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Discontinuous Interest Rate Processes: An Equilibrium Model for Bond Option Prices

Published online by Cambridge University Press:  06 April 2009

Mukarram Attari
Mukarram Attari
Affiliation:
University of Wisconsin-Madison, School of Business, Grainger Hall, 975 University Avenue, Madison, WI 53706
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Abstract

This paper obtains equilibrium interest rate option prices for discontinuous short-term interest rate processes. The prices are first obtained for a general distribution of jump sizes using a process with a number of fixed size jumps. The pricing formulas are then used to obtain option prices when the jump distribution is known to be one of the continuous distributions. The commonly used jump-diffusion, stochastic volatility jump-diffusion, and Gamma process option prices can be obtained as limiting cases. The methodology is also applied to obtain the prices of options on stocks. Finally, the paper shows how portfolios to hedge derivative securities can be built.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 34 , Issue 3 , September 1999 , pp. 293 - 322
Copyright
Copyright © School of Business Administration, University of Washington 1999

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References

Ahn, C.-M., and Thompson, H. E.. “Jump-Diffusion Processes and the Term-Structure of Interest Rates.” Journal of Finance, 43 (1988), 155–174.Google Scholar
Bakshi, G.; Cao, C.; and Chen, Z.. “Empirical Performance of Alternative Option Pricing Models.” Journal of Finance, 52 (1997), 2003–2049.CrossRefGoogle Scholar
Bates, D. S. “Pricing Options under Jump-Diffusion Processes.” Working Paper (37–88), The Wharton School (1988).Google Scholar
Bates, D. S. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” Review of Financial Studies, 9 (1996), 69–107.CrossRefGoogle Scholar
Bjork, T.; Masi, G. Di; Kabanov, Y.; and Runggaldier, W.. “Towards a General Theory of Bond Markets.” Finance and Stochastics, 1 (1997), 141–174.CrossRefGoogle Scholar
Bjork, T.; Kabanov, Y.; and Runggaldier, W.. “Bond Markets where Prices are Driven by a General Marked Point Process.” Working Paper, 67, Department of Finance, Stockholm School of Economics (1995).Google Scholar
Bjork, T.; Kabanov, Y.; and Runggaldier, W.. “Bond Market Structure in the Presence of Marked Point Processes.” Mathematical Finance, 7 (1997), 211–239.CrossRefGoogle Scholar
Constantinides, G. M. “A Theory of the Nominal Term Structure of Interest Rates.” Review of Financial Studies, 5 (1992), 531–552.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
Cox, J. C.; Ross, S. A.; and Rubinstein, M.. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, 3 (1979), 125–144.Google Scholar
Das, S. R. “Jump-Diffusion Processes and the Bond Markets.” Working Paper, Harvard Business School (1995).Google Scholar
Das, S. R., and Foresi, S.. “Exact Solutions for Band and Option Prices with Systematic Jump Risk.” Review of Derivative Studies, 1 (1996), 7–24.CrossRefGoogle Scholar
Dufresne, F.; Gerber, H. U.; and Shiu, E. S. W.. “Risk Theory with the Gamma Process.” Astin Bulletin, 21 (1991), 177–192.CrossRefGoogle Scholar
Dybvig, P. H., and Huang, C.-F.. “Non-Negative Wealth, Absence of Arbitrage and Feasible Consumption Plans.” Review of Financial Studies, 1 (1988), 377–401.CrossRefGoogle Scholar
Feller, W.An Introduction to Probability Theory and Its Applications: Volume I. New York, NY: John Wiley & Sons (1968).Google Scholar
Gihman, I. I., and Skorohod, A. V.. The Theory of Stochastic Processes: III. New York, NY: Springer-Verlag (1975).Google Scholar
Goldstein, R., and Zapatero, F.. “General Equilibrium with Constant Relative Risk Aversion and Vasicek Interest Rates.” Mathematical Finance, 6 (1996), 331–340.CrossRefGoogle Scholar
Heston, S. L. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” Review of Financial Studies, 6 (1993), 327–344.CrossRefGoogle Scholar
Heston, S. L. “A Model of Discontinuous Interest Rate Behavior, Yield Curves and Volatility.” Working Paper, Washington Univ. (1995).Google Scholar
Heston, S. L. “Option Pricing with Infinitely Divisible Distributions.” Working Paper, Washington Univ. (1997).Google Scholar
Hull, J., and White, A.. “The Pricing of Options on Assets with Stochastic Volatilities.” Journal of Finance, 42 (1987), 281–300.CrossRefGoogle Scholar
Hull, J., and White, A.. “One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities.” Journal of Financial and Quantitative Analysis, 28 (1992), 235–254.CrossRefGoogle Scholar
Ingersoll, J.; Skelton, J.; and Weil, R.. “Duration Forty Years Later.” Journal of Financial and Quantitative Analysis, 14 (1978), 627–650.CrossRefGoogle Scholar
Jarrow, R. A.; Jin, X.; and Madan, D.. “The Second Fundamental Theorem of Asset Pricing.” Working Paper Univ. of Maryland (1998).Google Scholar
Jarrow, R. A., and Madan, D.. “Option Pricing Using the Term-Structure to Hedge Systematic Discontinuities in Asset Returns.” Mathematical Finance, 5 (1995), 311–336.CrossRefGoogle Scholar
Jarrow, R. A., and Madan, D.. “Hedging Contingent Claims on Semi-Martingales.” Finance and Stochastics, 3 (1999), 111–134.CrossRefGoogle Scholar
Jones, E. P. “Option Arbitrage and Strategies with Large Price Changes.” Journal of Financial Economics, 13 (1984), 91–113.CrossRefGoogle Scholar
Karlin, S., and Taylor, H. M.. A Second Course in Stochastic Processes. New York, NY: Academic Press (1981).Google Scholar
Longstaff, F. A. “Multiple Equilibria and Term Structure Models.” Journal of Financial Economics, 32 (1992), 333–344.CrossRefGoogle Scholar
Lucas, R. E. “Asset Prices in an Exchange Economy.” Econometrica, 46 (1978), 1429–1445.CrossRefGoogle Scholar
Madan, D.; Carr, P.; and Chang, E. C.. “The Variance Gamma Process and Option Pricing.” European Finance Review, 2 (1998), 79–105.CrossRefGoogle Scholar
Madan, D., and Milne, F.. “Option Pricing with VG Martingale Components.” Mathematical Finance, 1 (1991), 39–55.CrossRefGoogle Scholar
Madan, D., and Milne, F.. “Contingent Claims Valued and Hedged by Pricing and Investing in a Basis.” Mathematical Finance, 4 (1994), 223–245.CrossRefGoogle Scholar
Madan, D.; Milne, F.; and Shefrin, H.. “The Multinomial Option Pricing Model and Its Brownian and Poisson Limits.” Review of Financial Studies, 2 (1989), 251–265.CrossRefGoogle Scholar
Madan, D., and Seneta, E.. “The Variance Gamma (VG) Model for Share Market Returns.” Journal of Business, 63 (1990), 511–524.CrossRefGoogle Scholar
McCulloch, J. H. “Continuous Time Processes with Stable Increments.” Journal of Business, 51 (1978), 601–619.CrossRefGoogle Scholar
Merton, R. C. “Optimum Consumption and Portfolio Rules in a Continuous Time Model.” Journal of Economic Theory, 3 (1971), 373–413.CrossRefGoogle Scholar
Merton, R. C. “Option Pricing when the Underlying Stock Returns are Discontinuous.” Journal of Financial Economics, 3 (1976), 125–144.CrossRefGoogle Scholar
Naik, V. “Option Valuation and Hedging Strategies with Jumps in the Volatility of Asset Returns.” Journal of Finance, 48 (1993), 1969–1984.CrossRefGoogle Scholar
Naik, V., and Lee, M.. “General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns.” Review of Financial Studies, 3 (1990), 493–521.CrossRefGoogle Scholar
Naik, V., and Lee, M.. “The Yield Curve and Bond Option Prices with Discrete Shifts in Economic Regimes.” Working Paper Univ. of British Columbia (1995).Google Scholar
Royden, H. L.Real Analysis. London, England: Macmillan (1988).Google Scholar
Shirakawa, H. “Interest Rate Option Pricing with Poisson-Gaussian Forward Rate Curve Processes.” Mathematical Finance, 1 (1991), 77–94.CrossRefGoogle Scholar
Turnbull, S. M., and Milne, F.. “A Simple Approach to Interest-Rate Option Pricing.” Review of Financial Studies, 4 (1991), 87–120.CrossRefGoogle Scholar
Vasicek, O. “An Equilibrium Characterization of the Term Structure.” Journal of Financial Economics, 5 (1977), 177–188.CrossRefGoogle Scholar