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A Multifactor Spot Rate Model for the Pricing of Interest Rate Derivatives

Published online by Cambridge University Press:  06 April 2009

Sandra Peterson ,
Richard C. Stapleton  and
Marti G. Subrahmanyam
Sandra Peterson
Affiliation:
drsjp@btopenworld.com, Scottish Institute for Research in Investment and Finance, Strathclyde University, Glasgow, UK;
Richard C. Stapleton
Affiliation:
richard.stapleton@btinternet.com, Manchester School of Accounting and Finance, University of Manchester, Manchester, UK, and University of Melbourne, Australia;
Marti G. Subrahmanyam
Affiliation:
msubrahm@stern.nyu.edu, Stern School of Business, New York University, Management Education Center, New York, NY 10012.
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Abstract

We propose a multifactor model in which the spot rate, LIBOR, follows a lognormal process, with a stochastic conditional mean, under the risk-neutral measure. In addition to the spot rate factor, the second factor is related to the premium of the first futures rate over the spot LIBOR. Similarly, the third factor is related to the premium of the second futures rate over the first futures rate. We calibrate the model to the initial term structure of futures rates and to the implied volatilities of interest rate caplets. We then apply the model to price interest rate derivatives such as European and Bermudan-style swaptions, and yieldspread options. The model can be employed to price more complex interest rate derivatives such as path-dependent derivatives or multi-currency-dependent derivatives because of its Markovian property.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 38 , Issue 4 , December 2003 , pp. 847 - 880
Copyright
Copyright © School of Business Administration, University of Washington 2003

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References

Ahn, D.; Dittmar, R.; and Gallant, A. R.. “Quadratic Term Structure Models: Theory and Evidence.” Review of Financial Studies, 15 (2002), 243–288.CrossRefGoogle Scholar
Amin, K. I. “On the Computation of Continuous-Time Option Prices using Discrete Approximations.” Journal of Financial and Quantitative Analysis, 26 (1991), 447–495.CrossRefGoogle Scholar
Andersen, L. “A Simple Approach to the Pricing of Bermudan Swaptions in the Multifactor LIBOR Market Model.” Journal of Computational Finance, 3 (2000), 5–32.CrossRefGoogle Scholar
Andersen, L., and Andreasen, J.. “Factor Dependence of Bermudan Swaptions: Fact or Fiction?” Journal of Financial Economics, 62 (2001), 3–38.CrossRefGoogle Scholar
Andersen, L., and Broadie, M.. “A Primal-Dual Simulation Algorithm for Pricing Multi-Dimensional American Options.” Working Paper, Columbia Univ. (2001).Google Scholar
Balduzzi, P.; Das, S. R.; and Foresi, S.. “The Central Tendency: A Second Factor in Bond Yields.” The Review of Economics and Statistics, 80 (1998), 62–72.CrossRefGoogle Scholar
Black, F.; Derman, E.; and Toy, W.. “A One-Factor Model of Interest Rates and its Application to Treasury Bond Options.” Financial Analysts Journal, 46 (1990), 33–39.CrossRefGoogle Scholar
Black, F., and Karasinski, P.. “Bond and Option Pricing when Short Rates are Lognormal.” Financial Analysts Journal, 47 (1991), 52–59.CrossRefGoogle Scholar
Boyle, P.; Broadie, M.; and Glasserman, P.. “Simulation Methods for Security Pricing.” Journal of Economic Dynamics and Control, 21 (1997), 1267–1321.CrossRefGoogle Scholar
Brace, A.; Gatarek, D.; and Musiela, M.. “The Market Model of Interest Rate Dynamics.” Mathematical Finance, 7 (1997), 127–155.CrossRefGoogle Scholar
Brennan, M. J., and Schwartz, E. S.. “A Continuous-Time Approach to the Pricing of Bonds.” Journal of Banking and Finance, 3 (1979), 135–155.CrossRefGoogle Scholar
Brennan, M. J., and Schwartz, E. S.. “An Equilibrium Model of Bond Pricing and a Test of Market Efficiency.” Journal of Financial and Quantitative Analysis, 17 (1982), 301–329.CrossRefGoogle Scholar
Broadie, M., and Glasserman, P.. “Pricing American-Style Securities Using Simulation.” Journal of Economic Dynamics and Control, 21 (1997), 1323–1352.CrossRefGoogle Scholar
Collin-Dufresne, P., and Goldstein, R. S.. “Pricing Swaptions within an Affine Framework.” Working Paper, Carnegie-Mellon Univ. (2001a).CrossRefGoogle Scholar
Collin-Dufresne, P., and Goldstein, R. S.. “Stochastic Correlation and the Relative Pricing of Caps and Swaptions in a Generalized-affine Framework.” Working Paper, Carnegie-Mellon Univ. (2001b).CrossRefGoogle Scholar
Courtadon, G. “The Pricing of Options on Default-Free Bonds.” Journal of Financial and Quantitative Analysis, 17 (1982), 75–100.CrossRefGoogle Scholar
Cox, J. C.; Ingersoll, J. E.; and Ross, S. A.. “The Relationship between Forward Prices and Futures Prices.” Journal of Financial Economics, 9 (1981), 321–346.CrossRefGoogle Scholar
Cox, J. C.; Ingersoll, J. E.; and Ross, S. A.. “A Theory of the Term Structure of Interest Rates.” Econometrica, 53 (1985), 385–407.CrossRefGoogle Scholar
Dai, Q., and Singleton, K. J.. “Specification Analysis of Affine Term Structure.” Journal of Finance, 55 (2000), 1943–1978CrossRefGoogle Scholar
Driessen, J.; Klaassen, P.; and Melenberg, B.. “The Performance of Multi-Factor Term Structure Models for Pricing and Hedging Caps and Swaptions.” Journal of Financial and Quantitative Analysis, 38 (2003), 635–672.CrossRefGoogle Scholar
Duffee, G. “Term Premia and Interest Rate Forecasts in Affine Models.” Journal of Finance, 57 (2002), 406–444.CrossRefGoogle Scholar
Fan, R.; Gupta, A.; and Ritchken, P.. “Hedging in the Possible Presence of Unspanned Stochastic Volatility: Evidence from Swaption Markets.” Journal of Finance, 58 (2003), 2219–2248.CrossRefGoogle Scholar
Gupta, A., and Subrahmanyam, M. G.. “An Empirical Examination of the Convexity Bias in the Pricing of Interest Rate Swaps.” Journal of Financial Economics, 55 (2000), 239–279.CrossRefGoogle Scholar
Gupta, A., and Subrahmanyam, M. G.. “Pricing and Hedging Interest Rate Options: Evidence from Cap-Floor Markets.” Working Paper, New York Univ. (2002).Google Scholar
Heath, D. “Some New Term Structure Models.” Working Paper, Carnegie-Mellon Univ. (1998).Google Scholar
Heath, D.; Jarrow, R. A.; and Morton, A.. “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation.” Econometrica, 60 (1992), 77–105.CrossRefGoogle Scholar
Ho, T. S. Y., and Lee, S. B.. “Term Structure Movements and Pricing of Interest Rate Claims.” Journal of Finance, 41 (1986), 1011–1029.CrossRefGoogle Scholar
Ho, T. S.; Stapleton, R. C.; and Subrahmanyam, M. G.. “Multivariate Binomial Approximations for Asset Prices with Non-Stationary Variance and Covariance Characteristics.” Review of Financial Studies, 8 (1995), 1125–1152.CrossRefGoogle Scholar
Hull, J., and White, A.. “Numerical Procedures for Implementing Term Structure Models II: Two Factor Models.” Journal of Derivatives, 2 (1994), 37–48.CrossRefGoogle Scholar
Jagannathan, R.; Kaplin, A.; and Sun, S. G.. “An Evaluation of Multi-Factor CIR Models Using LIBOR, Swap Rates, and Cap and Swaption Prices.” Working Paper, Northwestern Univ. (2002).Google Scholar
Jamshidian, F. “An Exact Bond Option Formula.” Journal of Finance, 44 (1989), 205–209.CrossRefGoogle Scholar
Jegadeesh, N., and Pennacchi, G. G.. “The Behavior of Interest Rates Implied by the Term Structure of Eurodollar Futures.” Journal of Money, Credit and Banking, 28 (1996), 426–451.CrossRefGoogle Scholar
Li, A.; Ritchken, P.; and Sankasubramanian, L.. “Lattice Models for Pricing American Interest Rate Claims.” Journal of Finance, 50 (1995), 55–72.CrossRefGoogle Scholar
Longstaff, F., and Schwartz, E.. “Valuation American Options By Simulation: A Simple Least Squares Approach.” Review of Financial Studies, 14 (2001), 113–148.CrossRefGoogle Scholar
Longstaff, F.; Santa-Clara, P.; and Schwartz, E. S.. “Throwing Away a Billion Dollars: The Cost of Suboptimal Exercise Strategies in the Swaptions Market.” Journal of Financial Economics, 62 (2001a), 39–66.CrossRefGoogle Scholar
Longstaff, F.; Santa-Clara, P.; and Schwartz, E. S.. “The Relative Value of Caps and Swaptions: Theory and Empirical Evidence.” Journal of Finance, 61 (2001b), 2067–2109.CrossRefGoogle Scholar
Miltersen, K. R.; Sandmann, K.; and Sondermann, D.. “Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates.” Journal of Finance, 57 (1997), 409–430.Google Scholar
Moraleda, J. M., and Pelsser, A.. “Forward versus Spot Interest Rate Models of the Term Structure: An Empirical Comparison.” Journal of Derivatives, 8 (2000), 9–21.CrossRefGoogle Scholar
Nelson, D. B., and Ramaswamy, K.. “Simple Binomial Processes as Diffusion Approximations in Financial Models.” Review of Financial Studies, 3 (1990), 393–430.CrossRefGoogle Scholar
Pliska, S.Introduction to Mathematical Finance: Discrete Time Models. Oxford: Blackwell (1997).Google Scholar
Rebonato, R. “On the Simultaneous Calibration of Multifactor Lognormal Interest Rate Models to Black Volatilities and to the Correlation Matrix.” Journal of Computational Finance, 4 (1999), 5–27.CrossRefGoogle Scholar
Rebonato, R., and Cooper, I.. “Limitations of Simple Two-Factor Interest Rate Models.” Journal of Financial Engineering, 5 (1995), 1–16.Google Scholar
Ritchken, P., and Sankasubramanian, L.. “Volatility Structures of Forward Rates and the Dynamics of the Term Structure.” Mathematical Finance, 5 (1995), 55–72.CrossRefGoogle Scholar
Rogers, C. “Monte Carlo Valuation of American Options.” Working Paper, Univ. of Bath (2001).Google Scholar
Schmidt, J. W. “Asymptotische Einschliebung bei Konvergengenzbeschlenigenden Verfahren. II.” Numerical Mathematics, 11 (1968), 53–56.CrossRefGoogle Scholar
Sidenius, J. “LIBOR Market Models in Practice.” Journal of Computational Finance, 3 (2000), 5–26.CrossRefGoogle Scholar
Singleton, K., and Umantsev, L.. “Pricing Coupon-Bond Options and Swaptions in Affine Term Structure Models.” Working Paper, Stanford Univ. (2001).Google Scholar
Stapleton, R. C., and Subrahmanyam, M. G.. “The Term Structure of Interest-Rate Futures Prices.” Working Paper, New York Univ. (2001).Google Scholar
Sundaresan, S. “Futures Prices on Yields, Forward Prices, and Implied Forward Prices from Term Structure.” Journal of Financial and Quantitative Analysis, 26 (1991), 409–424.CrossRefGoogle Scholar
Vasicek, O. “An Equilibrium Characterization of the Term Structure.” Journal of Financial Economics, 5 (1977), 177–188.CrossRefGoogle Scholar