Arbitrage-Free Discretization of Lognormal Forward Libor and Swap Rate Models

Posted: 17 Nov 1999

See all articles by Paul Glasserman

Paul Glasserman

Columbia Business School

Xiaoliang Zhao

Columbia University - Department of Statistics

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.

JEL Classification: G14, E43

Suggested Citation

Glasserman, Paul and Zhao, Xiaoliang, Arbitrage-Free Discretization of Lognormal Forward Libor and Swap Rate Models. Available at SSRN: https://ssrn.com/abstract=191609

Paul Glasserman (Contact Author)

Columbia Business School ( email )

3022 Broadway
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New York, NY 10027
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212-854-4102 (Phone)
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Xiaoliang Zhao

Columbia University - Department of Statistics ( email )

Mail Code 4403
New York, NY 10027
United States

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