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On the computation of option prices and sensitivities in the Black-Scholes-Merton model

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  • B. A. Shadwick
  • W. F. Shadwick

Abstract

We consider the maximal extension (excluding jump processes) of the one-factor Black-Scholes-Merton option pricing model. We argue that this model and the local volatility approach provide the optimal one-factor option pricing model. We present a formalism for determining sensitivities to variations in volatility, interest rate or dividend rate when these quantities are functions of any or all of the independent variables. This formalism results in partial differential equations for the sensitivities which have gamma or delta (depending on the particular case) as source terms. We show that these new expressions for the sensitivities reduce to the usual partial derivatives in the case where the relevant parameter is constant. We include various numerical examples to illustrate these ideas.

Suggested Citation

  • B. A. Shadwick & W. F. Shadwick, 2002. "On the computation of option prices and sensitivities in the Black-Scholes-Merton model," Quantitative Finance, Taylor & Francis Journals, vol. 2(2), pages 158-166.
    • Handle: RePEc:taf:quantf:v:2:y:2002:i:2:p:158-166
    DOI: 10.1088/1469-7688/2/2/307
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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    3. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    4. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    5. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
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