Recent Developments in Option Pricing

ABSTRACT

In this paper, we investigate recent developments in option pricing based on Black-Scholes processes, pure jump processes, jump diffusion process, and stochastic volatility processes. Results on Black-Scholes model with GARCH volatility (Gong, Thavaneswaran and Singh [1]) and Black-Scholes model with stochastic volatility (Gong, Thavaneswaran and Singh [2]) are studied. Also, recent results on option pricing for jump diffusion processes, partial differential equation (PDE) method together with FFT (fast Fourier transform) approximations of Pillay and O’ Hara [3] and a recently proposed method based on moments of truncated lognormal distribution (Thavaneswaran and Singh [4]) are also discussed in some detail.

In this paper, we investigate recent developments in option pricing based on Black-Scholes processes, pure jump processes, jump diffusion process, and stochastic volatility processes. Results on Black-Scholes model with GARCH volatility (Gong, Thavaneswaran and Singh [1]) and Black-Scholes model with stochastic volatility (Gong, Thavaneswaran and Singh [2]) are studied. Also, recent results on option pricing for jump diffusion processes, partial differential equation (PDE) method together with FFT (fast Fourier transform) approximations of Pillay and O’ Hara [3] and a recently proposed method based on moments of truncated lognormal distribution (Thavaneswaran and Singh [4]) are also discussed in some detail.

KEYWORDS

Stochastic Volatility, Black-Scholes Partial Differential Equations, Option Pricing, Monte Carlo

Stochastic Volatility, Black-Scholes Partial Differential Equations, Option Pricing, Monte Carlo

Cite this paper

nullH. Gong, Y. Liang and A. Thavaneswaran, "Recent Developments in Option Pricing,"*Journal of Mathematical Finance*, Vol. 1 No. 3, 2011, pp. 63-71. doi: 10.4236/jmf.2011.13009.

nullH. Gong, Y. Liang and A. Thavaneswaran, "Recent Developments in Option Pricing,"

References

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[2] H. Gong, A. Thavaneswaran and J. Singh, “Stochastic Volatility Models with Application in Option Pricing,” Journal of Statistical Theory and Practice, Vol. 4, No. 4, 2010.

[3] E. Pillay and J. G. O’ Hara, “FFT Based Option Pricing under a Mean Reverting Process with Stochastic Volatility and Jumps,” Journal of Computational and Applied Mathematics, Vol. 235, No. 12, 2011, pp. 3378-3384. doi:10.1016/j.cam.2010.10.024

[4] A. Thavaneswaran and J. Singh, “Option Pricing for Jump Diffusion Model with Random Volatility,” The Journal of Risk Finance, Vol. 11, No. 5, 2010, pp. 496-507. doi:10.1108/15265941011092077

[5] F. Black and M. Scholes, “The Valuation of Option Con- tracts and a Test of Market Efficiency,” Journal of Fi- nance, Vol. 27, No. 2, 1972, pp. 99-417. doi:10.2307/2978484

[6] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062

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

[8] S. L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” The Review of Financial Studies, Vol. 6, No. 2, 1992, pp. 327-343. doi:10.1093/rfs/6.2.327

[9] S. L. Heston, “Option Pricing with Infinitely Divisible Distributions,” Quantitative Finance, Vol. 4, No. 1, 2004, pp. 515-524. doi:10.1080/14697680400000035

[10] J. C. Cox, J. E. Ingersoll and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407. doi:10.2307/1911242

[11] G. Bakshi, C. Cao and Z. Chen, “Empirical Performance of Altemative Option Pricing Models,” Journal of Finance, Vol. 52, No. 5, 1997, pp. 2003-2049. doi:10.2307/2329472

[12] L. O. Scott, “Option Pricing When the Variance Changes Randomly: Theory, Estimation, and an Application,” Journal of Financial and Quantitative Analysis, Vol. 22, No. 4, 1987, pp. 419-438. doi:10.2307/2330793

[13] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.2307/2328253

[14] J. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,” Journal of In- ternational Economics, Vol. 24, No. 4, 1988, pp. 129- 145.

[15] P. Ritchken and R. Trevor, “Pricing Options under Generalized GARCH and Stochastic Volatility Processes,” Journal of Finance, Vol. 59, No. 1, 1999, pp. 377-402. doi:10.1111/0022-1082.00109

[16] J. B. Wiggins, “Option Values under Stochastic Volatilities,” Journal of Financial Economics, Vol. 19, 1987, pp. 351-372. doi:10.1016/0304-405X(87)90009-2

[17] S. L. Heston and S. Nandi, “A Closed-Form GARCH Option Valuation Model,” The Review of Financial Studies, Vol. 13, No. 1, 2000, pp. 585-625. doi:10.1093/rfs/13.3.585

[18] R. J. Elliot, T. K. Siu and L. Chan, “Option Pricing for GARCH Models with Markov Switching,” International Journal of Theoretical and Applied Finance, Vol. 9, No. 6, 2006, pp. 825-841. doi:10.1142/S0219024906003846

[19] P. Christoffersen, S. Heston and K. Jacobs, “Option Valuation with Conditional Skewness,” Journal of Eco- nomtrics, Vol. 131, No. 1-2, 2006, pp. 253-284.

[20] L. Mercuri, “Option Pricing in a GARCH Model with Tempered Stable Innovations,” Finance Research Letter, Vol. 5, No. 3, 2008, pp. 172-182. doi:10.1016/j.frl.2008.05.003

[21] A. M. Badescu and R. J. Kulperger, “GARCH Option Oricing: a Semiparametric Approach,” Insurance Mathe- matics and Economics, Vol. 43, No. 1, 2008, pp. 69-84. doi:10.1016/j.insmatheco.2007.09.011

[22] G. Barone-Adesi, H. Rasmussen and C. Ravanelli, “An Option Pricing Formula for the GARCH Diffusion Model,” Computational Statistics and Data Analysis, Vol. 49, No. 2, 2005, pp. 287-310. doi:10.1016/j.csda.2004.05.014

[23] G. Barone-Adesi, R. F. Engle and L. Mancini, “A GARCH Option Pricing Model with Filtered Historical Simulation,” Review of Financial Studies, Vol. 21, No. 3, 2008, pp. 1223-1258. doi:10.1093/rfs/hhn031

[24] A. Thavaneswaran, S. Peiris and J. Singh, “Derivation of Kurtosis and Option Pricing Formulas for Popular Vola- tility Models with Applications in Finance,” Communications in Statistics.Theory and Methods, Vol. 37, 2008, pp. 1799-1814. doi:10.1080/03610920701826435

[25] A. Thavaneswaran, J. Singh and S. S. Appadoo, “Option Pricing for some Stochastic Volatility Models,” Journal of Risk Finance, Vol. 7, No. 4, 2006, pp. 425-445. doi:10.1108/15265940610688982

[26] S. J. Taylor, “Asset Price Dynamics, Volatility, and Pre- diction,” Princeton University Press, Princeton, 2005.

[27] J. M. Steele, “Stochastic Calculus and Financial Applica- tions,” Springer, New York, 2001.

[28] J. Vecer, “Stochastic Finance: A Numeraire Approach,” Chapman & Hall/CRC Press, London, 2011.

[29] H. Y. Wong and Y. W. Lo, “Option Pricing with Mean Reversion and Stochastic Volatility,” European Journal of Operational Research, Vol. 197, No. 1, 2009, pp. 179- 187. doi:10.1016/j.ejor.2008.05.014

[30] D. Duffie, J. Pan and K. Singleton, “Transform analysis and Asset Pricing for Affine Jump-Diffusion,” Econo- metrica, Vol. 68, No. 6, 2000, pp. 1343-1376. doi:10.1111/1468-0262.00164

[31] 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, Vol. 6, No. 2, 2011, pp. 12-18.

[32] L. Cao and Z. F. Guo, “Applying Variance Gamma Correlated to Estimate Optimal Portfolio of Variance Swap,” Proceedings of International Conference of Financial Engineering, London, 6-8 July 2011.

[1] [1] H. Gong, A. Thavaneswaran and J. Singh, “A Black- Scholes Model with GARCH Volatility,” The Mathematical Scientist, Vol. 35, No. 1, 2010, pp. 37-42.

[2] H. Gong, A. Thavaneswaran and J. Singh, “Stochastic Volatility Models with Application in Option Pricing,” Journal of Statistical Theory and Practice, Vol. 4, No. 4, 2010.

[3] E. Pillay and J. G. O’ Hara, “FFT Based Option Pricing under a Mean Reverting Process with Stochastic Volatility and Jumps,” Journal of Computational and Applied Mathematics, Vol. 235, No. 12, 2011, pp. 3378-3384. doi:10.1016/j.cam.2010.10.024

[4] A. Thavaneswaran and J. Singh, “Option Pricing for Jump Diffusion Model with Random Volatility,” The Journal of Risk Finance, Vol. 11, No. 5, 2010, pp. 496-507. doi:10.1108/15265941011092077

[5] F. Black and M. Scholes, “The Valuation of Option Con- tracts and a Test of Market Efficiency,” Journal of Fi- nance, Vol. 27, No. 2, 1972, pp. 99-417. doi:10.2307/2978484

[6] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062

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

[8] S. L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” The Review of Financial Studies, Vol. 6, No. 2, 1992, pp. 327-343. doi:10.1093/rfs/6.2.327

[9] S. L. Heston, “Option Pricing with Infinitely Divisible Distributions,” Quantitative Finance, Vol. 4, No. 1, 2004, pp. 515-524. doi:10.1080/14697680400000035

[10] J. C. Cox, J. E. Ingersoll and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407. doi:10.2307/1911242

[11] G. Bakshi, C. Cao and Z. Chen, “Empirical Performance of Altemative Option Pricing Models,” Journal of Finance, Vol. 52, No. 5, 1997, pp. 2003-2049. doi:10.2307/2329472

[12] L. O. Scott, “Option Pricing When the Variance Changes Randomly: Theory, Estimation, and an Application,” Journal of Financial and Quantitative Analysis, Vol. 22, No. 4, 1987, pp. 419-438. doi:10.2307/2330793

[13] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.2307/2328253

[14] J. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,” Journal of In- ternational Economics, Vol. 24, No. 4, 1988, pp. 129- 145.

[15] P. Ritchken and R. Trevor, “Pricing Options under Generalized GARCH and Stochastic Volatility Processes,” Journal of Finance, Vol. 59, No. 1, 1999, pp. 377-402. doi:10.1111/0022-1082.00109

[16] J. B. Wiggins, “Option Values under Stochastic Volatilities,” Journal of Financial Economics, Vol. 19, 1987, pp. 351-372. doi:10.1016/0304-405X(87)90009-2

[17] S. L. Heston and S. Nandi, “A Closed-Form GARCH Option Valuation Model,” The Review of Financial Studies, Vol. 13, No. 1, 2000, pp. 585-625. doi:10.1093/rfs/13.3.585

[18] R. J. Elliot, T. K. Siu and L. Chan, “Option Pricing for GARCH Models with Markov Switching,” International Journal of Theoretical and Applied Finance, Vol. 9, No. 6, 2006, pp. 825-841. doi:10.1142/S0219024906003846

[19] P. Christoffersen, S. Heston and K. Jacobs, “Option Valuation with Conditional Skewness,” Journal of Eco- nomtrics, Vol. 131, No. 1-2, 2006, pp. 253-284.

[20] L. Mercuri, “Option Pricing in a GARCH Model with Tempered Stable Innovations,” Finance Research Letter, Vol. 5, No. 3, 2008, pp. 172-182. doi:10.1016/j.frl.2008.05.003

[21] A. M. Badescu and R. J. Kulperger, “GARCH Option Oricing: a Semiparametric Approach,” Insurance Mathe- matics and Economics, Vol. 43, No. 1, 2008, pp. 69-84. doi:10.1016/j.insmatheco.2007.09.011

[22] G. Barone-Adesi, H. Rasmussen and C. Ravanelli, “An Option Pricing Formula for the GARCH Diffusion Model,” Computational Statistics and Data Analysis, Vol. 49, No. 2, 2005, pp. 287-310. doi:10.1016/j.csda.2004.05.014

[23] G. Barone-Adesi, R. F. Engle and L. Mancini, “A GARCH Option Pricing Model with Filtered Historical Simulation,” Review of Financial Studies, Vol. 21, No. 3, 2008, pp. 1223-1258. doi:10.1093/rfs/hhn031

[24] A. Thavaneswaran, S. Peiris and J. Singh, “Derivation of Kurtosis and Option Pricing Formulas for Popular Vola- tility Models with Applications in Finance,” Communications in Statistics.Theory and Methods, Vol. 37, 2008, pp. 1799-1814. doi:10.1080/03610920701826435

[25] A. Thavaneswaran, J. Singh and S. S. Appadoo, “Option Pricing for some Stochastic Volatility Models,” Journal of Risk Finance, Vol. 7, No. 4, 2006, pp. 425-445. doi:10.1108/15265940610688982

[26] S. J. Taylor, “Asset Price Dynamics, Volatility, and Pre- diction,” Princeton University Press, Princeton, 2005.

[27] J. M. Steele, “Stochastic Calculus and Financial Applica- tions,” Springer, New York, 2001.

[28] J. Vecer, “Stochastic Finance: A Numeraire Approach,” Chapman & Hall/CRC Press, London, 2011.

[29] H. Y. Wong and Y. W. Lo, “Option Pricing with Mean Reversion and Stochastic Volatility,” European Journal of Operational Research, Vol. 197, No. 1, 2009, pp. 179- 187. doi:10.1016/j.ejor.2008.05.014

[30] D. Duffie, J. Pan and K. Singleton, “Transform analysis and Asset Pricing for Affine Jump-Diffusion,” Econo- metrica, Vol. 68, No. 6, 2000, pp. 1343-1376. doi:10.1111/1468-0262.00164

[31] 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, Vol. 6, No. 2, 2011, pp. 12-18.

[32] L. Cao and Z. F. Guo, “Applying Variance Gamma Correlated to Estimate Optimal Portfolio of Variance Swap,” Proceedings of International Conference of Financial Engineering, London, 6-8 July 2011.