Arbitrage-Free Discretization of Lognormal Forward Libor and Swap Rate Models
AbstractAn 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.
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Bibliographic InfoPaper provided by Columbia - Graduate School of Business in its series Papers with number 98-09.
Length: 33 pages
Date of creation: 1998
Date of revision:
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Postal: U.S.A.; COLUMBIA UNIVERSITY, GRADUATE SCHOOL OF BUSINESS, PAINE WEBBER , New York, NY 10027 U.S.A
Phone: (212) 854-5553
Web page: http://www.columbia.edu/cu/business/
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SIMULATION ; INTEREST RATE ; ECONOMIC MODELS ; FINANCIAL MARKET;
Find related papers by JEL classification:
- G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies
- E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects
- C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
- C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General
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- Marek Rutkowski, 1999. "Models of forward Libor and swap rates," Applied Mathematical Finance, Taylor and Francis Journals, vol. 6(1), pages 29-60.
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