The Parallel Solution of Early-exercise Asian Options with Stochastic Volatility

This paper describes an parallel semi-Lagrangian nite diierence approach to the pricing of early exercise Asian Options on assets with a stochastic volatility. A multigrid procedure is described for the fast iterative solution of the discrete linear complementarity problems that result. The accuracy and performance of this approach is improved considerably by a strike-price related analytic transformation of asset prices. Asian options are contingent claims with payoos that depend on the average price of an asset over some time interval. The payoo may depend on this average and a xed strike price (Fixed Strike Asians) or it may depend on the average and the asset price (Floating Strike Asians). The option may also permit early exercise (American contract) or connne the holder to a xed exercise date (European contract). The Fixed Strike Asian with early exercise is considered here where continuous arithmetic averaging has been used. Pricing such an option where the asset price has a stochastic volatility leads to the requirement to solve a tri-variate partial diierential inequation in the three state variables of asset price, average price and volatility (or equivalently, variance). The similarity transformations 6] used with Floating Strike Asian options to reduce the dimensionality of the problem are not applicable to Fixed Strikes and so the numerical solution of a tri-variate problem is necessary. The computational challenge is to provide accurate solutions suuciently quickly to support real-time trading activities at a reasonable cost in terms of hardware requirements. 1.2 American options with stochastic volatility American Asian options with a stochastic volatility contain the vanilla American stochastic volatility option as a sub-problem; consequently the solution of this sub-problem (see 4] for more details) is summarised here. A standard American option with a stochastic volatility has a share price process S t and its variance process Y t (the variance has been used rather than the volatility p Y t) which satisfy dS t = S t dt + p Y t S t dB 1 t (1) dY t = (? Y t)dt + p Y t dB 2 t (2)