  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 Fournie et al [13] and Davis and Johansson [9] 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, Lutkebohmert 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.
