Probabilistic Numerical Approach for PDE and Its Application in the Valuation of European Options

Abstract

This paper suggests a probabilistic numerical approach for a class of PDE. First of all, by simulating Brownian motion and using Monte-Carlo method, we obtain a probabilistic numerical solution for the PDE. Then, we prove that the probabilistic numerical solution converges in probability to its solution. At the end of this paper, as an application, we give a probabilistic numerical approach for the valuation of European Options, where we see volatility $\sigma$, interest rate $r$ and divident rate $D_0$ as functions of stock $S$, respectively.