Monte Carlo Variance Reduction by Conditioning for Pricing with Underlying a Continuous-Time Finite State Markov Process

Abstract

We consider the pricing of derivatives when the evolution of the underlying is given by a continuous time finite state Markov chain. We present a semianalytic approach that consists in (i) simulating the number of transitions of the underlying up to a given time horizon, (ii) computing via an explicit analytic formula the derivative price for each simulated number of transitions, and (iii) approximating the actual price by the empirical average over the values computed in (ii). This corresponds to a Monte Carlo approach with variance reduction by conditioning and, with respect to a plain Monte Carlo, it thus leads to a smaller variance in addition to more precise values. The method is applied, in particular, to path dependent derivatives, and numerical results are presented and discussed.