Randomisation and recursion methods for mixed-exponential Levy models, with financial applications

TL;DR

This paper introduces a recursive Monte Carlo variance reduction technique for efficiently estimating path-dependent functionals like first-passage and occupation times in Levy processes, with applications in financial derivatives valuation.

Contribution

It develops a new recursive approximation method for Levy bridges, specifically for mixed-exponential jump-diffusions, improving computational efficiency in financial modeling.

Findings

Method significantly outperforms standard Monte Carlo in efficiency.

Accurate error estimates and numerical implementation details provided.

Applied successfully to valuation of range accruals and barrier options.

Abstract

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Levy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Levy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Levy processes, which includes Brownian motion with drift, Kou's double-exponential model and hyper-exponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Levy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.