Skip to main content
SpringerLink
Log in

Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: An Arrow-Debreu Pricing Approach

  • Published:
Review of Derivatives Research Aims and scope Submit manuscript

Abstract

Simple analytical pricing formulae have been derived, by different authors and for several derivatives, under the Gaussian Langetieg (1980) model. The purpose of this paper is to use such exact Gaussian solutions in order to obtain approximate analytical pricing formulas under the most general stochastic volatility specification of the Duffie and Kan (1996) model. Using Gaussian Arrow-Debreu state prices, first order stochastic volatility approximate pricing solutions will be derived only involving one integral with respect to the time-to-maturity of the contingent claim under valuation. Such approximations will be shown to be much faster than the existing exact numerical solutions, as well as accurate.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Arnold, L. (1992). Stochastic Differential Equations: Theory and Applications. Florida: Krieger Publishing Company.

    Google Scholar 

  • Barone, E., and L. Mengoni. (1997). “Futures-Style Options on Euro-Deposit Futures: Nihil Sub Sole Novi?,” European Financial Management 3, 99-126.

    Google Scholar 

  • Baxter, M., and A. Rennie. (1996). Financial Calculus: An Introduction to Derivative Pricing. Cambridge: Cambridge University Press.

    Google Scholar 

  • Beaglehole, D., and M. Tenney. (1991). “General Solutions of Some Interest Rate-Contingent Claim Pricing Equations,” Journal of Fixed Income (September), 69-83.

  • Brace, A., and M. Musiela. (1994). “A Multifactor Gauss Markov Implementation of Heath, Jarrow, and Morton,” Mathematical Finance 4, 259-283.

    Google Scholar 

  • Chen, L. (1994). “Stochastic Mean and Stochastic Volatility: A Three-Factor Model of the Term Structure and its Application in Pricing of Interest Rate Derivatives,” Working Paper. Federal Reserve Board.

  • Chen, L. (1996). Interest Rate Dynamics, Derivatives Pricing and Risk Management. Lecture Notes in Economics and Mathematical Systems 435. New York: Spinger-Verlag.

    Google Scholar 

  • Chen, R.-R., and L. Scott. (1992). “Pricing Interest Rate Options in a Two-Factor Cox-Ingersoll-Ross Model of the Term Structure,” The Review of Financial Studies 5, 613-636.

    Google Scholar 

  • Chen, R.-R., and L. Scott. (1993). “Pricing Interest Rate Futures Options with Futures-Style Margining,” Journal of Futures Markets 13, 15-22.

    Google Scholar 

  • Chen, R.-R., and L. Scott. (1995). “Interest Rate Options in Multifactor Cox-Ingersoll-Ross Models of the Term Structure,” Journal of Derivatives (Winter), 53-72.

  • Cox, J., J. Ingersoll, and S. Ross. (1981). “The Relation Between Forward Prices and Futures Prices,” Journal of Financial Economics 9, 321-346.

    Google Scholar 

  • Cox, J., J. Ingersoll, and S. Ross. (1985). “A Theory of the Term Structure of Interest Rates,” Econometrica 53, 385-407.

    Google Scholar 

  • Dai, Q., and K. Singleton. (1998). “Specification Analysis of Affine Term Structure Models,” Working Paper. New York University and Stanford University.

  • Duan, J.-C., and J.-G. Simonato. (1995). “Estimating and Testing Exponential-Affine Term Structure Models by Kalman Filter,” Working Paper. McGill University.

  • Duffie, D. (1989). Futures Markets. Englewood Cliffs, New Jersey: Prentice-Hall.

    Google Scholar 

  • Duffie, D., and K. Singleton. (1997). “An Econometric Model of the Term Structure of Interest-Rate Swap Yields,” Journal of Finance 52, 1287-1321.

    Google Scholar 

  • Duffie, D., and R. Kan. (1994). “Multi-Factor Term Structure Models,” Phil. Trans. R. Soc. Lond. 347, 577-586.

    Google Scholar 

  • Duffie, D., and R. Kan. (1996). “A Yield-Factor Model of Interest Rates,” Mathematical Finance 6, 379-406.

    Google Scholar 

  • Duffie, D., J. Pan, and K. Singleton. (1998). “Transform Analysis and Option Pricing for Affine Jump-Diffusions,” Working Paper. Graduate School of Business, Stanford University.

  • El Karoui, N., and J.-C. Rochet. (1989). “A Pricing Formula for Options on Coupon Bonds,” Working Paper 72. SEEDS.

  • El Karoui, N., C. Lepage, R. Myneni, N. Roseau, and R. Viswanathan. (1991). “The Valuation and Hedging of Contingent Claims with Markovian Interest Rates,” Working Paper. Universitè de Paris VI.

  • Fong, H., and O. Vasicek. (1991). “Interest Rate Volatility as a Stochastic Factor,” Working Paper. Gifford Fong Associates.

  • Jamshidian, F. (1991). “Bond and Option Evaluation in the Gaussian Interest Rate Model,” Research in Finance 9, 131-170.

    Google Scholar 

  • Jamshidian, F. (1993). “Option and Futures Evaluation with Deterministic Volatilities,” Mathematical Finance 3, 149-159.

    Google Scholar 

  • Kloeden, P., and E. Platen. (1992). Numerical Solution of Stochastic Differential Equations. New York: Springer-Verlag.

    Google Scholar 

  • Langetieg, T. (1980). “A Multivariate Model of the Term Structure,” Journal of Finance 35, 71-97.

    Google Scholar 

  • Leblanc, B., and O. Scaillet. (1998). “Path Dependent Options on Yields in the Affine Term Structure Model,” Finance and Stochastics 2, 349-367.

    Google Scholar 

  • Longstaff, F., and E. Schwartz. (1992). “Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model,” Journal of Finance 47, 1259-1282.

    Google Scholar 

  • Lund, J. (1994). “Econometric Analysis of Continuous-Time Arbitrage-Free Models of the Term Structure of Interest Rates,” Working paper. The Aarhus School of Business.

  • Munk, C. (1998). “Stochastic Duration and Fast Coupon Bond Option Pricing in Multi-Factor Models,”Working Paper. Odense University.

  • Murdock, J. (1991). Perturbations: Theory and Methods. New York: John Wiley & Sons.

    Google Scholar 

  • Nayfeh, A. (1973). Perturbation Methods. New York: John Wiley & Sons.

    Google Scholar 

  • Nunes, J. (1998). “Interest Rate Options in a Duffie-Kan Model with DeterministicVolatility,” Revista de Mercados e Activos Financeiros 1, 63-101.

    Google Scholar 

  • Press, W., B. Flannery, S. Teukolsky, and W. Vetterling. (1994). Numerical Recipes in Pascal: The Art of Scientific Computing. Cambridge: Cambridge University Press.

    Google Scholar 

  • Schlogl, E., and D. Sommer. (1997). “Factor Models and the Shape of the Term Structure,” Discussion Paper B-395. University of Bonn.

  • Schlogl, E., and D. Sommer. (1998). “Factor Models and the Shape of the Term Structure,” Journal of Financial Engineering 7, 79-88.

    Google Scholar 

  • Schobel, R. (1990). “Options on Short Term Interest Rate Futures,” Working Paper. Universitat Luneburg.

  • Shephard, N. (1991). “From Characteristic Function to Distribution Function: A Simple Framework for the Theory,” Econometric Theory 7, 519-529.

    Google Scholar 

  • Van Loan, C. (1978). “Computing Integrals Involving the Matrix Exponential,” IEEE Transactions on Automatic Control 23, 395-404.

    Google Scholar 

  • Wei, J. (1997). “A Simple Approach to Bond Option Pricing,” Journal of Futures Markets 17, 131-160.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Warwick Business School, University of Warwick, U.K.

    João Pedro Vidal Nunes

  2. CEMAF, Edif. INDEG/ISCTE, Av. Prof. Aníbal Bettencourt, 1600, Lisboa, Portugal

    João Pedro Vidal Nunes

  3. Financial Options Research Centre, University of Warwick, U.K.

    Les Clewlow & Stewart Hodges

  4. School of Finance and Economics, University of Technology, Australia

    Les Clewlow

  5. Centre for Financial Mathematics, Australian National University, Australia

    Les Clewlow

Authors
  1. João Pedro Vidal Nunes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Les Clewlow
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stewart Hodges
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nunes, J.P.V., Clewlow, L. & Hodges, S. Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: An Arrow-Debreu Pricing Approach. Review of Derivatives Research 3, 5–66 (1999). https://doi.org/10.1023/A:1009646430215

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009646430215

Search

Navigation