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The Journal of Derivatives

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Displaced Jump-Diffusion Option Valuation

António Câmara, Tim Krehbiel and Weiping Li
The Journal of Derivatives Winter 2009, 17 (2) 41-58; DOI: https://doi.org/10.3905/JOD.2009.17.2.041
António Câmara
is the Watson Family Chair in Commodity and Financial Risk Management and an associate professor of finance at the Spears School of Business at Oklahoma State University in Stillwater, OK.
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  • For correspondence: acamara@okstate.edu
Tim Krehbiel
is a professor of finance at the Spears School of Business at Oklahoma State University in Stillwater, OK.
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  • For correspondence: tlk@okstate.edu
Weiping Li
is the South western Bell Professor and a professor of mathematic s at Oklahoma State University in Stillwater, OK.
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  • For correspondence: wli@math.okstate.edu
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Abstract

The lognormal diffusion process was the most convenient assumption Black and Scholes could make to capture the general features of stock price movements; it allows stochastic evolution of returns in continuous-time, and stock prices are bounded below by zero. But once we gained more empirical knowledge about returns distributions and observed the persistent volatility skew in real world option prices, alternative processes such as jump-diffusions were introduced. In this article, Câmara, Krehbiel, and Li note that, unlike an individual stock, a stock index will have a minimum value that is strictly positive, because a component stock whose price is going towards zero will be replaced in the index by a different stock. Thus a stock index should follow a displaced jump-diffusion. With this assumption, the authors derive an option pricing formula and test it on 10 years of S&P 500 Index and index option data. The model’s behavior with regard to implied jump intensity and frequency, the shape of the volatility skew, and the existence of a positive lower bound on the index are quite plausible and the goodness of fit is better than either Rubinstein’s displaced diffusion model (without jumps) or Merton’s jump-diffusion model (with lower bound of zero).

TOPICS: Options, simulations, volatility measures

  • © 2009 Pageant Media Ltd
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The Journal of Derivatives: 17 (2)
The Journal of Derivatives
Vol. 17, Issue 2
Winter 2009
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Displaced Jump-Diffusion Option Valuation
António Câmara, Tim Krehbiel, Weiping Li
The Journal of Derivatives Nov 2009, 17 (2) 41-58; DOI: 10.3905/JOD.2009.17.2.041

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Displaced Jump-Diffusion Option Valuation
António Câmara, Tim Krehbiel, Weiping Li
The Journal of Derivatives Nov 2009, 17 (2) 41-58; DOI: 10.3905/JOD.2009.17.2.041
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  • Article
    • Abstract
    • THE MINIMAL STOCK INDEX VALUE
    • EQUILIBRIUM RELATIONS
    • THE DJD OPTION PRICING MODEL
    • EMPIRICAL TESTS
    • APPENDIX A
    • APPENDIX B
    • ENDNOTES
    • REFERENCES
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