Exact simulation of final, minimal and maximal values of Brownian motion and jump-diffusions with applications to option pricing

Abstract

We introduce a method for generating $(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)})$ , where $W_{x,T}^{(\mu,\sigma)}$ denotes the final value of a Brownian motion starting in x with drift μ and volatility σ at some final time T, $m_{x,T}^{(\mu,\sigma)} = {\rm inf}_{0\leq t \leq T}W_{x,t}^{(\mu,\sigma)}$ and $M_{x,T}^{(\mu,\sigma)} = {\rm sup}_{0\leq t \leq T} W_{x,t}^{(\mu,\sigma)}$ . By using the trivariate distribution of $(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)})$ , we obtain a fast method which is unaffected by the well-known random walk approximation errors. The method is extended to jump-diffusion models. As sample applications we include Monte Carlo pricing methods for European double barrier knock-out calls with continuous reset conditions under both models. The proposed methods feature simple importance sampling techniques for variance reduction.