Analysis of Hedging Profits Under Two Stock Pricing Models

ABSTRACT

In this paper, we employ two stock pricing models: a Black-Scholes (BS) model and a Variance Gamma (VG) model, and apply the maximum likelihood method (MLE) to estimate corresponding parameters in each model. With the estimated parameters, we conduct Monte Carlo simulations to simulate spot prices and deltas of the European call option at different time spots over different sample paths. We focus on calculating the deltas for the two models by the method of the in?nitesimal perturbation analysis (IPA). Then, we-nally, hedging profits under these two models are calculated and analyzed.

In this paper, we employ two stock pricing models: a Black-Scholes (BS) model and a Variance Gamma (VG) model, and apply the maximum likelihood method (MLE) to estimate corresponding parameters in each model. With the estimated parameters, we conduct Monte Carlo simulations to simulate spot prices and deltas of the European call option at different time spots over different sample paths. We focus on calculating the deltas for the two models by the method of the in?nitesimal perturbation analysis (IPA). Then, we-nally, hedging profits under these two models are calculated and analyzed.

Cite this paper

nullL. Cao and Z. Guo, "Analysis of Hedging Profits Under Two Stock Pricing Models,"*Journal of Mathematical Finance*, Vol. 1 No. 3, 2011, pp. 120-124. doi: 10.4236/jmf.2011.13015.

nullL. Cao and Z. Guo, "Analysis of Hedging Profits Under Two Stock Pricing Models,"

References

[1] J. Hull, “Options, Futures, and Other Derivatives,” 5th Edition, Prentice Hall, Upper Saddle River, 2003.

[2] R. Jarrow and S. Turnbull, “Derivative Securities,” Thomson Learning Company, Belmont, 1999.

[3] M. C. Fu and J. Q. Hu, “Sensitivity Analysis for Monte Carlo Simulation of Option Pricing,” Engineering and Informational Sciences, Vol. 9, No. 3, 1995, pp. 417-446. doi:10.1017/S0269964800003958

[4] P. Glasserman, “Monte Carlo Methods in Financial Engineering,” Springer, New York, 2004.

[5] M. C. Fu, “What you should know about simulation and derivatives,” Naval Research Logistics, Vol. 55, No. 8, 2006, pp. 723-736. doi:10.1002/nav.20313

[6] M. C. Fu, “Stochastic Gradient Estimation,” In: S. G. Henderson, B. L. Nelson, Eds., Handbooks in Operations Research and Management Science, Elsevier, Amsterdam, 2008, pp. 575-616.

[7] M. Broadie and P. Glasserman, “Estimating Security Price Derivatives Using Simulation,” Management Sci- ence, Vol. 42, No. 2, 1996, pp. 269-285. doi:10.1287/mnsc.42.2.269

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

[9] R. C. Merton, “Theory of Rational Option Pricing,” Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183.

[10] D. Madan and E. Seneta, “The Variance Gamma(VG) Model for Share Market Returns,” Journal of Business, Vol. 63, No. 4, 1990, pp. 511-524. doi:10.1086/296519

[11] D. Madan and F. Milne, “Option Pricing with V.G. Martingale Components,” Mathematical Finance, Vol. 1, 1991, pp. 39-55. doi:10.1111/j.1467-9965.1991.tb00018.x

[12] D. Madan, P. Carr and E. Chang. “The Variance Gamma Processes and Option Pricing,” European Finance Review, Vol. 2, No. 1, 1998, pp. 79-10. doi:10.1023/A:1009703431535

[13] M. C. Fu, “Variance-Gamma and Monte Carlo,” Advances in Mathematical Finance, Springer, 2007, pp. 21-35. doi:10.1007/978-0-8176-4545-8_2

[14] L. Cao and M. C. Fu, “Estimating Greeks for Variance Gamma,” Proceedings of the 2010 Winter Simulation Conference, Baltimore, 5-8 December 2010, pp. 2620- 2628. doi:10.1109/WSC.2010.5678958

[15] L. Cao and Z. F. Guo, “Applying Gradient Estimation Technique to Estimate Gradients of European call following Variance-Gamma,” Proceedings of Global Conference on Business and Finance, Nevada, 2-5 January 2011.

[16] L. Cao and Z. F. Guo, “A Comparison of Gradient Estimation Techniques for European Call Options,” Accounting & Taxation, Forthcoming, 2011.

[17] L. Cao and Z. F. Guo, “A Comparison of Delta Hedging under Two Price Distribution Assumptions by Likelihood Ratio,” International Journal of Business and Finance Research, Forthcoming, 2011.

[18] L. Cao and Z. F. Guo, “Delta Hedging with Deltas from a Geometric Brownian Motion Process,” Proceedings of International Conference on Applied Financial Economic, Samos Island, March 2011.

[1] J. Hull, “Options, Futures, and Other Derivatives,” 5th Edition, Prentice Hall, Upper Saddle River, 2003.

[2] R. Jarrow and S. Turnbull, “Derivative Securities,” Thomson Learning Company, Belmont, 1999.

[3] M. C. Fu and J. Q. Hu, “Sensitivity Analysis for Monte Carlo Simulation of Option Pricing,” Engineering and Informational Sciences, Vol. 9, No. 3, 1995, pp. 417-446. doi:10.1017/S0269964800003958

[4] P. Glasserman, “Monte Carlo Methods in Financial Engineering,” Springer, New York, 2004.

[5] M. C. Fu, “What you should know about simulation and derivatives,” Naval Research Logistics, Vol. 55, No. 8, 2006, pp. 723-736. doi:10.1002/nav.20313

[6] M. C. Fu, “Stochastic Gradient Estimation,” In: S. G. Henderson, B. L. Nelson, Eds., Handbooks in Operations Research and Management Science, Elsevier, Amsterdam, 2008, pp. 575-616.

[7] M. Broadie and P. Glasserman, “Estimating Security Price Derivatives Using Simulation,” Management Sci- ence, Vol. 42, No. 2, 1996, pp. 269-285. doi:10.1287/mnsc.42.2.269

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

[9] R. C. Merton, “Theory of Rational Option Pricing,” Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183.

[10] D. Madan and E. Seneta, “The Variance Gamma(VG) Model for Share Market Returns,” Journal of Business, Vol. 63, No. 4, 1990, pp. 511-524. doi:10.1086/296519

[11] D. Madan and F. Milne, “Option Pricing with V.G. Martingale Components,” Mathematical Finance, Vol. 1, 1991, pp. 39-55. doi:10.1111/j.1467-9965.1991.tb00018.x

[12] D. Madan, P. Carr and E. Chang. “The Variance Gamma Processes and Option Pricing,” European Finance Review, Vol. 2, No. 1, 1998, pp. 79-10. doi:10.1023/A:1009703431535

[13] M. C. Fu, “Variance-Gamma and Monte Carlo,” Advances in Mathematical Finance, Springer, 2007, pp. 21-35. doi:10.1007/978-0-8176-4545-8_2

[14] L. Cao and M. C. Fu, “Estimating Greeks for Variance Gamma,” Proceedings of the 2010 Winter Simulation Conference, Baltimore, 5-8 December 2010, pp. 2620- 2628. doi:10.1109/WSC.2010.5678958

[15] L. Cao and Z. F. Guo, “Applying Gradient Estimation Technique to Estimate Gradients of European call following Variance-Gamma,” Proceedings of Global Conference on Business and Finance, Nevada, 2-5 January 2011.

[16] L. Cao and Z. F. Guo, “A Comparison of Gradient Estimation Techniques for European Call Options,” Accounting & Taxation, Forthcoming, 2011.

[17] L. Cao and Z. F. Guo, “A Comparison of Delta Hedging under Two Price Distribution Assumptions by Likelihood Ratio,” International Journal of Business and Finance Research, Forthcoming, 2011.

[18] L. Cao and Z. F. Guo, “Delta Hedging with Deltas from a Geometric Brownian Motion Process,” Proceedings of International Conference on Applied Financial Economic, Samos Island, March 2011.