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The Random Yield Curve and Interest Rate Options

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  • Meifang Chu

    (CQF - Imperial College - London)

Abstract

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.

Suggested Citation

  • Meifang Chu, 1997. "The Random Yield Curve and Interest Rate Options," Finance 9710003, EconWPA.
  • Handle: RePEc:wpa:wuwpfi:9710003 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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    References listed on IDEAS

    as
    1. D. P. Kennedy, 1994. "The Term Structure Of Interest Rates As A Gaussian Random Field," Mathematical Finance, Wiley Blackwell, pages 247-258.
    2. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305 World Scientific Publishing Co. Pte. Ltd..
    3. 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.
    4. 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.
    5. 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 637-654, May-June.
    6. Alan Brace & Marek Musiela, 1994. "A Multifactor Gauss Markov Implementation Of Heath, Jarrow, And Morton," Mathematical Finance, Wiley Blackwell, pages 259-283.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    Kolmogorov Field Equation; Brownian Sheet; Arbitrage Pricing Theory; Self-Financing Strategy; Heath-Jarrow-Morton Framework;

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G1 - Financial Economics - - General Financial Markets
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C0 - Mathematical and Quantitative Methods - - General
    • E4 - Macroeconomics and Monetary Economics - - Money and Interest Rates

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