Fries, Christian P.; Kampen, Jörg: Proxy Simulation Schemes for generic robust Monte-Carlo sensitivities, process oriented importance sampling and high accuracy drift approximation (with applications to the LIBOR Market Model). (PDF, 5.3 MB). (Version 1.2.1 - August 24th, 2006).
We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation paths, where we use one simulation scheme K° (proxy scheme) to generate realizations and then reinterpret them as realizations of another scheme K* (target scheme) by adjusting measure (via likelihood ratio) to match the distribution of K* such that
E( f(K*) | Ft ) = E( f(K°) • w | Ft ).
This is done numerically in every time step, on every path.
This makes the approach independent of the product (the function f) and even of the model, it only depends on the numerical scheme.
The approach is essentially a numerical version of the likelihood ratio method [Broadie & Glasserman, 1996] and Malliavin's Calculus [Fournie et al., 1999; Malliavin, 1997] reconsidered on the level of the discrete numerical simulation scheme.
Since the numerical scheme represents a time discrete stochastic process sampled on a discrete probability space the essence of the method may be motivated without a deeper mathematical understanding of the time continuous theory (e.g. Malliavin's Calculus).
We present numerical results using a Monte-Carlo simulation of the LIBOR Market Model for benchmarking.
Fries, Christian P.; Mark, Joshi S.: Partial Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks. 2006. Version 1.1.0.
Download form SSRN.
We consider a generic framework which allows to calculate robust Monte-Carlo sensitivities seamlessly through simple finite difference approximation. The method proposed is a generalization and improvement of the proxy simulation scheme method (Fries and Kampen, 2005). As a benchmark we apply the method to the pricing of digital caplets and target redemption notes using LIBOR and CMS indices under a LIBOR Market Model. The framework is generic in the sense that it is model and almost product independent. The only product dependent part is the specification of the proxy constraint. This allows for an elegant implementation, where new products may be included at small additional costs.
Fries, Christian P.: Localized Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks. (2007)
Download form SSRN.
For the numerical calculation of partial derivatives (aka.~sensitivites or greeks) from a Monte-Carlo 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 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.
I will post more numerical results here soon.