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.

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Correspondence to Martin Becker.

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Becker, M. Exact simulation of final, minimal and maximal values of Brownian motion and jump-diffusions with applications to option pricing. Comput Manag Sci 7, 1 (2010). https://doi.org/10.1007/s10287-007-0065-9

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Keywords

  • Brownian motion
  • Monte Carlo simulation
  • Jump-diffusions
  • Double barrier options
  • Importance sampling