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The Bismut-Elworthy-Li formula for jump-diffusions and applications to Monte (2007)
| Citations: | 2 - 0 self |
BibTeX
@MISC{Cass07thebismut-elworthy-li,
author = {Thomas R. Cass and Peter K. Friz},
title = {The Bismut-Elworthy-Li formula for jump-diffusions and applications to Monte},
year = {2007}
}
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Abstract
We extend the Bismut-Elworthy-Li formula to non-degenerate jump diffusions and ”payoff ” functions depending on the process at multiple future times. In the spirit of Fournié et al [14] and Davis and Johansson [10] this can improve Monte Carlo numerics for stochastic volatility models with jumps. To this end one needs so-called Malliavin weights and we give explicit formulae valid in presence of jumps: (a) In a non-degenerate situation, the extended BEL formula represents possible Malliavin weights as Ito integrals with explicit integrands; (b) in a hypoelliptic setting we review work of Arnaudon and Thalmaier [1] and also find explicit weights, now involving the Malliavin covariance matrix, but still straight-forward to implement. (This is in contrast to recent work by Forster, Lütkebohmert and Teichmann where weights are constructed as anticipating Skorohod integrals.) We give some financial examples covered by (b) but note that most practical cases of poor Monte Carlo performance, Digital Cliquet contracts for instance, can be dealt with by the extended BEL formula and hence without any reliance on Malliavin calculus at all. We then discuss some of the approximations, often ignored in the literature, needed to justify the use of the Malliavin weights in the context of standard jump diffusion models. Finally, as all this is meant to improve numerics, we give some numerical results with focus on Cliquets under the Heston model with jumps. 1 1
Keyphrases
bismut-elworthy-li formula extended bel formula explicit formula valid payoff function numerical result financial example multiple future time poor monte carlo performance recent work hypoelliptic setting malliavin covariance matrix explicit weight stochastic volatility model standard jump diffusion model heston model possible malliavin weight monte carlo numerics ito integral non-degenerate jump diffusion so-called malliavin weight explicit integrands malliavin weight practical case non-degenerate situation digital cliquet contract skorohod integral malliavin calculus
