 415 Gronwall’s inequality, 390, 394 Heavy-tailed distributions, 164 Heavy-tailed skewed t distribution, 181–182 Hedging portfolio standard, 385 H estimates, 150 Heston model, 280–281 Higher-order GARCH models, 181. See also Generalized autoregressive conditionally heteroskedastic (GARCH) methodology High-frequency data, xi, 120, 272, 345 from the Bear Stearns crash week, 148–160 corresponding to Bear Stearns crash, 131–132 modeling, 364 multivariate volatility estimation with, 243–294 simulating, 280–281 from a typical day, 129–131 in volatility computing, 243–244 High-frequency financial data, 27–46 High-frequency tick data, 147–148 High-frequency time series, analyzing, 258 High-frequency tranche price histograms, 93 High-frequency tranche prices quantile–quantile plot of, 92, 94 sensitivity to default correlation, 88–89 sensitivity to dynamic default correlation, 89–92 High-frequency tranches, default correlation and, 87–92 High frequency tranching, 76 High-frequency transaction data, 6 High-low frequency, vs. fixed frequency, 208–212 High-low frequency approach, 185–186, 212 High-low frequency density, 210 High-low frequency method, 200, 212, 215–216 limits of, 195

Index High-low frequency VaR forecast, 186. See also Value at risk (VaR) High parameter values, 136 High trading activity, 42 Hilbert space, 387 Hillebrand, Eric, xiii, 75 HL estimator, 263. See also VaRHL Hölder constants, 358 Hölder continuous real-valued function, 350 Hölder continuous real-valued function with exponent δ, 351 Hölder’s inequality, 391, 395 Hölder spaces, 349, 355, 367, 388–390, 411 Homotopy perturbation method, 379–380 Housing crisis, 136 Hu–Kercheval method, 164 Hull–White process, 400 Hurst analysis, 130–131, 132 results of, 141 Hurst exponent, 125, 126, 132 values of, 138 Hurst index, 221. See also Implied Hurst index Hurst index estimation, Whittle-based approach for, 225–226 Hurst parameter(s), 121–122, 125, 135, 220, 221 Hurst parameter analysis, 136 Hurst parameter estimates, 132 Hurst parameter estimators, 229 Hurst regression plots, 137 Hyperbolic distributions, 171 IBM DFA and Hurst methods applied to, 147 Lévy flight parameter for, 343 IBM time series, 257 i.i.d. data, 172. See also Independent and identically distributed (i.i.d.) sample Implicit functions theorem, 321 Implied Hurst index, 226–227 Implied Hurst parameter, finding, 228 Implied volatility, 114–115 Improved regularity, 397 Increased noise term, 268

429 Independent and identically distributed (iid) sample, 171. See also i.i.d. data Independent identically distributed (IID) random variables, 334 Independent ownership structure, 59 Index option market, 105 Index variants, 107–108 Indicator variables, 188, 189 Indices, predictive power of, 107–110 Induction argument, 358, 359 Inequalities, 390–391 Infinite horizon case, 305–324 Infinite horizon problem, 307 Infinite jump activity, 4 Infinite time horizon, 311 Initial-boundary-value problem, 355, 362–363, 366, 369, 410 Innovations, 178, 180 Student t, 182 Insider ownership, 53–54, 59 Insider ownership variables, 55 ‘‘Inside spread’’, 239 Instantaneous covariance, computing, 252 Instantaneous volatility process, 253 Institutional brokers’ estimate system (IBES), 62 INTC data series, DFA and Hurst methods applied to, 152. See also Intel (INTC) stock INTC histograms, 22 INTC return histograms, logarithm of, 23 Integral operator, 363 Integral representation, 360 Integrated covariance, Fourier estimator of, 263–272 Integrated covariance estimators, forecasting power of, 280–285 Integrated covolatility, 248 Integrated quarticity (IQ), 255. See also IQ estimates Integrated time series, 127 Integrated volatility computation of, 258 forecasting, 273 Fourier estimator of, 252–263 Integrated volatility estimators comparison of, 270–271 optimized, 262

430 Integrated volatility/covolatilities, computing, 248 Integrating factor, 328, 331, 332–333 Integration by parts, 328 Integro-differential equations, in a Lévy market, 375–380 Integro-differential model, 365 Integro-differential operator, 367 Integro-differential parabolic problems, 347–381 Integro-differential problem, 362–364 Intel Corporation, Lévy flight parameter for, 345 Intel (INTC) stock, 18. See also INTC entries Internal processes perspective, 52, 55 International indices, 135 International market indices, 120, 128 Interpolating formula, 405 Interquartile range (IQR) rule, 31–32 Intraday data, 4, 202 Inverse Fourier transform, 336 InverseGamma distribution, 170, 171 density of, 172 Inverse Gaussian distribution, 8 Investment bank industry, risk management meltdown of, 121 Investor fear gauge, 98 IQ estimates, 258. See also Integrated quarticity (IQ) iShares MSCI EAFE Index (EFA), 128. See also EAFE index; EFA entries Ising model, 64 Iterated two-step process, 172–175 Iterative equations, 208 Iterative method, 364–375 Itö process, 297 Itö’s formula, 401, 405 one-dimensional, 329 two-dimensional, 328–329 Itö’s lemma, 354 Itö’s rule, 307, 315, 320 IV estimates, 258 Jensen’s inequality, 179 Joint density, 172 Jointly Gaussian variables, 78 JPM data series, DFA and Hurst methods applied to, 158 JP Morgan, Lévy flight parameter for, 341

Index Jump activity, 4 Jump diffusion models, 148 Jump intensity, 354 Jumps Black–Scholes models with, 353, 364 integro-differential operator modeling, 365 modeling, 375–380 Kercheval, Alec N., xiii, 163. See also Hu–Kercheval method Kernels Bartlett-type, 261, 263 cubic-type, 261, 263 Dirichlet, 255, 261 estimator for, 279 Fejer, 261, 263 flat-top realized, 261 multivariate realized, 267, 259, 280 Parzen, 269 TH2 -type, 261, 263 Khashanah, Khaldoun, xiii, 27, 97 Koponen model, 124 Kurtosis, 11 of innovations, 180 Kurtosis estimator, 5 Kurtosis parameter, 24 Lagrange multiplier, 301 Lancette, Steven R., xiii, 3 Laplace transform, 169 Large capitalization equities, 34 Large market movements, 375 Large price movement, 29 Large-volume stocks, 34–35 Last-tick interpolation, 267 Latent mixing variables, 172 Latent variable trajectory, recovering, 245 LBC data series, DFA and Hurst methods applied to, 159 Leading indicators, economic models with, 67 Lead-lag realized covariance, 272 Learning algorithms, 64, 66–67 Learning and growth perspective, 52 Least squares regression, 126 Lebesgue measure, 297, 306 Lee, Kiseop, xiii, 3 Legendre–Fenchel transform, 299, 308, 321

Index Lehman bankruptcy, 150 Leland model, 384 Leverage, volatility of, 250–252 Lévy distributions, 336, 346 Lévy flight, 125 Lévy flight models, 336–340 Lévy flight parameter estimating, 135 values of, 136, 138 Lévy-like stochastic process, 364 Lévy market, integro-differential equations in, 375–380 Lévy model(s), 4–5 for describing log returns, 22 log return process increments under, 13 motivations of, 5 numerical simulations and, 340–345 suitability assessment of, 23 Lévy processes, 148 Lévy–Smirnov distribution, 122, 337 Lévy-stable distribution, 337 Likelihood function, 13, 171 Likelihood ratio process, 297 Likelihood ratio test (LRT), 192 acceptable band of, 202 stability of, 199–202 Likelihood ratio test statistic, 190–191 asymptotic distribution of, 191 Limit orders, elasticity/plasticity of, 28–29 Linear discriminant analysis, 47 Linear models, statistical significance of, 47 Linear transformations, of GH distributions, 169–170 Link mining algorithm, 62 Lipschitz constant, 358 Lipschitz continuous function, 356 Liquidity, increased, 236 ‘‘Liquidity bottleneck’’, 236 Liquidity costs, 236 Liu, Yang, xiii, 163 Location parameter, 337 Locked-in interest rate process, 296 Logarithmic utility functions, 321 Logistic regression, 47 Logistic transitional default correlation, 84–87 Logitboost, 49, 62 Log likelihood, maximizing, 172

431 Log-normal diffusion process, 275 Log-periodogram regression, 221 Hurst parameter estimator for, 222–225 Log-price process, 247, 253 Log-return process, 7, 8, 9 discretizing, 222 Log return process increments, 13 Log returns, 5 Log squared returns, 222 Long correlations data related to, 128–132 persistence of, 141 results and discussions of, 132–150 Long memory, in financial datasets, 220 Long-memory effects, 120 analyzing, 135 Long-memory parameter, determining, 226 Long-memory stochastic volatility (LMSV) models, 221 application to S&P index, 228–229 continuous-time, 220 parameter/estimation/calibration for, 219–231 parameter estimation under, 221 simulation results of, 227 statistical inference under, 222–227 Long-range correlations, 120, 127 Long-range dependence, xi, 220 Long-term-assets-to-sales ratio, 58 Long-term behavior, methods of estimating, 150 Long-term GARCH, 203–216. See also Generalized autoregressive conditionally heteroskedastic (GARCH) methodology Long-term investments, 135 Long-term memory effects, 119, 150 Lorentz(ian) distribution, 122, 337 Lorentzian random variable, 335 Lower solution, 356, 357, 358, 364. See also Ordered lower-upper solution pair Low-frequency tranches, default correlation and, 79–87 Low-order GARCH models, 179. See also Generalized autoregressive conditionally heteroskedastic (GARCH) methodology LP spaces, 386

432 LRT failure, 196. See also Likelihood ratio test (LRT) LRT p-values, 192–195 Lunch-time trader activity, 42 Machine learning methods, 48, 64–65 calibration of, 68 Machine learning perspective, 62 Machine-readable news, 64 Major financial events observations centered on, 107 probability curves for, 108 Mancino, Maria Elvira, xiv, 243 Marginal utility function, 299 Mariani, Maria C., xiv, 347, 383 Market capitalization index, 128 Market completeness assumption, 302 Market complexity, modeling of, 99 Market crash, 346 2008, 136 Market index (indices) exponents calculated for, 345 squared returns of, 220 technique for producing, 110 Market index decrease, spread and, 105 Market inefficiencies, for small-space and mid-volume classes, 44 Market microstructure effects, 263 Market microstructure, effects on Fourier estimator, 245 Market microstructure contaminations, 273 Market microstructure model, of ultra high frequency trading, 235–242 Market model, 296–297 Market movement, indicators of, 110 Market reaction, to abnormal price movements, 45 Market-traded option prices, 219 Markov chain, stochastic volatility process with, 401 Markowitz-type optimization, 286 Martingale-difference process, 178. See also Continuous semimartingales; Equivalent martingale measure; Exponential martingale process Supermartingale Matlab, 14, 257 Matlab module, 125, 339 Maximum likelihood estimation (MLE), 13–14, 185

Index finite-sample performance of, 14–17 performance of, 23–24 Maximum likelihood estimators (MLEs, mles), 4, 6, 172–175, 190, 225. See also MLE entries; NIG MLE; VG MLE Maximum likelihood method, 183 Maximum price movement, 30 Maximum principle, 359, 360 MBS portfolio, 77. See also Mortgage-backed securities (MBSs); Subprime MBS portfolios slicing into tranches, 88–89 MBS tranches, 76 MBS units, 79 MBS vehicle, function of, 77 m-dimensional Brownian motion, 311, 312 Mean squared error (MSE), 245, 254–256. See also MSE entries cutting frequency and, 259, 260 Mean–variance mixture definition, 170 Mean-variance optimization, 286 Mean-variance utility, 287 Medium-volume stocks, 34–35 Memory effects, 135 Method of moment estimators (MMEs), 4, 5–6, 10–13. See also MME entries; VG MME finite-sample performance of, 14–17 performance of, 23–24 Method of upper and lower solutions, 351–364 MFA data series, DFA and Hurst methods applied to, 156 Mi, Yanhui, xiv, 3 Microstructural model, 237–239 future research on, 241 Microstructure effects, 19, 21, 22 Microstructure noise Fourier estimator and, 252–263, 263–272 impact of, 244 Microstructure noise component, 275–276 Microstructure noise variance, 276 Midrange frequencies, 19 Mincer–Zarnowitz-style regression, 276 Minimum variance estimators, 274

Index MLE estimator, increase of, 20. See also Maximum likelihood estimators (MLEs, mles) MLE results, for NIG and VG model estimation, 18–19 MME estimator, increase of, 20, 21. See also Method of moment estimators (MMEs) MME results, for NIG and VG model estimation, 18–19 Model-free statistical analysis, 29 Modeling, popular distributions used in, 165 Model selection problem, 5 Modified Bessel function of the third kind, 166, 167 Modulated realized covariation, 267 Moment estimators, 24 Moment formulas, 166 Monopolistic competition, 238 Monopolistic competition models, 237 Monotone convergence theorem, 315 Monte Carlo analysis, 256–263, 266–272, 275–285 Monte Carlo replications, 269 Monte Carlo (MC) simulation(s), 6, 76, 186, 200, 206 violation count stability in, 201 Monthly forecast horizon, 196–199 Morgan Stanley Capital International, 128 Morrey imbedding, 361 Mortgage, default probability of, 77 Mortgage-backed securities (MBSs), 75. See also MBS entries; Subprime MBS portfolios Mortgage vintages, 89–90 MSCI EAFE stock index, 128. See also EAFE index MSCI Emerging Markets Index, 128 MSE-based estimators, 245, 261. See also Mean squared error (MSE) MSE-based parameter values, 263 MSE computation, of the Fourier estimator, 264–266 MSE estimates, computing, 258 MSFT data series, DFA and Hurst methods applied to, 151 M-step, of an iterated two-step process, 172–175

433 Multiagent portfolio management system, 64 Multinomial recombining tree algorithm, 221, 226 Multinomial tree approximation method, 97–115 Multiple timescale forecasts, 185–188 Multiscale method, 217 Multiscale VaR forecast backtest results, 202. See also Value at risk (VaR) Multiscale volume classification, 33–35 Multistock automated trading system, 66 Multivariate normal distribution, 170 Multivariate normal mean–variance mixture distribution, 165–166 Multivariate realized kernel, 280 estimator for, 267 implementing, 269 Multivariate spot volatility, Fourier estimator of, 246–252 Multivariate volatility based on Fourier series, 244–245 Fourier estimator of, 266 Multivariate volatility estimation, with high-frequency data, 243–294 N -asset portfolios, 217 Ncheuguim, Emmanuel K., xiv, 383 n-day horizon, 192–195 Near-term call options, 101 Near-term/next-term options chains, 100, 101 Negative correlation, 148 Negative log returns, 189, 208, 209–210 filtering, 177 Negative log returns process, 181 Newey–West covariance matrix, 282 New volatility index calculation, 113–114 New York Stock Exchange index, 128 Next-term call options chain, 101 NIG MLE, 6. See also Maximum likelihood estimators (MLEs); Normal inverse Gaussian (NIG) model NLY data series, DFA and Hurst methods applied to, 159 NMR data series, DFA and Hurst methods applied to, 157 Noise variance, 268. See also Increased noise term; Microstructure noise

434 Noise variance (continued ) entries; Simultaneous correlated noise; Strict white noise process No-leverage hypothesis, 255, 274 Non-Gaussian processes, 120 Nonlinear parabolic PDEs, 348. See also Partial differential equation (PDE) methods Nonlinear partial differential equations, 384 Nonnegative integers, 349 Nonoverlapping windows, 31 ‘‘Nonparametric’’ methods, 67 Nonstationarity, types of, 136 Nonstationary datasets, 127 Norm, 351, 390 Normal distribution, 164, 181, 337 Normal inverse Gaussian (NIG) distributions, 171 Normal inverse Gaussian (NIG) model, 4, 8–9. See also NIG MLE computing MME for, 12–13 empirical results for, 18–22 MME and MLE finite-sample performance for, 16–17 Normality hypothesis, 138–139 Normality test results, 138 Normality tests, 144 Normalized truncated Lévy model, 125 Normalizing constant, 167, 170 Normal mean–variance mixture distributions, 165–166, 167–168 Null hypothesis, 128–129, 192 unit-root tests rejection of, 145 Numerical simulations, Lévy models and, 340–345 Nyquist frequency, 257, 269 NYSE TAQ database, 18 Objective function, computing, 173 One-day return forecasting, 195–196 One-dimensional diffusions, optimal stopping for, 311–318 One-dimensional GH distribution, 169 One-dimensional hyperbolic distributions, 171 One-dimensional integrals, 372, 414–415 One-dimensional Itö’s formula, 329 One-factor Gaussian copula model, 78

Index One-sided stable distribution, 337 Operating-expenses-to-sales ratio, 58 Operating-income-to-sales ratio, 58 Optimal α level, for equity classes, 37. See also Optimal level α/window size trading rule Optimal after-event window size, for equity classes, 40–41 Optimal level α/window size trading rule, 33. See also Optimal α level, for equity classes Optimal MSE-based covariance estimator, 269. See also Mean squared error (MSE) Optimal MSE-based Fourier estimator, 269 Optimal portfolio/consumption process, 322 Optimal portfolio process, 321 Optimal stopping, for one-dimensional diffusions, 311–318 Optimal stopping boundary, 322 Optimal stopping time, 313, 314, 322 Optimal trading parameters, 45 Optimal values, calculating, 37, 39, 40 Optimal wealth, 324 Optimal wealth process, 319–320, 322 Optimization problem, 299–300, 307 Optimized integrated volatility estimators, 262 Optional sampling theorem, 316 Option chain values, 99 updating, 403 Option price(s), 406 discrepancies among, 219 in stochastic volatility models, 401 Option price evolution model, 120 Option price formula, 384 Option price valuation, in the geometric Brownian motion case, 384–386 Option pricing algorithm, 226 Options, 348. See also Call options chain; European option entries; European call option; Put options chains; Stock options compensation based on, 59 as given assets, 401–404 market volatility and, 100–101 maturity date of, 99 path-dependent, 226 Options chains, selecting, 110

Index Order arrivals, simulating, 240 Ordered lower-upper solution pair, 360. See also Lower solution; Upper solution Organizational variables, optimal values of, 54 Ornstein–Uhlenbeck process, 219 Osborne model, 27 Outliers, types of, 28 Out-of-money call option, 105–106 Out-of-money options, 100 Out-of-money put option, 105–106 Out-of-sample forecast, 287 Out-of-the-money SPX, 98. See also Standard and Poor Index (SPX) Overfitting, 67 Ownership–performance relationship, 54 Parabolic distance, 350, 389 Parabolic domain, 348 Parabolic equation, 409, 417 Parabolic integro-differential problem, 364 Parabolic operator, 370 Parabolic problem, 360–361 Parameter estimates, crisis-related, 150 Parameter estimation, 67 under the LMSV model, 221 techniques for, 229–230 Parameter/estimation/calibration, for long-memory stochastic volatility models, 219–231 Parameters optimal choice of, 224 values of, 14 Parametric detection rule, 31 Parametric estimation methods, 9–14 Parametric estimators, performance of, 23–24 Parametric exponential Lévy models (ELMs), 4 consistency of, 5 parametric classes of, 22–23 Parametric families, heavy-tailed, 164 Parsimonious model, 5, 22–23 Partial differential equation (PDE) methods, 295. See also Black–Scholes PDE; Nonlinear parabolic PDEs; PDE entries Partial differential equations (PDEs). See also PDE entries

435 analysis of, 408–417 under transactions costs and stochastic volatility, 407–408 Partial integral-differential equations (PIDEs), 348, 353, 354, 364, 375 Particle filtering algorithm, 226 Parzen kernel, 269 Parzen weight function, 267 Pasarica, Cristian, xiv, 295 Path-dependent options, 226 PDE derivation. See also Partial differential equations (PDEs) given asset option and, 401–404 traded asset volatility and, 405–408 PDE problems, solving, 352 pdf forecasting, 176. See also Forecast pdfs; Probability density function (pdf) Peaks, in rare-events distribution, 42 Penn–Lehman Automated Trading (PLAT) Project competition, 65 Percentage excess kurtosis, 12 Performance, insider ownership and, 54 Performance analysis, of S&P500 companies, 54–60 Performance evaluation, 53–60 Periodogram, 223. See also Log-periodogram regression entries Persistent time series, 126 Phillips–Peron (PP) test, 128–129 Poincare’s inequality, 391, 395 Point estimates, stability of, 4–5 Point estimators, 19 Pointwise limit, 359 Poisson order-arrival process, 239 Poisson probabilities, 240 Poisson process, 237, 354 Poisson random variables, 238 Poisson trading, 268, 272 Population skewness, 11 Portfolio/consumption process, 298 Portfolio/consumption strategy, 300 Portfolio diversification, 135 Portfolio insurers/hedgers, 105 Portfolio management, 169 time horizon for, 185 Portfolio processes, 297–299, 305–307 Portfolio rebalancing, 402 Portfolio risk management method, 170 Portfolios. See also Constant rebalanced portfolio technical analysis

436 Portfolios (continued ) (CRP-TA) trading algorithm; Multiagent portfolio management system; Subprime MBS portfolios MBS, 77 tranches of, 77 vintage of, 77 Portfolios value, expected change in, 385 Portfolio utility, 286 Position strategy, 33 Positive process, 310 Powell’s method, 6, 14, 19 Power-type utility functions, 305 Preaveraging technique, 267 Prediction nodes, 50, 51 Prediction rule, 48, 49 Prespecified terminal time, 295 Price behavior, analyzing after rare events, 28 Price change distributions, 31 Price distribution distortion, 91 Price evolution in time, 30 Price movement(s) corresponding to small volume, 30 detecting and evaluating, 44 persistence of, 27–46 Price movement methodology, results of, 35–41 Price process, 121 Price recovery probability of, 44 after rare events, 45 Price volatility, UHFT and, 241 Price–volume relationship, 27–28 outlying observations of, 28 Principal–agent conflict, 53 Principal–agent problem, 60 Probability of favorable price movement, 35–36 Poisson, 240 Probability density, 13–14 Probability density function (pdf), 119, 120, 163, 171, 335. See also Forecast pdfs; pdf forecasting; Sample pdfs Probability distributions, 165 Probability mass function (pmf), 171 Probability surfaces, 35, 37 Proportionality constant, 402 Pure optimal stopping problems, 311 Put options, demand for, 106

Index Put options chains, constructed VIX using, 105–106 p-values, 138–139, 204–205 pVIX-b, 102–103, 105. See also Volatility index (VIX) pVIX cVIX spread, 106 Qiu, Hongwei, xiv, 97 Q-learning algorithm, 65 Quadratic covariation formula, 244 Quadratic covariation-realized covariance estimator, 266 Quadratic utility function, 286 Quadratic variation, estimate of, 224 Quadrinomial tree method, 99–100 volatility index convergence and, 105 vs. CBOE procedure, 100–101 Quantile–quantile (QQ) plots, 80 of empirical CDF, 136 of high-frequency tranche prices, 92, 94 of tranche prices, 83–84 ‘‘Quantile type’’ rule, 30 Quantum mechanics, 385 Quote-to-quote returns, 258, 260 Random variables, 334–336 Random walk, 126 Rare-event analysis, 32–33 Rare-event detection, 28, 30–32 Rare events detecting and evaluating, 29–35 equity price and, 44 trades profile and, 42, 43 Rare-events distribution, 41–44 peaks in, 42 Real daily integrated covariance, regressing, 281 Real integrated covariance regressions, results of, 282–285 Realized covariance (RC), 269 estimator for, 280 measures of, 272 Realized covariance plus leads and lags (RCLL), 266 estimator for, 280, 290 Realized covariance–quadratic variation estimator, 244 Realized variance, 12 Realized volatility, microstructure noise and, 274

Index Realized volatility estimator, 253–254, 256 results of, 276–279 Realized volatility estimator performance, ranking, 279 Realized-volatility-type measures, 275 Real-valued functions, 350, 351, 388–389 Refresh time, 267 Refresh time procedure, 244 Refresh time synchronization method, 268 Regime-switching default correlation, 81–84 Regime-switching default correlation model, 76 Regime-switching model, drawback of, 84–85 ‘‘Regret-free’’ prices, 238 Regular asynchronous trading, 264 Regular nonsynchronous trading, 268 Regular synchronous trading, 268 Relative risk process, 296 Rellich’s theorem, 398 Representative ADT algorithm, 52–53, 54. See also Alternating decision trees (ADTs) Representative ADTs, 56–57, 67 Rescaled range (R/S) analysis, 120, 121, 125–126, 140 Retirement problem, 295–326 explicit formulas for, 318–324 Risk, defined, 163 Risk adjustment, standardization and, 124 Risk aversion levels, 287–290 Risk-factor returns, modeling, 166 Risk forecasting, 163–218 Risk forecasts on a fixed timescale, 176–185 weekly or monthly, 164 Risk-free portfolio, 404, 407 Risk management, 68, 93 Risk models, 163–164 Risky asset, price process of, 6–7, 8 Root mean square fluctuation, 127 Rule of detecting rare events, 31–32 Sabr process, 400 Salas, Marc, xiv, 347

437 Sample pdfs, theoretical pdf vs., 184. See also Probability density function (pdf) Sample size EM algorithm dependence on, 183 GARCH calibration dependence on, 185 Sampling frequency, 5 Sanfelici, Simona, xiv, 243 S&P500 companies, corporate governance and performance analysis of, 54–60. See also Standard and Poor entries S&P500 index, 137, 138, 139. See also Standard and Poor Index (SPX); Standard and Poor’s 500 equity index (SPX) analysis results for, 143 application of LMSV model to, 228–229 correlation with VIX/S&P500, 106–107 index variants and, 108 S&P500 prices, volatility increase and, 107–110 S&P500 representative ADTs, 56–57. See also Alternating decision trees (ADTs) interpreting, 58–59 S&P500 representative board scorecard, 61 Santa Fe stock market model, 63 Sarbanes–Oxley Act of 2002, 53 Scale-invariant truncated Lévy (STL) process, 124 Schaefer’s fixed-point theorem, 391 applying, 398–399 existence based on, 397 SCHW data series, DFA and Hurst methods applied to, 157 Second-by-second return path, 275 Securities and Exchange Commission (SEC), 53, 241 Securitized structures, impact of correlation fluctuations on, 75–95 Self-similarity, 127 Semilinear parabolic problem, 355–362 Seminorm, 351, 388, 389, 390 Seneta approximation method, 12 Sengupta, Ambar N., xiv, 75 Sengupta, Indranil, xiv, 347, 383

438 Senior tranche, 79 default risk of, 93 prices of, 82, 83, 86 Serial correlation, 83, 84, 87, 90 behavior of, 78 slowly decaying, 7 Shareholder–manager conflict, 60 Short-term memory models, 121 Simulated daily returns scenario, 215–216 Simulated weekly returns scenario, 212–215 Simulations, finite-sample performance via, 14–17 Simultaneous correlated noise, 282 Skewed t distributions, 165–175 algorithm for, 175 density of, 170 simulation of, 171 Skewness parameter, 337 Small parameter, 379 Small-volume stocks, 34–35 ‘‘Small-world’’ model, 63 Smooth-fit principle, 319 Sobolev spaces, 349, 352, 387 Sobolev space solutions, 391–400 Social networks, 62, 63 Spaces, involving time, 387–388. See also Banach spaces; Function space; Hilbert space; Hölder spaces; LP spaces; Sobolev spaces Sparse estimator, 279 Spectral density, 225 Spectral density function, 223 Spin model, 64 Splitter nodes, 50, 51 Spot variance, 251 Spot volatility, 248 Spot volatility model, 273 Spread, between indices, 110 SPY, 97 Stability exponent, 337 Stable distributions, 334–336 Stable Lévy distribution, 339 Stable Lévy processes, 340 Stakeholder perspective, 59 Standard and Poor Index (SPX), 405. See also S&P entries Standard and Poor’s 500 equity index (SPX), 97–98

Index Standard and Poor’s Governance Services, 51 Standard deviation (StD), 163 Standard diagonal argument, 375, 417. See also Cantor diagonal argument Standardized Lévy models, 125, 340, 346 Standardized truncated Lévy flight model, 124 Standardized truncated Lévy model, 339 Standardized value, 114 State-price-density process, 297 State variables, 79, 82, 90 Static comparisons,’’239–241 Stationarity tests, 129–131. See also Covariance stationarity Stationarity/unit-root test, 127–128 Statistical inference, under the LMSV model, 222–227 Statistical models, 6–9 Statistical tests, 190–192 Stochastic differential equations (SDEs), 327–334 Stochastic differential equation solution, Lévy flight parameter for, 340 Stochastic-Dirichlet problem, 317 Stochastic function of time, 245 Stochastic order flow process, 237 Stochastic processes, 352, 400 empirical characterization of, 119 Lévy-like, 364 Stochastic recurrence equation (SRE), 179 Stochastic variable, 129 Stochastic volatility, 348, 354 financial models with, 400–408 Stochastic volatility models, 148, 250–251, 401 problem with, 100 Stochastic volatility process, 100 with Markov chain, 401 Stochastic volatility quadrinomial tree method, 99–100 VIX construction using, 114–115 Stock index, monthly returns for, 164. See also Standard and Poor Index (SPX); Volatility index (VIX) Stock market volatility, 97–98. See also Volatility index (VIX) Stock options, compensation based on, 53 Stock price, relationship to volume, 27

Index Strict white noise process, 177 Strike price, 98, 99, 112 selecting, 111 Strong Markov Property, 317, 320 Strong prediction rule, 49 Strong solutions, 351–352, 355, 361, 362, 364, 368, 369, 374, 412 Student t innovations, 182 ‘‘Stylized facts’’, 176–177 Subprime MBS portfolios, 87. See also Mortgage-backed securities (MBSs) Subprime mortgage fiasco, 75 Subseries, 125–126 Super equity, 34–35 Supermartingale, 307, 315 Surfaces, 2D plots of, 39–40 Suspicious events, 45 ‘‘Symmetric case’’, 4 Symmetric Lévy distribution, 338 Synchronization bias, 248 Target expected returns, 287–290 Taylor’s formula, 403, 406 Technical indicators, 65 Technical trading strategies, 64 Temporal time series, statistical properties of, 120 Terminal condition, 348, 353 TH2 -type kernels, 261, 263 Tick-by-tick data, 29, 244 Time. See also Calendar time sampling; Continuous-time entries; Discrete time model; Exit time; Fixed stopping time; Fixed time interval; Infinite time horizon; Lunch-time trader activity; Optimal stopping time; Prespecified terminal time; Refresh time entries price evolution in, 30 spaces involving, 387–388 stochastic function of, 245 Time consistency of Lévy processes, 5 Time-dependent volatility matrix, 246 Time distribution, of rare events, 41–44 Time lag, 339 Timescale forecasts, multiple, 185–188. See also Fixed timescale Time-scaling problems, 236 Time series, 125, 126. See also Classical time series analysis; Financial time

439 series; High-frequency time series; IBM time series; Integrated time series; Temporal time series; Weekly returns time series Time series data, filtering, 176 Time series forecasting, 68 Time series stationarity, 127–128 investigating, 141 Timestep, 403, 404, 406 Time to expiration, 111 Time to maturity, 112 TLF analysis, 140. See also Truncated Lévy flight (TLF) TLF distribution, 123, 338 TLF model, 120, 345 Tobin’s Q, values of, 55 Trade activity, rare events distribution and, 44 Traded assets, 401 Traded-asset volatility, 405–408 Trades, distribution of, 41–42 Trades profile, 42, 43 Trading, using boosting for, 47–74 Trading activity heightened, 30 increase in, 42 Trading horizon, 237 Trading rules, learning algorithms for generating, 64 Trading strategies, activation of, 42 Trading system optimization, 66 Traditional quantile rule, 31 Tranche price convergence, 91 Tranche price distribution QQ plot, 83–84 Tranche price histograms, 80 across vintages, 82–83, 85 Tranche prices, 76, 77, 80 across vintages, 90–91 default correlation and, 82 default correlation dynamics and, 92 unconditional distribution of, 80–81 Tranches, of a portfolio, 77 Tranche seniority, 82, 89, 93 Transaction costs, 402–404, 406–407 financial models with, 383–408 in the geometric Brownian motion case, 384–386 Transition level, 89 Truncated Lévy flight (TLF), 120, 122–125, 338

440 Two-dimensional Itö’s formula, 328–329 Two-factor affine process, 275 Two-scaled adjusted estimators, 279 Two-scale estimator, 261, 279 Two-scale ZMA estimator, 263 UHFT market activities, 236. See also Ultra high frequency trading (UHFT) UHFT market restrictions, 237 UHFT regulation, 241 UHFT transaction cost, 241 UHFT volume, 235 Ulibarri, Carlos A., xiv, 235 Ultra high frequency traders, 235 Ultra high frequency trading (UHFT). See also UHFT entries impacts of, 236 market microstructure model of, 235–242 Unbounded parabolic domain, 352 Unconditional default probability, 79, 89 Uniform convergence, 374 Unit-root stationarity tests, results of, 135 Unit-root tests, 121, 127–128, 141 results of, 130–131 Upper solution, 356, 357, 364. See also Ordered lower-upper solution pair U-shape, of trade distributions, 42 Utility after retirement, 321 Utility estimations, 287 Utility functions, 296, 299 of power type, 305 Utility loss, 290 Value at risk (VaR), 163, 165, 176. See also VaR entries Value function, 304, 307, 312, 313 for the constant coefficients case, 318 VaR error, 201. See also Value at risk (VaR) VaR estimates, based on Monte Carlo simulation, 199 VaRFixed , 213, 214, 215 VaR forecast(s), 210, 212 high-low frequency, 186 intraday, 202–203 VaR forecasting, 182 VaRHL , 213, 214, 215. See also HL estimator

Index Variance, volatility of, 250–252 Variance estimator optimization, 286 Variance forecast, 206 Variance gamma (VG) distributions, 171 Variance-gamma (VG) model, 4, 8–9. See also VG MLE computing MME for, 10–11 empirical results for, 18–22 VaRTrue , 213, 214 VaR violations, 210 counting, 191–192 VG MLE, 6. See also Maximum likelihood estimators (MLEs); Variance-gamma (VG) model finite-sample performance of, 15–16 VG MME, finite-sample performance of, 14–15. See also Method of moment estimators (MMEs); Variance-gamma (VG) model Vintage, of a portfolio, 77 Vintage correlation, 76, 79 Violation count stability, in Monte Carlo simulations, 201 Violation indicators, independence of, 188–189 Violation ratio tables, 192–195, 196–199 VIX construction, using stochastic volatility quadrinomial tree method, 114–115. See also Volatility index (VIX); Volatility indices Volatilities (volatility) forecasting, 273–275 nonconstant, 352 options maturity date and, 99 spread between, 106 Volatility changes, 212 Volatility clusters, 176, 180 Volatility distribution, 113 Volatility function, 248 Volatility index (VIX), 97–98, 405. See also Chicago Board Options Exchange (CBOE) Market Volatility Index (VIX); cVIX entries; pVIX entries; VIX construction; Volatility indices CBOE calculation of, 98–99, 110 Volatility index convergence, using quadrinomial tree method, 105

441

Index Volatility indices. See also Volatility index entries constructing, 97–115 new methodology related to, 99–100 predictive power of, 107–110 using different inputs for, 101–110 Volatility matrix, 246 Fourier coefficients of, 247 Volatility measurement/forecasting, as a key issue in finance, 243 Volatility measures, 213 Volatility models, long-memory stochastic, 219–231 Volatility parameter, 5, 6, 24 Volatility particle filter, 226 Volatility process, 255 Volatility smiles (smirks), 219, 220 VolAvg, 213, 214 VolStD, 213, 214 Volume constant in time, 30 relationship to stock price, 27 Volume window, limited, 32 Walmart, Lévy flight parameter for, 344 Walt Disney Company, Lévy flight parameter for, 342, 344 Wang, Jim, xiv, 27 Weak derivatives, 349, 386–387 Weak hypothesis, 51

Weak learner, 48, 49 Weak prediction rules, 49, 51 Weak solution, 399 Wealth processes, 297–299, 305–307 Week-based forecasts, 210, 211 Weekly returns scenario, 212–215 Weekly returns time series, 212 Weekly return/volatility, 211–212 Weighted options, 101 Weighting, 48–49 Whittle-based approach, for Hurst index estimation, 225–226 Whittle contrast function, 225 Whittle estimator, 227 Whittle maximum likelihood estimate, 225 Whittle-type criterion, 221 Whole real line, solution construction in, 399–400 Wiener process, 3, 7, 8 WMT data series, DFA and Hurst methods applied to, 151 XOM data series, DFA and Hurst methods applied to, 152 Xu, Junyue, xiv, 75 Zero autocorrelation, 178 Zero-boundary condition, 369, 412 Zero Dirichlet condition, 393, 399 ZMA estimator, 263

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