A Monte Carlo approach to value exchange options
using a single stochastic factor
Giovanni VillaniAbstract.This article describes an important sampling regarding modification of the Monte
Carlo method in order to minimise the variance of simulations. In a particular way, we propose
a generalisation of the antithetic method and a newa-sampling of stratified procedure with
a=^12 to value exchange options using a single stochastic factor. As is well known, exchange
options give the holder the right to exchange one risky assetVfor another risky assetDand
therefore, when an exchange option is valued, we generally are exposed to two sources of
uncertainity. The reduction of the bi-dimensionality of valuation problem to a single stochastic
factor implies a new stratification procedure to improve the Monte Carlo method. We also
provide a set of numerical experiments to verify the accuracy derived bya-sampling.Key words:exchange options, Monte Carlo simulations, variance reduction1 Introduction
Simulations are widely used to solve option pricing. With the arrival of ever faster
computers coupled with the development of new numerical methods, we are able to
numerically solve an increasing number of important security pricing models. Even
where we appear to have analytical solutions it is often desirable to have an alternative
implementation that is supposed to give the same answer. Simulation methods for
asset pricing were introduced in finance by Boyle [3]. Since that time simulation has
been successfully applied to a wide range of pricing problems, particularly to value
American options, as witnessed by the contributions of Tilley [10], Barraquand and
Martineau [2], Broadie and Glasserman [4], Raymar and Zwecher [9].
The aim of this paper is to improve the Monte Carlo procedure in order to evaluate
exchange options generalizing the antithetic variate methodology and
proposing a new stratification procedure. To realise this objective, we price the most
important exchange options using a single stochastic factorPthat is the ratio between
the underlying assetVand the delivery oneD. For this reason, we need a particular
sampling to concentrate the simulations in the range in which the functionPis more
sensitive.M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010