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

Xiaoliang Zhao () and Paul Glasserman ()
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Xiaoliang Zhao: Department of Statistics, Columbia University, New York, NY 10027, USA Manuscript
Paul Glasserman: Graduate School of Business, Columbia University, Uris Hall, 3022 Broadway, Room 403, New York, NY 10027-6902, USA

Finance and Stochastics, 2000, vol. 4, issue 1, 35-68

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.

Keywords: Interest rate models; Monte Carlo simulation; market models (search for similar items in EconPapers)
JEL-codes: E43 G14 (search for similar items in EconPapers)
Date: 1999-10-29
Note: received: March 1998; final version received: January 1999
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Citations: View citations in EconPapers (3)

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