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


  • 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)


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.

Suggested Citation

  • Xiaoliang Zhao & Paul Glasserman, 2000. "Arbitrage-free discretization of lognormal forward Libor and swap rate models," Finance and Stochastics, Springer, vol. 4(1), pages 35-68.
  • Handle: RePEc:spr:finsto:v:4:y:2000:i:1:p:35-68
    Note: received: March 1998; final version received: January 1999

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    References listed on IDEAS

    1. Martin Kulldorff & Ajay Khanna, 1999. "A generalization of the mutual fund theorem," Finance and Stochastics, Springer, vol. 3(2), pages 167-185.
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    Cited by:

    1. Pietersz, R. & Pelsser, A.A.J., 2003. "Risk managing bermudan swaptions in the libor BGM model," Econometric Institute Research Papers EI 2003-33, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Christian Z├╝hlsdorff, 2002. "Extended Libor Market Models with Affine and Quadratic Volatility," Bonn Econ Discussion Papers bgse6_2002, University of Bonn, Germany.
    3. Frank De Jong & Joost Driessen & Antoon Pelsser, 2001. "Libor Market Models versus Swap Market Models for Pricing Interest Rate Derivatives: An Empirical Analysis," Review of Finance, European Finance Association, vol. 5(3), pages 201-237.
    4. Philipp J. Sch├Ânbucher, 2000. "A Libor Market Model with Default Risk," Bonn Econ Discussion Papers bgse15_2001, University of Bonn, Germany.

    More about this item


    Interest rate models; Monte Carlo simulation; market models;

    JEL classification:

    • G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading
    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects


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