Development of Computational Algorithms for Evaluating Option Prices Associated with Square-Root Volatility Processes

The stochastic volatility model of Heston (Rev Financ Stud 6:327–343, 1993) has been accepted by many practitioners for pricing various financial derivatives, because of its capability to explain the smile curve of the implied volatility. While analytical results are available for pricing plain Vanilla European options based on the Heston model, there hardly exist any closed form solutions for exotic options. The purpose of this paper is to develop computational algorithms for evaluating the prices of such exotic options based on a bivariate birth-death approximation approach. Given the underlying price process S t , the logarithmic process U t = logS t is first approximated by a birth-death process $B^U_t $ via moment matching. A second birth-death process $B^V_t $ is then constructed for approximating the stochastic volatility process V t through infinitesimal generator matching. Efficient numerical procedures are developed for capturing the dynamic behavior of $\{ B^U_t , B^V_t \} $ . Consequently, the prices of any exotic options based on the Heston model can be computed as long as such prices are expressed in terms of the joint distribution of { S t ,V t } and the associated first passage times. As an example, the prices of down-and-out call options are evaluated explicitly, demonstrating speed and fair accuracy of the proposed algorithms.