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A Jump Diffusion Model for Option Pricing with Three Properties: Leptokurtic Feature, Volatility Smile, and Analytical Tractability

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  • Steven Kou

    (Columbia University)

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    Abstract

    Brownian motion and normal distribution have been widely used, for example, in the Black-Scholes-Merton option pricing framework, to study the return of assets. However, two puzzles, emerged from many empirical investigations, have got much attention recently, namely (a) the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and (b) an empirical abnormity called ``volatility smile'' in option pricing. To incorporate both the leptokurtic feature and ``volatility smile'', this paper proposes, for the purpose of studying option pricing, a jump diffusion model, in which the price of the underlying asset is modeled by two parts, a continuous part driven by Brownian motion, and a jump part with the logarithm of the jump sizes having a double exponential distribution. In addition to the above two desirable properties, leptokurtic feature and ``volatility smile'', the model is simple enough to produce analytical solutions for a variety of option pricing problems, including options, future options, and interest rate derivatives, such as caps and floors, in terms of the $Hh$ function. Although there are many models can incorporate some of the three properties (the leptokurtic feature, ``volatility smile'', and analytical tractability), the current model can incorporate all three under a unified framework.

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    Bibliographic Info

    Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 0062.

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    Date of creation: 01 Aug 2000
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    Handle: RePEc:ecm:wc2000:0062

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    1. Robert Jarrow & Dilip Madan, 1995. "Option Pricing Using The Term Structure Of Interest Rates To Hedge Systematic Discontinuities In Asset Returns," Mathematical Finance, Wiley Blackwell, vol. 5(4), pages 311-336.
    2. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-55, January.
    3. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    4. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    5. Simon Hurst & Eckhard Platen & Svetlozar Rachev, 1997. "Subordinated Market Index Models: A Comparison," Asia-Pacific Financial Markets, Springer, vol. 4(2), pages 97-124, May.
    6. Lina El-Jahel & Hans Lindberg & William Perraudin, 1996. "Interest Rate Distributions, Yield Curve Modelling and Monetary Policy," Archive Working Papers 021, Birkbeck, Department of Economics, Mathematics & Statistics.
    7. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, vol. 53(2), pages 385-407, March.
    8. 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.
    9. Jarrow, Robert A & Turnbull, Stuart M, 1995. " Pricing Derivatives on Financial Securities Subject to Credit Risk," Journal of Finance, American Finance Association, vol. 50(1), pages 53-85, March.
    10. Tomas Björk & Yuri Kabanov & Wolfgang Runggaldier, 1997. "Bond Market Structure in the Presence of Marked Point Processes," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 211-239.
    11. 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.
    12. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
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    Cited by:
    1. Fajardo, J. & Mordecki, E., 2003. "Put-Call Duality and Symmetry," Finance Lab Working Papers flwp_54, Finance Lab, Insper Instituto de Ensino e Pesquisa.
    2. Fajardo, J. & Mordeckiz, E., 2004. "Duality and Derivative Pricing with Lévy Processes," Finance Lab Working Papers flwp_71, Finance Lab, Insper Instituto de Ensino e Pesquisa.
    3. Ernesto Mordecki & José Fajardo, 2004. "Pricing Derivatives on Two Lé}vy-driven Stocks," Econometric Society 2004 North American Winter Meetings 139, Econometric Society.

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