A Mixed PDE/Monte Carlo approach as an efficient way to price under high-dimensional systems

Abstract

We propose to price derivatives modelled by multi-dimensional systems of stochastic di�fferential\ud equations using a mixed PDE/Monte Carlo approach. We derive a stochastic PDE where some of the coeffi�cients are conditional on stochastic ancillary factors. The stochastic\ud PDE is solved with either analytical or �finite diff�erence methods, where we simulate all the ancillary processes using Monte Carlo. The multilevel technique has also been introduced to further reduce the variance. The combined method showed over 80% cost reduction for the same accuracy, in pricing a barrier option in an FX market with stochastic interest rate and volatility (which is usually expensive to work with) , when compared to the pure Monte\ud Carlo simulation