I found a paper devoted to my question: The working paper is at
http://www.lehrstab-statistik.de/downlo ... rownWP.pdf .
Computational Management Science
January 2010, 7:1
Exact simulation of final, minimal and maximal values of Brownian motion and jump-diffusions with applications to option pricing
- 1.
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Original Paper
First Online: 28 November 2007[/font]
Abstract
We introduce a method for generating
[ltr]
(W(μ,σ)x,T,m(μ,σ)x,T,M(μ,σ)x,T)(Wx,T(μ,σ),mx,T(μ,σ),Mx,T(μ,σ))[/ltr]
, where
[ltr]
W(μ,σ)x,TWx,T(μ,σ)[/ltr]
denotes the final value of a Brownian motion starting in x with drift μ and volatility σ at some final time T,
[ltr]
m(μ,σ)x,T=inf0≤t≤TW(μ,σ)x,tmx,T(μ,σ)=inf0≤t≤TWx,t(μ,σ)[/ltr]
and
[ltr]
M(μ,σ)x,T=sup0≤t≤TW(μ,σ)x,tMx,T(μ,σ)=sup0≤t≤TWx,t(μ,σ)[/ltr]
. By using the trivariate distribution of
[ltr]
(W(μ,σ)x,T,m(μ,σ)x,T,M(μ,σ)x,T)(Wx,T(μ,σ),mx,T(μ,σ),Mx,T(μ,σ))[/ltr]
, 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.
Keywords
Brownian motion Monte Carlo simulation Jump-diffusions Double barrier options