文档介绍：The Pricing of Exotic Options by Monte-Carlo
Simulations in a L´evy Market with Stochastic
Volatility
Wim Schoutens∗and Stijn Symens†
October 17, 2002
∗, Celestijnenlaan 200 B, B-3001 Leuven, Belgium. E-mail:
Wim.******@
†University of Antwerp, Universiteitsplein 1, B-2610 Wilrijk, Belgium. E-mail:
stijn.******@
1
Abstract
Recently, stock price models based on L´evy processes with stochastic
volatility were introduced. The resulting vanilla option prices can be
calibrated almost perfectly to empirical prices. Under this model, we will
price exotic options, like the barrier, lookback and cliquet options, by
Monte-Carlo simulation. The sampling of paths is based on pound
Poisson approximation of the L´evy process involved. The precise choice
of the terms in the approximation is crucial and investigated in detail.
In order to reduce the standard error of the Monte-Carlo simulation, we
make use of the technique of control variates. It turns out that there are
significant differences with the classical Black-Scholes prices.
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1 Introduction
The most famous continuous-time model for stock prices or indices is the cel-
ebrated Black-Scholes model (BS-model) [11]. It uses the Normal distribution
to fit the log-returns of the underlying: the price process of the underlying is
given by the geometric Brownian Motion
σ2
S = S exp µ t + σW ,
t 0 − 2 t
where Wt, t 0 is standard Brownian motion, . Wt follows a Normal
distribution{ with≥ mean} 0 and variance t. Under this model pricing formulae for
a variety of options are available. We are particularly interested in the pricing
of so-called exotic options of European nature, . the payoff function can be
path-dependent, however there is a fix maturity date and no-early exercise is
allowed.
Path-dependent options have e popular in the OTC market in the last
decades. Examples of these exotic path-dependent options are lookback options