Monte Carlo Pricing Using Operator Methods and Measure Changes

Abstract

A large class of generic stochastic processes which are not necessarily analytically solvable but are still numerically tractable can be described by giving transition probability kernels over a contiguous set of time intervals. From the numerical viewpoint, this procedure is highly effective on current microchip architectures as kernels can be conveniently evaluated using GPU co-processors and then used for scenario generation while storing them in CPU caches. This paper describes the pricing methodology and a mathematical framework for Finance based on direct kernel manipulations, i.e. operator methods. We also discuss a number of techniques based on measure changes to accomplish tasks such as variance reduction and sensitivity calculations. Numerical experiments are included along with performance benchmarks. Source code is distributed separately online under GPL license in a library named OPLib.

Keywords: pricing theory, monte carlo algorithms, likelihood ratio method, variance reduction, fundamental theorem, operator methods