Alternating Direction Explicit Method for a Nonlinear Model in Finance
محورهای موضوعی : Financial Mathematics
Sima Mashayekhi
1
(Department of Mathematics, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran)
کلید واژه: Barles and Soner Model, Finite Difference Methods, Alternating Direction Explicit Methods, Black-Scholes Model,
چکیده مقاله :
In this article, at first standard linear Black-Scholes model and then some nonlinear Black-Scholes models will be considered and thereupon alternating direction explicit (ADE) method is applied firstly for solving the standard Black-Scholes model and then for Barles and Soner model which is one of the most complete and comprehensive nonlinear Black-Scholes models. Furthermore, the stability of this method has been considered and its accuracy will be compared with other numerical methods such as finite difference methods. Since in solving nonlinear Black-Scholes models by the ADE methods, we need to solve only some scalar nonlinear equations instead of a full nonlinear system of equations that we should solve in implicit methods, so this method can be a suitable choice for solving such models.
[1] Bakstein, D. and Howison, S., A Non-arbitrage Liquidity Model with Observable Parameters for Derivatives, 2003.
[2] Barles, G. and Soner, H. M., Option Pricing with Transaction Costs and a Nonlinear Black-Scholes Equation. Finance and Stochastics, 1998, 2(4), P. 369-397.
[3] Black, F. and Scholes, M., The Pricing of Options and Corporate Liabilities. Journal of political economy, 1973, 81(3), P. 637-654.
[4] Buckova, Z., Ehrhardt, M., Gunther, M., and Polvora, P., Alternating Direction Explicit Methods for Linear, Nonlinear and Multi-Dimensional Black-Scholes Models. Novel Methods in Computational Finance, Springer, 2017, P. 333-371.
[5] Company, R., Navarro, E., Pintos, J. R., and Ponsoda, E., Numerical Solution of Linear and Nonlinear Black-Scholes Option Pricing Equations. Computers & Mathematics with Applications, 2008, 56(3), P. 813-821.
[6] Dibachi, H., Behzadi, M.H., Izadikhah, M., Stochastic Modified MAJ Model for Measuring the Efficiency and Ranking of DMUs, Indian Journal of Science and Technology, 2015, 8(8), P. 1-7, Doi: 10.17485/ijst/2015/v8iS8/71505
[7] During, B., Hendricks, C., and Miles, J., Sparse Grid High-Order ADI Scheme for Option Pricing in Stochastic Volatility Models. Novel Methods in Computational Finance, Springer, 2017, P. 295-312.
[8] Ehrhardt, M., Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing. Nova Science Publishers, 2008.
[9] Frey, R. and Patie, P., Risk Management for Derivatives in Illiquid Markets: A simulation Study, In Advances in finance and stochastics, Springer, 2002, P. 137-159.
[10] Izadikhah, M., Azadi, M., Shokri Kahi, V., Farzipoor Saen, R., Developing a new chance constrained NDEA model to measure the performance of humanitarian supply chains, International Journal of Production Research, 2019, 57(3), P. 662-682, Doi: 10.1080/00207543.2018.1480840
[11] Jokar, H., Shamsaddini, K., Daneshi, V., Investigating the Effect of Investors' Behavior and Management on the Stock Returns: Evidence from Iran. Advances in Mathematical Finance and Applications, 2018, 3(3), P. 41-52. Doi: 10.22034/amfa.2018.544948
[12] Koleva, M. N., Positivity Preserving Numerical Method for Non-linear Black-Scholes Models. In International Conference on Numerical Analysis and Its Applications, Springer, 2012, P. 363-370.
[13] Kratka, M., No Mystery Behind the Smile. RISK-LONDON-RISK MAGAZINE LIMITED, 1998, 11, P. 67-71.
[14] Leland, H. E., Option Pricing and Replication with Transactions Costs. The journal of Finance, 1985, 40(5), P. 1283-1301.
[15] Lesmana, D. C. and Wang, S, An upwind Finite Difference Method for a Nonlinear Black-Scholes Equation Governing European Option Valuation under Transaction Costs. Applied Mathematics and Computation, 2013, 219(16), P. 8811-8828.
[16] Leung, S. and Osher, S., An Alternating Direction Explicit (ADE) Scheme for Time-Dependent Evolution Equations. Preprint UCLA June, 2005, 9, P. 2005.
[17] Liu, H. and Yong, J, Option Pricing with an Illiquid Underlying Asset Market. Journal of Economic Dynamics and Control, 2005, 29(12), P. 2125-2156.
[18] Mashayekhi, S. and Hugger, J, Finite Difference Schemes for a Nonlinear Black-Scholes Model with Transaction cost and Volatility risk. Acta Mathematica Universitatis Comenianae, 2015, 84(2), P.255-266.
[19] Merton, R. C., Theory of Rational Option Pricing, The Bell Journal of economics and management science. 1973, P.141-83.
[20] Parsa, B., Sarraf, F., Financial Statement Comparability and the Expected Crash Risk of Stock Prices. Advances in Mathematical Finance and Applications, 2018, 3(3), P. 77-93. Doi: 10.22034/amfa.2018.544951
[21] Piacsek, S. A. and Williams, G. P., Conservation Properties of Convection Difference Schemes, Journal of Computational Physics, 1970, 6(3), P. 392-405.
[22] Roberts, K. and Weiss, N., Convective Difference Schemes, Mathematics of Computation, 1966, 20(94), P. 272-299.
[23] Salehi, A., Mohammadi, S., Afshari, M., Impact of Institutional Ownership and Board Independence on the Relationship Between Excess Free Cash Flow and Earnings Management. Advances in Mathematical Finance and Applications, 2017, 2(3), P. 91-105. Doi: 10.22034/amfa.2017.533104
[24] Tone, K., Toloo, M., Izadikhah, M., A modified slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 2020, 287 (2), P. 560-571, Doi: 10.1016/j.ejor.2020.04.019.
[25] Towler, B. F. and Yang, R. Y., Numerical Stability of the Classical and the Modified Saul'yev's Finite Difference Methods. Computers & Chemical Engineering, 1978, 2(1), P. 45-51.
