Randomisation and recursion methods for mixed-exponential Levy models

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 L\'{e}vy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general L\'{e}vy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of L\'evy 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 L\'{e}vy 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.

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