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

摘要

We introduce a method for generating (W_(x,T)~((μ,σ)), m_(x,T)~((μ,σ)) , M_(x,T)~((μ,σ)) ), where W_(x,T)~((μ,σ)) 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)~((μ,σ)) =inf0≤t≤T W_(x,T)~((μ,σ)) and M_(x,T)~((μ,σ))=rnsup0≤t≤T W_(x,T)~((μ,σ)). By using the trivariate distribution of (W_(x,T)~((μ,σ)),m_(x,T)~((μ,σ)), M_(x,T)~((μ,σ))), 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.