We develop a new Monte Carlo variance reduction method to estimate theexpectation of two commonly encountered path-dependent functionals:first-passage times and occupation times of sets. The method is based on arecursive approximation of the first-passage time probability and expectedoccupation time of sets of a Lévy bridge process that relies in part ona randomisation of the time parameter. We establish this recursion for generalLévy processes and derive its explicit form for mixed-exponentialjump-diffusions, a dense subclass (in the sense of weak approximation) ofLévy processes, which includes Brownian motion with drift, Kou'sdouble-exponential model, and hyper-exponential jump-diffusion models. Wepresent a highly accurate numerical realisation and derive error estimates. Byway of illustration the method is applied to the valuation of range accrualsand barrier options under exponential Lévy models and Bates-typestochastic volatility models with exponential jumps. Compared with standardMonte Carlo methods, we find that the method is significantly more efficient.
"Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications." J. Appl. Probab. 52 (4) 1076 - 1096, December 2015. https://doi.org/10.1239/jap/1450802754