A Kurganov-Tadmor numerical method for option pricing under the constant elasticity of variance model

محورهای موضوعی : Financial Mathematics

Sakineh Ghiasi ¹ , Nouredin Parandin ²

1 - Department of Mathematics, Payame Noor University, Tehran, Iran
2 - Department of Mathematics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

تاریخ دریافت : 1399/11/01 تاریخ پذیرش : 1400/01/05 تاریخ انتشار : 1401/04/10

کلید واژه: Kurganov-Tadmor method, Constant elasticity of variance, option price,

چکیده مقاله :

The primary goal of option pricing theory is to calculate the probability that an option will be exercised at expiration and assign a dollar value to it. Options pricing theory also derives various risk factors or sensitivities based on those inputs, since market conditions are constantly changing, these factors provide traders with a means of determining how sensitive a specific trade is to price fluctuations, volatility fluctuations, and the passage of time. In this study, we derive a new exact solution for pricing European options using Kurganov-Tadmor when the underlying process follows the constant elasticity of variance model. This method was successfully applied to nonlinear convection-diffusion equations by Kurganov and Tadmor. Also, we provide computational results showing the performance of the method for European option pricing problems. The results showed that the proposed method is convenient to calculate the option price for K=3,β=(-3)/4,and N=200.

منابع و مأخذ:

[1] Bahirai. A., Alipour, M., Sadiq, R., Option pricing in the presence of operational risk page, Advances in Mathematical finance and applications, 2020, 5(4), P.437-448. Doi:10.22034/amfa.2020.1897687.1409

[2] Black-Scholes F., Black- Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ, 1973, 81, P. 637-659.

[3] Blattberg, R. C., Gonedes, N. J., A comparison of the stable and student distributions as stochastic models for stock prices, Journal of Business, 1974, 47, P. 244-280. Doi: 10.1086/295634

[4] C.D. Fuh, R.H. Wang, J.C. Cheng, Option pricing in a Black–Scholes model with Markov switching, Working Paper, 2002.

[5] Cox, J.C., and Ross, S.A., The valuation of options for alternative stochastic processes, Journal of Financial Economics, 1976, 4, P.145-166. Doi:10.1016/0304-405X(76)90023-4

[6] Cox, J.C., Ingersoll, J.E., and Ross, S.A., A theory of the term structure of interest rates, In Theory of valuation, 2005, P.129-164. Doi:10.2307/1911242.

[7] Cox, J., Notes on option pricing I: Constant elasticity of variance diffusions, Unpublished note Stanford University Graduate School of Business. 1975 Sep. Doi: 10.4236/jmf.2011.13009

[8] Davoodi Kasbi, A., Dadashi, I., Azinfar, K., Stock price analysis using machine learning method (Non-sensory-parametric backup regression algorithm in linear and nonlinear mode), 2021, 6(2), P.285-301.Doi: 10.22034/amfa.2019.1869838.1232

[9] Emanuel, D., Mac, J., Beth Further results on the constant elasticity of variance call option pricing model, J. Finance. Quant. Anal, 1982, 17(4), P.533-554.

[10] Esmaili, B., Souri, A., Mirlohi, S, M., Higher moment portfolio Optimization with unequal weights based on generalized Capital Asset pricing model with independent and identically asymmetric power Distribution. Advances in Mathematical Finance and Applications, 2021, 6(2), P.263-283.Doi:10.22034/amfa.2020.1909590.1484

[11] Buffington, R. J., Elliott American options with regime switching, International Journal of Theoretical and Applied Finance, 2002, 5, P. 497-514. Doi:10.1142/S0219024902001523.

[12] Buffington, R. J., Elliott Regime switching and European Options Stochastic Theory and Control, Lecture Notes in Control and Information Sciences,2002, 280, P.73-82. Doi.10.1007/3-540-48022-6_5

[13] Bouchaud, J.P., Sornette, D., The BlackScholes option pricing problem in mathematical finance: generalization and extensions for a large class of stochastic processes, Journal of Physique I ,1994, 4, P.863-881

[14] Jianwei Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Mathematics and Economics,2009, 45(1), P.9-18. Doi: 10.1016/j.insmatheco.2009.02.006.

[15] Khalegi, P., Aghai, M., Rezai, F., Salience theory and pricing stock of corporates in Tehran stock exchange, Advances in mathematical finance and applications, 2018, 3(4), P.1-16. Doi:10.22034/AMFA.2019.577140.1120

[16] Kurganov, A., Tadmor, E., New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection--Diffusion Equations, Journal of Computational Physics, 2000, 160, P.241-282.Doi:10.1006/jcph.2000.6459.

[17] Mac Beth, J. D., Merville, L. J., An empirical examination of the Black-Scholes call option pricing model, Journal of Finance, 1979, 34, P. 1173-1186. Doi.1540-6261. 1979.tb00063.x

[18] Rezai, N., Elmi, Z., Behaviour finance model and behaviour biases in stock price forecasting, Advances in mathematical finance and applications, 2018, 3(4), P.67-82. Doi:10.22034/AMFA.2019.576127.1118.

[19] Schroder, M., Computing the constant elasticity of variance option pricing formula, J. Finance, 1989, 44(1), P.211-219. Doi:10.1111/j.1540-6261. 1989.tb02414.x

[20] Thakoor, N., Tangman, D., Bhuruth, M., A new fourth-order numerical scheme for option pricing under the CEV model, Appl. Math. Lett, 2013, 26(1), P.160-164. Doi: 10.1016/j.aml.2012.08.002.

[21] Naik V., Option valuation and hedging strategies with jumps in the volatility of asset returns, Journal of Finance,1993, 48(5), P. 1969-1984. Doi: 10.1111/j.1540-6261. 1993.tb05137.

[22] Wong, H., Zhao, J., An artificial boundary method for American option pricing under the CEV model, SIAM J. Numer. Anal, 2008, 46(4), P. 2183-2209. Doi.101137/060671541.

[23] Guo, X., Information and option pricings, Journal of Quantitative Finance, 2001, 1, P.38-44. Doi:10.1080/713665550.

[24] Zhang, S., Yang, H., and Yang, Y., A multi quadric quasi-interpolations method for CEV option pricing model, Journal of Computational and Applied Mathematics, 2019, 347, P.1-11.

[25] Zomorodian, GH., Barzegar, L., Kazemi,S., Poortalebi, Effect of oil price volatility and petroleum Bloomberg index on stock market returns of Tehran stock exchange using EGRACH model, Advances in Mathematical Finance and Applications, 2016, 1(2), P.69-84. Doi:10.22034/amfa.2016.527821