Abstracts

Required in a wide range of applications in, e.g., finance, engineering, and physics, first-passage time problems have attracted considerable interest over the past decades. Since analytical solutions often do not exist, one strand of research focuses on fast and accurate numerical techniques. In this paper, we present an efficient and unbiased Monte-Carlo simulation to obtain double-barrier first-passage time probabilities of a jump-diffusion process with arbitrary jump size distribution.
We rely on numerical schemes and provide an unbiased, fast, and accurate Monte Carlo simulation based on the so-called ``Brownian bridge technique''. It proceeds as follows: First, the jump-instants of the process in consideration as well as the process immediately before and after the jump times are simulated. In between these generated points, one has a pure diffusion with fixed endpoints. Here, the so-called Brownian bridge probabilities provide an analytical expression for the first- passage time on a given threshold.
This simulation technique turns out to be (1) unbiased and (2) significantly faster than the standard Monte-Carlo simulation. We show how the Brownian bridge technique can be adapted to a large variety of exotic double barrier products. Those products are very flexible and thus allow investors to adapt to their specific hedging needs or speculative views. However, those contracts can hardly be traded without a fast and reliable pricing technique. Analytical solutions reach their limitations as they are often not flexible enough to adapt to complicated payoff streams and/or jump size distributions. To provide this flexibility for the Brownian bridge technique, we extend the existing algorithms and (1) allow to price double barrier derivatives that trigger different events depending on which barrier was hit first (a feature that is required to price, e.g., corridor bonus certificates) and (2) allow to evaluate payoff streams that depend on the first-passage time (a feature important in, e.g., structural credit risk models). Furthermore, (3) we show that time dependent barriers can easily be treated (a feature that is relevant for, e.g., window or step double barrier options). Finally, we discuss the implementation and show that -- in contrast to most alternative techniques -- the Brownian bridge algorithms are easy to understand and implement.