Financial Engineering and the Japanese Markets

, Volume 3, Issue 3, pp 217–238 | Cite as

A preference free partial differential equation for the term structure of interest rates

  • Carl Chiarella
  • Nadima El-Hassan

Abstract

The objectives of this paper are two-fold: the first is the reconciliation of the differences between the Vasicek and the Heath-Jarrow-Morton approaches to the modelling of term structure of interest rates. We demonstrate that under certain (not empirically unreasonable) assumptions prices of interest-rate sensitive claims within the Heath-Jarrow-Morton framework can be expressed as a partial differential equation which both is preference-free and matches the currently observed yield curve. This partial differential equation is shown to be equivalent to the extended Vasicek model of Hull and White. The second is the pricing of interest rate claims in this framework. The preference free partial differential equation that we obtain has the added advantage that it allows us to bring to bear on the problem of evaluating American style contingent claims in a stochastic interest rate environment the various numerical techniques for solving free boundary value problems which have been developed in recent years such as the method of lines.

Key words

Heath-Jarrow-Morton term structure of interest rates preference-free partial differential equation American options free boundary value problems method of lines 

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References

  1. Amin, K. (1995), ‘Option Pricing Trees’,Journal of Derivatives, 34–46, Summer.Google Scholar
  2. Bhar, R. and Chiarella, C. (1995), ‘Transformation of Heath-Jarrow-Morton Models to Markovian Systems’, presented at the conference onForecasting Financial Markets organised by Chemical Bank and Imperial College, London. March.Google Scholar
  3. Bhar, R. and Hunt, B.F. (1993), ‘Predicting the Short Term Forward Interest Rate Structure Using a Parsimonious Model’,International Review of Financial Economics and Finance, (2) 4.Google Scholar
  4. Brennan, M.J., and Schwartz E.S. (1978), ‘Finite Difference Method and Jump Processes Arising in the Pricing of Contingent Claims’,Journal of Financial and Quantitative Analysis,13, 461–474.CrossRefGoogle Scholar
  5. Carr, P. and Faguet, D. (1994), ‘Fast Accurate Valuation of American Options’,Working Paper Series, Cornell University. June.Google Scholar
  6. Carverhill, A. (1994), ‘When is the Short Rate Markovian’,Mathematical Finance,4, 305–312.Google Scholar
  7. Chesney, M., Elliott, R.J. and Gibson, R. (1993), ‘Analytical Solutions for the Pricing of American Bond and Yield Options’,Mathematical Finance,3, 3, 277–294.Google Scholar
  8. Chiarella, C. and El-Hassan, N. (1996), ‘Evaluation of Derivative Security Prices in the HJM Framework by Means Of Fast Fourier Transform Techniques’, Working Paper, UTS.Google Scholar
  9. Gihman, I.I. and Skorokhod, A.V. (1965), ‘Introduction to the Theory of Random Processes’,Saunders, Philadelphia.Google Scholar
  10. Goldenberg, D.H. and Schmidt, R.J. (1994), ‘Estimating the Early Exercise Boundary and Pricing American Options’,Working paper, Rensselaer Polytechnic Institute.Google Scholar
  11. Heath, D., Jarrow, R. and Morton, A. (1992), ‘Bond Pricing and the Term Structure of Interest Rates’,Review of Futures Markets, 54–82.Google Scholar
  12. Hull, J. (1993), ‘Options, Futures and Other Derivative Securities’,Prentice-Hall, Inc, Second Edition.Google Scholar
  13. Hull, J. and White, A. (1990), ‘Pricing Interest Rate Derivative Securities’,Review of Financial Studies,3, 4, 573–592.CrossRefGoogle Scholar
  14. Hull, J. and White, A. (1990), ‘Valuing Derivative Securities using the Explicit Finite Difference Method’,Journal of Financial and Quantitative Analysis,25, 1. March.CrossRefGoogle Scholar
  15. Hull, J. and White, A. (1993), ‘Efficient Procedures For Valuing European and American Path-Dependent Options’,The Journal of Derivatives, Fall.Google Scholar
  16. Jeffrey, A. (1995), ‘Single Factor Heath-Jarrow-Morton Term Structure Models Based on Markov Spot Interest Rate Dynamics’,Journal of Financial and Quantitative Analysis,30, 4, 619–642 December.CrossRefGoogle Scholar
  17. Kotlow, D.B. (1973), ‘A Free Boundary Connected with the Optimal Stopping Problem for Diffusion Processes’,Trans. Americ. Math. Soc.,184, 457–478.CrossRefGoogle Scholar
  18. Li, A., Ritchken, P. and Sankarasubramanian, L. (1995), ‘Lattice Models for Pricing American Interest Rate Claims’,The Journal of Finance,1, 2.Google Scholar
  19. Meyer, G.H. (1981), ‘The Method of Lines and Invariant Imbedding for Elliptic and Parabolic Free Boundary Problems’,SIAM Journal of Numerical Analysis,18, 150–164.CrossRefGoogle Scholar
  20. Meyer, G.H. and Van der Hoek, J. (1994), ‘The Evaluation of American Options Using the Method of Lines’, presented atThe First Asia Pacific Finance Association Conference, Sept.Google Scholar
  21. Meyer, G.H. (1977), ‘One-dimensional Parabolic Free Boundary Problems’,SIAM Review,19, 17–34.CrossRefGoogle Scholar
  22. Meyer, G.H. (1973), ‘Initial Value Methods for Boundary Value Problems-Theory and Applications of Invariant Imbedding’,Academic Press.Google Scholar
  23. Myneni, R. (1972), ‘The Pricing of the American Option’,Annals Applied Probability,1, 1–23.Google Scholar
  24. Ritchken, P. and Sankarasubramanian, L. (1995), ‘Volatilities Structures of Forward Rates and the Dynamics of Term Structure’,Mathematical Finance,5(1), 55–73.Google Scholar
  25. Vasicek, O. (1977), ‘An Equilibrium Characterisation of the Term Structure’,Journal of Financial Economics,5, 177–188.CrossRefGoogle Scholar
  26. Wilmott, P, Dewynne, J. and Howison, S. (1993), ‘Option Pricing: Mathematical Models and Computation’,Oxford Financial Press.Google Scholar
  27. Yu, G.G. (1993), ‘Valuation of American Bond Options’,Leonard N. Stern School of Business Working Paper Series, S-93-46.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Carl Chiarella
    • 1
  • Nadima El-Hassan
    • 1
  1. 1.School of Finance and EconomicsUniversity of TechnologySydneyAustralia

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