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**Using Brownian Bridge for Fast Simulation of Jump-Diffusion Processes and Barrier Options**
by Steve A.K. Metwally of Lehman Brothers, and Amir F. Atiya of Cairo University Fall 2002 **Abstract:** Barrier options are one of the most popular derivatives in the financial markets. The authors present a fast and unbiased Monte Carlo approach to pricing barrier options when the underlying security follows a simple jump-diffusion process with constant parameters and a continuously monitored barrier. Two algorithms are based on the Brownian bridge concept. The first one is based on a sampling approach to evaluate an integral that results from application of the Brownian bridge. The second approach approximates that integral using a Taylor series expansion.
Both methods significantly reduce bias and speed convergence compared to the standard Monte Carlo simulation approach. For example, the first method achieves zero bias. In addition, it is about 100 times faster than the conventional Monte Carlo method that achieves acceptable bias. In developing the second algorithm, the authors derive a novel approach for obtaining a first-passage time density integral using a Taylor series expansion. This approach is potentially useful in other applications, where the expectation of some function over the first-passage time distribution needs to be derived.
**Published in:** *Journal of Derivatives*, Vol. 10, No. 1, (Fall 2002), pp. 43-54.
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