Applied Maths Seminar: Griebsch -- The Evaluation of European Compound Option Prices under Stochastic Volatility using Fourier Transform Techniques

Susanne Griebsch, School of Finance and Economics, University of Technology Sydney 

This study focuses on European compound option pricing under stochastic volatility
dynamics.  Since a compound option is an option on an option, its value is not only
sensitive to future movements of the underlying asset price, but also to future changes
of volatility levels.  Despite the existence of an analytical valuation formula for this
type of option, the Black-Scholes model is not able to incorporate this sensitivity with
respect to volatility in the option value.  One approach to take the forward volatility
into account is to value the option within a stochastic volatility model.  Hence, the
aim of this work is to develop a pricing procedure for compound options in stochastic
volatility models, specifically focusing on the model of Heston (1993).  In this case,
the difficulty in the development of pricing methods compared to the Black-Scholes model
lies in the additional dimension of uncertainty coming from the stochastic volatility.  

Here, a numerical pricing algorithm is developed to solve this problem.  It exploits
that the representation of the compound option value can be divided into the difference
of three probabilities under two different probability measures.  These probabilities
depend on three random variables (the future volatility and two future spot values)
through a complex functional form.  The joint distribution of these random variables
under both measures can be uniquely determined by their joint characteristic function
for each measure and therefore the probabilities can each be expressed as a multiple
inverse Fourier transform.  Solving the inverse Fourier transform with respect to
volatility, the pricing problem is reduced to two dimensions and approximations of the
three probabilities are obtained through a numerical fast Fourier transform technique.
The resulting approximations are then compared with other numerical methods such as
Monte Carlo simulations, showing promising results.  

http://www.maths.usyd.edu.au/u/AppliedSeminar/abstracts/2010/griebsch.html