The Random Yield Curve and Interest Rate Options
This paper proposes a simple and unifying model to price the interest rate contingent claims in a complete market where trading can be made in continuous time. The underlying dynamics of the yield curve is modelled by a random string whose trajectory produces a random surface described by a Brownian sheet. Generalising Black-Scholes' PDE methodology, we derive the Kolmogorov field equation which describes the time-evolution of the contingent claims and obtain explicit pricing formulae for a large class of interest rate options including European calls, compound options, swaps, swaptions, caps and captions. This model can be thought of as an infinite-factor Gaussian model in the Heath-Jarrow-Morton framework and can be implemented without having to calibrate explicit parameters in the covariance function of the discount bond returns.
|Date of creation:||22 Oct 1997|
|Date of revision:|
|Note:||Type of Document - ps; prepared on UNIX Sparc TeX; to print on HP/PostScript; pages: 23; figures: none. This paper has been submitted for publication.|
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- Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
- D. P. Kennedy, 1994. "The Term Structure Of Interest Rates As A Gaussian Random Field," Mathematical Finance, Wiley Blackwell, vol. 4(3), pages 247-258.
- Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
- Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
- Alan Brace & Marek Musiela, 1994. "A Multifactor Gauss Markov Implementation Of Heath, Jarrow, And Morton," Mathematical Finance, Wiley Blackwell, vol. 4(3), pages 259-283.
- Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
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