Arbitrage-free discretization of lognormal forward Libor and swap rate models

Abstract

Abstract. An important recent development in the pricing of interest rate derivatives is the emergence of models that incorporate lognormal volatilities for forward Libor or forward swap rates while keeping interest rates stable. These market models have three attractive features: they preclude arbitrage among bonds, they keep rates positive, and, most distinctively, they price caps or swaptions according to Black's formula, thus allowing automatic calibration to market data. But these features of continuous-time formulations are easily lost when the models are discretized for simulation. We introduce methods for discretizing these models giving particular attention to precluding arbitrage among bonds and to keeping interest rates positive even after discretization. These methods transform the Libor or swap rates to positive martingales, discretize the martingales, and then recover the Libor and swap rates from these discretized variables, rather than discretizing the rates themselves. Choosing the martingales proportional to differences of ratios of bond prices to numeraire prices turns out to be particularly convenient and effective. We can choose the discretization to price one caplet of arbitrary maturity without discretization error. We numerically investigate the accuracy of other caplet and swaption prices as a gauge of how closely a model calibrated to implied volatilities reproduces market prices. Numerical results indicate that several of the methods proposed here often outperform more standard discretizations.

Fast greeks by simulation in forward LIBOR models

“…The question of discretization of (9) is investigated by Glasserman and Zhao (1999), where it is shown that there are advantages to discretizing SDEs for de¯ated bond prices (or their increments) rather than the forward LIBOR rates themselves. Dierentiating these SDEs leads to a set of derivative SDEs analogous to (10) and it is possible to simulate a discrete-time approximation to those.…”

A Libor Market Model with Default Risk

“…The question of calibration is addressed by Rebonato (1998;1999a;1999b), advanced techniques for Monte-Carlo simulation can be found e.g. in Glasserman and Zhao (2000) and the survey article by Broadie and Glasserman (1998). On the background of this large literature we restrict ourselves to the details of the implementation that are specific to the case of credit risk modelling.…”

A Comparison of Control Variate Methods for Pricing Interest Rate Derivatives in the LIBOR Market Model

“…Brace, Gatarek and Musiela [1] introduced the Frozen drift approximation. Glasserman and Zhao [3] studied the arbitrage-free discretizational approximations of the LMM. Hull and White [4] , Jackel and Rebonato [5] developed the methods of approximation for the LMM by high dimensional lognormal processes.…”