On the pricing and hedging of options for highly volatile periods

El-Khatib, Youssef and Hatemi-J, Abdulnasser (2013): On the pricing and hedging of options for highly volatile periods.

Abstract

Option pricing is an integral part of modern financial risk management. The well-known Black and Scholes (1973) formula is commonly used for this purpose. This paper is an attempt to extend their work to a situation in which the unconditional volatility of the original asset is increasing during a certain period of time. We consider a market suffering from a financial crisis. We provide the solution for the equation of the underlying asset price as well as finding the hedging strategy. In addition, a closed formula of the pricing problem is proved for a particular case. The suggested formulas are expected to make the valuation of options and the underlying hedging strategies during financial crisis more precise.

• Item Type: MPRA Paper
• Language: English
• Keywords: Asset Pricing and Hedging, Options, Financial Crisis, Black and Scholes formula.
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; G - Financial Economics > G0 - General > G01 - Financial Crises; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions

References

References

[1] F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81:637–654, 1973.

[2] E. Deeba, G. Dibeh, and S. Xie. An algorithm for solving bond pricing problem. Appl. Math. Comput., 128(81), 2002.

[3] G. Dibeh and G. Chahda. Option pricing in markets with noisy cyclical and crash dynamics. Finance Lett., 3(2):25–32, 2005.

[4] G. Dibeh and H.M. Harmanani. Option pricing during post-crash relaxation times. Physica A, 380:357–365, 2007.

[5] H. Gu, J.R Liang, and Y.X. Zhang. Time-changed geometric fractional brownian motion and option pricing with transaction costs. Physica A, 391(15):3971--3977, 2012.

[6] F. Lillo and F. Mantenga. Power-law relaxation in a complex system: Omori law after a financial market crash. Phys. Rev. E, 68, 2003.

[7] J. McCauley. The Dynamics of Markets: Econophysics and Finance. Cambridge University Press, Cambridge, 2004.

[8] B. Øksendal. An introduction to malliavin calculus with applications to economics. Working paper 3, Institute of Finance and Management Science, Norwegian School of Economics and Business Administration, 1996.

[9] R. Savit. Nonlinearities and chaotic effects in options prices. J. Futures Mark., 9(507), 1989.

[10] D. Sornette. Why Stock Markets Crash: Critical Events in Complex Financial Markets. Princeton University Press, Princeton, NJ, 2003.

[11] J. Tvedt. Valuation of european futures options in the bifex market. J. Futures Mark., 18:167–175, 1998.