Fast Monte Carlo Bermudan Greeks

In recent years, much effort has been devoted to improving the efficiency of the Libor market model. Matthias Leclerc, Qian Liang and Ingo Schneider extend the pioneering work of Giles & Glasserman (2006) and show how fast calculations of Monte Carlo Greeks are feasible even within the framework of Bermudan-style derivatives. The authors demonstrate the efficiency gains in detail

Fast pricing and calculating hedging parameters are still a challenge in the framework of the Libor market model (LMM), which has become the fundamental pricing model in the fixed-income environment. Traditionally, for fixed-income securities, Greeks are calculated by the so-called bump and revalue method: each initial forward rate is perturbed by a basis-point shift and then the security is valued again. Besides the simplicity, there is no further advantage. The LMM is usually implemented with Monte Carlo methods and this can be rather slow, especially using the perturbation described before. The simulation procedure in the LMM is done in a forward measure and so the natural way to calculate Greeks is to do them on the fly. Giles & Glasserman (2006) have shown that under specific circumstances, their adjoint method can be suitable to get the Greeks a lot faster and save a considerable amount of computation time.

The rest of this article is structured as follows. We describe the dynamics of the LMM and fix notations. We review the basic forward calculations of pathwise Greeks (delta and vega) and then describe the fundamental principles of the adjoint method, both for European-style derivatives only. Next, we describe how the usual forward framework can be extended to value Bermudan options: after a forward procedure we need to work backwards to calculate the optimal exercise times. Based on this idea, we develop the modifications necessary for the adjoint method. Numerical applications demonstrate the extended adjoint method. Interestingly, both adjoint extensions are based on the originally developed pathwise forward method.

Matthias Leclerc is executive director at Value&Risk, Frankfurt and professor of mathematics at the University of Augsburg. Qian Liang is a graduate student in mathematics at the University of Kaiserslautern & Fraunhofer ITWM. Ingo Schneider is head of financial engineering at DekaBank. The result of this article is based on the diploma thesis of the second author at the University of Augsburg. Email: [e-mail]

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