Black-Scholes Option Pricing Model Modified to Admit a Miniscule Drift Can Reproduce the Volatility Smile

Affiliation(s)

Financial Guard Ltd., London, UK.

Department of Mathematics and Statistics, Utah State University, Logan, USA.

Financial Guard Ltd., London, UK.

Department of Mathematics and Statistics, Utah State University, Logan, USA.

ABSTRACT

This paper develops a closed-form solution to an extended Black-Scholes (EBS) pricing formula which admits an implied drift parameter alongside the standard implied volatility. The market volatility smiles for vanilla call options on the S&P 500 index are recreated fitting the best volatility-drift combination in this new EBS. Using a likelihood ratio test, the implied drift parameter is seen to be quite significant in explaining volatility smiles. The implied drift parameter is sufficiently small to be undetectable via historical pricing analysis, suggesting that drift is best considered as an implied parameter rather than a historically-fit one. An overview of option-pricing models is provided as background.

This paper develops a closed-form solution to an extended Black-Scholes (EBS) pricing formula which admits an implied drift parameter alongside the standard implied volatility. The market volatility smiles for vanilla call options on the S&P 500 index are recreated fitting the best volatility-drift combination in this new EBS. Using a likelihood ratio test, the implied drift parameter is seen to be quite significant in explaining volatility smiles. The implied drift parameter is sufficiently small to be undetectable via historical pricing analysis, suggesting that drift is best considered as an implied parameter rather than a historically-fit one. An overview of option-pricing models is provided as background.

Cite this paper

M. Modisett and J. Powell, "Black-Scholes Option Pricing Model Modified to Admit a Miniscule Drift Can Reproduce the Volatility Smile,"*Applied Mathematics*, Vol. 3 No. 6, 2012, pp. 597-605. doi: 10.4236/am.2012.36093.

M. Modisett and J. Powell, "Black-Scholes Option Pricing Model Modified to Admit a Miniscule Drift Can Reproduce the Volatility Smile,"

References

[1] B. Dupire, “Pricing with a Smile,” Risk, Vol. 7, No. 1, 1994, pp. 18-20.

[2] F. Black and M. S. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

[3] R. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 125-144. doi:10.1016/0304-405X(76)90022-2

[4] F. Black, “The Pricing of Commodity Contracts,” Journal of Financial Economics, Vol. 3, 1976, pp. 167-179. doi:10.1016/0304-405X(76)90024-6

[5] J.-P. Bouchaud and M. Potters, “Back to Basics: Historical Option Pricing Revisited,” Philosophical Transactions of the Royal Society, Vol. 357, No. 1735, 1999, pp. 2019-2028.

[6] E. Derman, “Regimes of Volatility,” Risk, Vol. 4, 1999, pp. 55-59.

[7] E. Derman and I. Kani, “Riding on a Smile,” Risk, Vol. 7, No. 2, 1994, pp. 32-39.

[8] M. Rubinstein, “Implied Binomial Trees,” Journal of Finance, Vol. 49, No. 3, 1994, pp. 771-818.

[9] L. Andersen and R. Brotherton-Ratcliffe, “The Equity Option Volatility Smile: An Implicit Finite-Difference Approach,” Journal of Computational Finance, Vol. 1, No. 2, 1997, pp. 5-38.

[10] J. C. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,” Advances in Futures and Options Research, Vol. 3, 1988, pp. 29-61.

[11] S. L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327-343. doi:10.1093/rfs/6.2.327

[12] D. S. Bates, “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options,” Review of Financial Studies, Vol. 9, No. 1, 1996, pp. 69107. doi:10.1093/rfs/9.1.69

[13] B. Dupire, “A Unified Theory of Volatility,” Working Paper, 1996.

[14] A. Lipton and W. McGhee, “Universal Barriers,” Risk, Vol. 15, No. 5, 2002, pp. 81-85.

[15] M. Britten-Jones and A. Neuberger, “Option Prices, Implied Prices Processes, and Stochastic Volatility,” Journal of Finance, Vol. 55, No. 2, 2000, pp. 839-866. doi:10.1111/0022-1082.00228

[16] G. Blacher, “A New Approach for Designing and Calibrating Stochastic Volatility Models for Optimal DeltaVega Hedging of Exotic Options,” Conference presentation at Global Derivatives and Risk Management, Juanles-Pins, 26 June 2002.

[17] D. Brigo and F. Mercurio, “A Mixed-Up Smile,” Risk, Vol. 13, No. 9, 2000, pp. 123-126.

[18] R. Hillborn and M. Mangel, “The Ecological Detective: Confronting Models with Data,” Princeton University Press, Princeton, 1997

[19] R. L. McDonald, “Derivative Markets,” 2nd Edition, Addison Wesley, New York, 2006.

[1] B. Dupire, “Pricing with a Smile,” Risk, Vol. 7, No. 1, 1994, pp. 18-20.

[2] F. Black and M. S. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

[3] R. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 125-144. doi:10.1016/0304-405X(76)90022-2

[4] F. Black, “The Pricing of Commodity Contracts,” Journal of Financial Economics, Vol. 3, 1976, pp. 167-179. doi:10.1016/0304-405X(76)90024-6

[5] J.-P. Bouchaud and M. Potters, “Back to Basics: Historical Option Pricing Revisited,” Philosophical Transactions of the Royal Society, Vol. 357, No. 1735, 1999, pp. 2019-2028.

[6] E. Derman, “Regimes of Volatility,” Risk, Vol. 4, 1999, pp. 55-59.

[7] E. Derman and I. Kani, “Riding on a Smile,” Risk, Vol. 7, No. 2, 1994, pp. 32-39.

[8] M. Rubinstein, “Implied Binomial Trees,” Journal of Finance, Vol. 49, No. 3, 1994, pp. 771-818.

[9] L. Andersen and R. Brotherton-Ratcliffe, “The Equity Option Volatility Smile: An Implicit Finite-Difference Approach,” Journal of Computational Finance, Vol. 1, No. 2, 1997, pp. 5-38.

[10] J. C. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,” Advances in Futures and Options Research, Vol. 3, 1988, pp. 29-61.

[11] S. L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327-343. doi:10.1093/rfs/6.2.327

[12] D. S. Bates, “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options,” Review of Financial Studies, Vol. 9, No. 1, 1996, pp. 69107. doi:10.1093/rfs/9.1.69

[13] B. Dupire, “A Unified Theory of Volatility,” Working Paper, 1996.

[14] A. Lipton and W. McGhee, “Universal Barriers,” Risk, Vol. 15, No. 5, 2002, pp. 81-85.

[15] M. Britten-Jones and A. Neuberger, “Option Prices, Implied Prices Processes, and Stochastic Volatility,” Journal of Finance, Vol. 55, No. 2, 2000, pp. 839-866. doi:10.1111/0022-1082.00228

[16] G. Blacher, “A New Approach for Designing and Calibrating Stochastic Volatility Models for Optimal DeltaVega Hedging of Exotic Options,” Conference presentation at Global Derivatives and Risk Management, Juanles-Pins, 26 June 2002.

[17] D. Brigo and F. Mercurio, “A Mixed-Up Smile,” Risk, Vol. 13, No. 9, 2000, pp. 123-126.

[18] R. Hillborn and M. Mangel, “The Ecological Detective: Confronting Models with Data,” Princeton University Press, Princeton, 1997

[19] R. L. McDonald, “Derivative Markets,” 2nd Edition, Addison Wesley, New York, 2006.