Localized Proxy Simulation Schemes

For the numerical calculation of partial derivatives (aka. sensitivites or greeks) from a MonteCarlo simulation there are essentially two possible approaches: The pathwise method and the likelihood ratio method. Both methods have their shortcomings: While the pathwise method works very well for smooth payouts it fails for discontinuous payouts. On the other hand, the likelihood ratio gives much better results on discontinuous payouts, but falls short of the pathwise method if smooth payouts are considered. In this paper, we present a modification to the (partial) proxy simulation scheme framework, resulting in a per-path selection of either the pathwise method or the likelihood ratio method. This allows us to chose the optimal simulation method on a path-by-path basis. Since the method is implemented as a proxy simulation scheme as well, the sensitivities can be calculated from simple finite differences applied to the pricing engine.