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The Statistical Properties of the Black–Scholes Option Price

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  • Mthuli Ncube
  • Stephen Satchell

Abstract

This paper investigates the statistical properties of the Black–Scholes option price, considered as a random variable. The option is conditioned on the current price and/or the estimated volatility of the underlying security. In both cases, some exact results for the distribution functions of the true option price and the predicted option price are derived. Extensions to puts and American contracts are considered. Numerical results are presented for option prices based on parameters appropriate for the FTSE 100 Index.

Suggested Citation

  • Mthuli Ncube & Stephen Satchell, 1997. "The Statistical Properties of the Black–Scholes Option Price," Mathematical Finance, Wiley Blackwell, vol. 7(3), pages 287-305, July.
    • Handle: RePEc:bla:mathfi:v:7:y:1997:i:3:p:287-305
    DOI: 10.1111/1467-9965.00033
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    Cited by:

    1. Cangrejo Esquivel, Álvaro Javier & Tovar Cuevas, José Rafael & García, Isabel Cristina & Manotas Duque, Diego Fernando, 2022. "Estimación clásica y bayesiana de la volatilidad en el modelo de Black-Scholes [Classical and Bayesian estimation of volatility in the Black-Scholes model]," Revista de Métodos Cuantitativos para la Economía y la Empresa = Journal of Quantitative Methods for Economics and Business Administration, Universidad Pablo de Olavide, Department of Quantitative Methods for Economics and Business Administration, vol. 34(1), pages 237-262, December .
    2. Kristensen, Dennis, 2008. "Estimation of partial differential equations with applications in finance," Journal of Econometrics, Elsevier, vol. 144(2), pages 392-408, June.
    3. Darsinos, T. & Satchell, S.E., 2001. "Bayesian Analysis of the Black-Scholes Option Price," Cambridge Working Papers in Economics 0102, Faculty of Economics, University of Cambridge.
    4. Popovic, Ray & Goldsman, David, 2012. "On valuing and hedging European options when volatility is estimated directly," European Journal of Operational Research, Elsevier, vol. 218(1), pages 124-131.
    5. Vagnani, Gianluca, 2009. "The Black-Scholes model as a determinant of the implied volatility smile: A simulation study," Journal of Economic Behavior & Organization, Elsevier, vol. 72(1), pages 103-118, October.
    6. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.

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