Table Of ContentQuantitative 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 New 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
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QUANTITATIVE ANALYSIS IN FINANCIAL MARKETS:
Collected Papers of the New York University Mathematical Finance Seminar, Volume II
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INTRODUCTION
It is a pleasure to edit the second volume of papers presented at the Mathema
tical 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 deriva
tives. 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 minimum-
variance 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 non-
parametric 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 im
plied 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 op
tion 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 dis
cusses 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 pro
gram" . 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 excel
lence 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 re
ceived 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 Na
tional 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 quantita
tive 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,
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