Quantitative Analysis in Financial Markets ASSET-PRICING AND RISK MANAGEMENT DATA-DRIVEN FINANCIAL MODELS MODEL CALIBRATION AND VOLATILITY SMILES Marco Avellaneda Editor Collected papers of the N e w York University Mathematical Finance Seminar, Volume II World Scientific Quantitative Analysis in Financial Markets Collected papers of the New York University Mathematical Finance Seminar, Volume II QUANTITATIVE ANALYSIS IN FINANCIAL MARKETS: Collected Papers of the New York University Mathematical Finance Seminar Editor: Marco Avellaneda (New York University) Published Vol 1: ISBN 981-02-3788-X ISBN 981-02-3789-8 (pbk) Quantitative Analysis in Financial Markets Collected papers of the New York University Mathematical Finance Seminar, Volume II Editor Marco Avellaneda Professor of Mathematics Director, Division of Quantitative Finance Courant Institute New York University m World Scientific II Singapore • New Jersey •London • Hong Kong Published by World Scientific Publishing Co Pte Ltd P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library QUANTITATIVE ANALYSIS IN FINANCIAL MARKETS: Collected Papers of the New York University Mathematical Finance Seminar, Volume II Copyright © 2001 by World Scientific Publishing Co Pte Ltd All rights reserved This book or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 981-02-4225-5 ISBN 981-02-4226-3 (pbk) Printed in Singapore by Fulsland Offset Printing INTRODUCTION It is a pleasure to edit the second volume of papers presented at the Mathematical Finance Seminar of New York University These articles, written by some of the leading experts in financial modeling cover a variety of topics in this field The volume is divided into three parts: (I) Estimation and Data-Driven Models, (II) Model Calibration and Option Volatility and (III) Pricing and Hedging The papers in the section on "Estimation and Data-Driven Models" develop new econometric techniques for finance and, in some cases, apply them to derivatives They showcase several ways in which mathematical models can interact with data Andrew Lo and his collaborators study the problem of dynamic hedging of contingent claims in incomplete markets They explore techniques of minimumvariance hedging and apply them to real data, taking into account transaction costs and discrete portfolio rebalancing These dynamic hedging techniques are called "epsilon-arbitrage" strategies The contribution of Yacine Ait-Sahalia describes the estimation of stochastic processes for financial time-series in the presence of missing data Andreas Weigend and Shanming Shi describe recent advances in nonparametric estimation based on Neural Networks They propose new techniques for characterizing time-series in terms of Hidden Markov Experts In their contribution on the statistics of prices, Geman, Madan and Yor argue that asset price processes arising from market clearing conditions should be modeled as pure jump processes, with no continuous martingale component However, they show that continuity and normality can always be obtained after a time change Kaushik Ronnie Sircar studies dynamic hedging in markets with stochastic volatility He presents a set of strategies that are robust with respect to the specification of the volatility process The paper tests his theoretical results on market data The second section deals with the calibration of asset-pricing models The authors develop different approaches to model the so-called "volatility skew" or "volatility smile" observed in most option markets In many cases, the techniques can be applied to fitting prices of more general instruments Peter Carr and Dilip Madan develop a model for pricing options based on the observation of the implied volatilities of a series of options with the same expiration date Using their vi Introduction model, they obtain closed-form solutions for pricing plain-vanilla and exotic options in markets with a volatility skew Thomas Coleman and collaborators attack the problem of the volatility smile in a different way Their method combines the use of numerical optimization, spline approximations, and automatic differentiation They illustrate the effectiveness of their approach on both synthetic and real data for option pricing and hedging Leisen and Laurent consider a discrete model for option pricing based on Markov chains Their approach is based on finding a probability measure on the Markov chain which satisfies an optimality criterion Avellaneda, Buff, Friedman, Kruk and Newman develop a methodology for calibrating Monte Carlo models They show how their method can be used to calibrate models to the prices of traded options in equity and FX markets and to calibrate models of the term-structure of interest rates In the section entitled "Pricing and Risk-Management" Alexander Levin discusses a lattice-based methodology for pricing mortgage-backed securities Peter Carr and Guang Yang consider the problem of pricing Bermudan-style interest rate options using Monte Carlo simulation Alexander Lipton studies the symmetries and scaling relations that exist in the Black-Scholes equation and applies them to the valuation of path-dependent options Cardenas and Picron, from Summit Systems, describe accelerated methods for computing the Value-at-Risk of large portfolios using Monte Carlo simulation The closing paper, by Katherine Wyatt, discusses algorithms for portfolio optimization under structural requirements, such as trade amount limits, restrictions on industry sector, or regulatory requirements Under such restrictions, the optimization problem often leads to a "disjunctive program" An example of a disjunctive program is the problem to select a portfolio that optimally tracks a benchmark, subject to trading amount requirements I hope that you will find this collection of papers interesting and intellectually stimulating, as I did Marco Avellaneda New York, October 1999 ACKNOWLEDGEMENTS The Mathematical Finance Seminar was supported by the New York University Board of Trustees and by a grant from the Belibtreu Foundation It is a pleasure to thank these individuals and organizations for their support We are also grateful to the editorial staff of World Scientific Publishing Co., and especially to Ms Yubing Zhai THE CONTRIBUTORS Yacine Ait-Sahalia is Professor of Economics and Finance and Director of the Bendheim Center for Finance at Princeton University He was previously an Assistant Professor (1993-1996), Associate Professor (1996-1998) and Professor of Finance (1998) at the University of Chicago's Graduate School of Business, where he has been teaching MBA, executive MBA and Ph.D courses in investments and financial engineering He received the University of Chicago's GSB award for excellence in teaching and has been consistently ranked as one of the best instructors He was named an outstanding faculty by Business Week's 1997 Guide to the Best Business Schools Outside the GSB, Professor Ait-Sahalia has conducted seminars in finance for investment bankers and corporate managers, both in Europe and the United States He has also consulted for financial firms and derivatives exchanges in Europe, Asia and the United States His research concentrates on investments, fixed-income and derivative securities, and has been published in leading academic journals Professor Ait-Sahalia is a Sloan Foundation Research Fellow and has received grants from the National Science Foundation He is also an associate editor for a number of academic finance journals, and a Research Associate for the National Bureau of Economic Research He received his Ph.D in Economics from the Massachusetts Institute of Technology in 1993 and is a graduate of France's Ecole Polytechnique Marco Avellaneda is Professor of Mathematics and Director of the Division of Financial Mathematics at the Courant Institute of Mathematical Sciences of New York University He earned his Ph.D in 1985 from the University of Minnesota His research interests center around pricing derivative securities and in quantitative trading strategies He has also published extensively in applied mathematics, most notably in the fields of partial differential equations, the design of composite materials and hydrodynamic turbulence He was consultant for Banque Indosuez, New York, where he established a quantitative modeling group in FX options in 1996 Subsequently, he moved to Morgan Stanley & Co., as Vice-President in the Fixed-Income Division's Derivatives Products Group, where he remained until 1998, IX D E C O M P O S I T I O N A N D SEARCH T E C H N I Q U E S I N D I S J U N C T I V E P R O G R A M S FOR PORTFOLIO SELECTION KATHERINE WYATT Logic Based Systems Lab, Brooklyn College, City University of New York Received 30 August 1999 Introduction There are many problems in financial portfolio selection in which the primary objective of maximizing return and minimizing risk must be pursued subject to various constraints Some examples of these portfolio selection problems are analysis of different types of portfoios vis a vis benchmarks in historical simulations or stress tests; designing portfolios with particular characteristics that track an index across different scenarios; and making an optimal hedging designation for items and derivatives under SFAS 133 requirements ' These problems can be formulated as disjunctive programs, where the desired portfolio is described by means of logical disjunctions For example, some portfolio requirements that can be modeled with logical disjunctions are that the number of securities is an integer, or bounded by an integer; securities have to be purchased in set denominations; securities have to have certain credit ratings; or there is a minimum investment requirement for a portfolio In many portfolio selection problems, the function to be optimized can be expressed in terms of the absolute value of the difference between a portfolio's return and the return of some known collection of securities Using the L\ distance instead of variance produces disjunctive linear programs Finally, because analysis takes place across many scenarios, these disjunctive linear programs can be very large This paper presents a disjunctive formulation for a portfolio selection problem and an algorithm for decomposing the large linear programs that result with this model 20 A main feature of the algorithm is that variable decomposition of the linear programs is combined with branch and bound search among the disjunctive sets that describe the desired portfolio Background Markowitz ' 17 pioneered portfolio analysis using variance of returns as the definition of risk In mean variance portfolio analysis, the ratio of expected return 346 Decomposition and Search Techniques in Disjunctive Programs 347 to variance is maximized, usually as a quadratic programming problem Recently, Konno 4,5 advocated using mean absolute deviation of returns in portfolio selection, since mean absolute deviation can be represented by a piecewise linear risk function One method of portfolio optimization involves picking the best "tracking" portfolio, i.e., one whose returns are closest to those of a benchmark portfolio or index Worzel et al minimized the mean absolute deviation of portfolio and benchmark returns The absolute deviation trade-off model we present uses mean absolute deviation from a benchmark on interest rate scenarios as the measure of risk, and has as objective maximizing the difference between expected return and expected risk Optimizing mean absolute deviation can be modeled with linear constraints and a linear objective function, thus selecting the optimal portfolio is now a large linear problem The choice of an optimal portfolio is often subject to structural constraints There may be integral trading amount requirements for certain instruments, e.g., Treasury bills have to be purchased in increments of $5,000, after an initial investment of $10,000 Investors may demand a certain level of diversification or a certain credit grade for their portfolios Dedicated bond portfolios often have sector specifications that have to be satisfied All these are logical requirements; they can be modeled with disjunctions that describe allowable choices The complexity of quadratic programming makes adding disjunctive requirements to a traditional mean-variance model computationally prohibitive However, if absolute deviation is used to measure dispersion of returns instead of variance, 18,19 then structural requirements can be added to a portfolio selection model with linear constraints and a linear objective function The portfolio selection problem can then be represented as a disjunctive linear program An additional advantage of this formulation is that optimal portfolios for LP-based models have fewer nonzero holdings than portfolios found optimal for quadratic models, ' thereby reducing transaction costs Model Formulation The absolute deviation trade-off model is an extension of earlier mean absolute deviation models 4,5 ' We use the L\ distance from a given benchmark as the measure of risk and maximize the trade-off between the expected present value of reinvested cash flows and this L\ distance The objective function is similar in spirit to one utilized by Hiller and Eckstein;9 they used downside deviation from a liability stream as the measure of risk In this context, let Rij be the sample returns for asset j on scenario i; let benchi be the known returns for the benchmark on scenario i Then minimizing the risk for the absolute deviation trade-off model is equivalent to minimizing the function s N f(x) = 2_] benchi —2_.RiJxi i=l j=l • 348 Quantitative Analysis in Financial Markets As an example of this technique, we present a solution to the problem of selecting a portfolio from a universe of N securities that maximizes the trade-off between expected return over a set of scenarios and expected L\ distance from a benchmark's return over the scenarios, with the additional requirement that investment be at least v The objective function for this problem is to maximize Expected portfolio returns - | Expected difference of portfolio and benchmark returns \ This can be expressed as a linear function if we introduce variables for the difference between benchmark and portfolio returns on each scenario, and add constraints to force these variables to the absolute value of the difference We can then decompose the set of variables into the variables for securities, which may have disjunctive requirements, and the variables for the L\ distance on each scenario 3.1 The absolute deviation portfolio selection trade-off model for fixed-income We first present the linear model without logical requirements For our model we assume that we have a fixed universe of available instruments and that there is a cashflow model available to calculate the present value of each security's cash payments and embedded options Further, we have fixed a benchmark with known returns on the set of scenarios We assume that the only constraints are that all asset holdings are non-negative, i.e., there is no short selling, and there is a set budget for the investment If there are N available assets and the holding period for the portfolio comprises M dates on which cash flows are collected, then there are M scenarios If the number of scenarios becomes too large, sampling techniques or Monte Carlo simulation methods can be used in conjunction with the basic model In this research, we have used a discrete model of the future evolution of spot rates 10 For every asset, the present value of investing the cash flows at the current spot rate over the holding period of the portfolio is calculated A feasible portfolio is a linear combination of the assets for which the price does not exceed the budget A variable yi is introduced for every scenario i to model the absolute value of the difference between a portfolio and the benchmark yielded by this scenario Two constraints are added to the constraint set for every scenario: N Vi J^ryj yi < benchmarks u"=i ^ Vi [ -^njXj ) - V% < —benchmark^ • The objective function then is the difference between expected return and expected risk: Decomposition and Search Techniques in Disjunctive Programs 349 S N S Ti x Pi Vi max ^2 Pi Yl i i ~ ^ where pi is the probability associated with scenario i At optimality, since the yi are non-negative, subtracting the expected yi is equivalent to minimizing the yi, or forcing yi to the absolute value of the difference between benchmark and portfolio on each scenario We now have a linear problem with linear constraints instead of a piecewise linear problem The definitions and constraints for the model are as follows: Definitions N M Bi rij Pj Xj yi = = = = = = = = number of scenarios number of securities available maximum to be invested expected return of benchmark on scenario i expected return from security j on scenario i current price of security j amount of security j held in portfolio absolute value of the difference between expected portfolio and benchmark returns on scenario i v = minimum trade amount w = minimum investment amount Pi = probability associated with scenario i • The linear program that models the problem of selecting the best tracking portfolio is: { S N S ^2 Pi X] Tiixi ~ Y2 PM \ =ZP i=l s.t j'=l i=l pixi -\ — XI Vj Vi Vi "J —••• — -Xj rnxi + • • • + riNxN -rnxi —• Vi J +PN%N v and d € n r Ua Drs if for all j , dj = V dj = 3.2 Step-shaped programs The programs that correspond to the absolute deviation models share several family characteristics First, auxiliary variables are introduced to represent the absolute value of the difference between portfolio and benchmark returns; second, there are constraints on the primary variables in which the auxiliary variables don't appear; third, the constraints in which both the primary and auxiliary variables Decomposition and Search Techniques in Disjunctive Programs 351 appear not further constrain the primary variables; and, finally, the only constraints on the auxiliary variables alone are bounding constraints restricting their values to the non-negative orthant We will call the primary variables portfolio variables and the auxiliary variables risk variables These characteristics motivate the following definitions Definition Let R = {x € 5ftn : Ax < b} and let Dx + Ey < / be a system of linear inequalities Then Dx + Ey < / is free for x € R if for every x € R3y >0 such that (a;, y) satisfies Dx + Ey < / D Let x e 5K+ and y SR+ Consider the following linear program (P): (P) max s.t {ex + dy} = zp A\x + A\y < t A\x + A\y < b A\x + Aly < Definition Let R = {x € 5ft+ : A\x < i) A program (P) is a step-shaped program if the coefficient matrix for (P) can be written as above and if, in this formulation for (P), (1) A\ and A\ are zero submatrices, and (2) the constraints A\x + A\y < b are free for x € R D 3.3 Decomposition for step-shaped programs The absolute deviation trade-off model produces large programs with two kinds of variables, portfolio variables and risk variables, so it seems useful to explore variable decomposition methods for these programs Benders 11 used a partition of the variable set to solve mixed integer problems Van Slyke and Wets 12 reported on a method called L-shaped decomposition, which is akin to Benders decomposition, for large-scale and stochastic linear problems The form of the coefficient matrix for step-shaped programs is close to the Van Slyke and Wets description of L-shaped programs The shape of the non-zero submatrices we are considering differs from the L-shaped form, however, in that we need to explicitly consider the bounding constraints on y since we will be interested in dual multipliers for the constraints In fact, our definition of free makes precise an informal property 12 We will first outline variable decomposition of linear step-shaped programs 3.4 Variable decomposition for linear step-shaped Let (P) be a linear step-shaped program: (P) max s.t {ex + dy} = zp A\x + A\y < t A\x + A\yo}1 s/eG J G = {y£^s+:A\y