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

Part of: Probabilistic methods, simulation and stochastic differential equations Mathematical finance

Published online by Cambridge University Press:  30 March 2016

Aleksandar Mijatović ,
Martijn R. Pistorius  and
Johannes Stolte
Aleksandar Mijatović*
Affiliation:
Imperial College London
Martijn R. Pistorius*
Affiliation:
Imperial College London
Johannes Stolte*
Affiliation:
Imperial College London
*
∗Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
∗Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
∗Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
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Abstract

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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 Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential 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 Lévy 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.

Keywords

Lévy bridge processmixed-exponential jump-diffusionfirst-passage timeoccupation timeMarkov bridge sampling.

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 91G60: Numerical methods (including Monte Carlo methods)
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1076 - 1096
Copyright
Copyright © Applied Probability Trust 2015 

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