A Monte Carlo Method using PDE Expansions for a Diversifed Equity Index Model

This paper considers a new class of Monte Carlo methods that are combined with PDE expansions for the pricing and hedging of derivative securities for multidimensional diffusion models. The proposed method combines the advantages of both PDE and Monte Carlo methods and can be directly applied to models with more than two state variables. The pricing procedure is illustrated using a three-component index model that captures some of the key features of a diversied stock index over long time periods. The method is widely applicable and is demonstrated here in the general setting of the benchmark approach, where spatial boundary limiting conditions for the PDE need to be appropriately chosen and approximated. The PDE expansion is based on a Taylor series approximation for the underlying three-component PDE. A Monte Carlo method with variance reduction is then formulated to approximate the true solution. Almost exact simulation schemes are described for the given state variables in the model. Numerical results are presented that demonstrate the effectiveness and tractability of the proposed pricing and hedging methodology.