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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)

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.

Suggested Citation

  • Steven Kou, 2000. "A Jump Diffusion Model for Option Pricing with Three Properties: Leptokurtic Feature, Volatility Smile, and Analytical Tractability," Econometric Society World Congress 2000 Contributed Papers 0062, Econometric Society.
    • Handle: RePEc:ecm:wc2000:0062
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    Cited by:

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    3. Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
    4. Fajardo, J. & Mordecki, E., 2003. "Put-Call Duality and Symmetry," Finance Lab Working Papers flwp_54, Finance Lab, Insper Instituto de Ensino e Pesquisa.
    5. Islyaev, Suren & Date, Paresh, 2015. "Electricity futures price models: Calibration and forecasting," European Journal of Operational Research, Elsevier, vol. 247(1), pages 144-154.

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