Mehdizadeh Khalsaraei, M., Osmani, N. (2017). A family of positive nonstandard numerical methods with application to Black-Scholes equation. Sahand Communications in Mathematical Analysis, 05(1), 31-40. doi: 10.22130/scma.2017.19335

Mohammad Mehdizadeh Khalsaraei; Nashmil Osmani. "A family of positive nonstandard numerical methods with application to Black-Scholes equation". Sahand Communications in Mathematical Analysis, 05, 1, 2017, 31-40. doi: 10.22130/scma.2017.19335

Mehdizadeh Khalsaraei, M., Osmani, N. (2017). 'A family of positive nonstandard numerical methods with application to Black-Scholes equation', Sahand Communications in Mathematical Analysis, 05(1), pp. 31-40. doi: 10.22130/scma.2017.19335

Mehdizadeh Khalsaraei, M., Osmani, N. A family of positive nonstandard numerical methods with application to Black-Scholes equation. Sahand Communications in Mathematical Analysis, 2017; 05(1): 31-40. doi: 10.22130/scma.2017.19335

A family of positive nonstandard numerical methods with application to Black-Scholes equation

^{}Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran.

Abstract

Nonstandard finite difference schemes for the Black-Scholes partial differential equation preserving the positivity property are proposed. Computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the Black-Scholes equation. Unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

[1] A.J. Arenas, G. Gonzalez-Parra, and B. M. Caraballo, A nonstandard finite difference scheme for a nonlinear Black-Scholes equation, Math. Comput. Model., 57 (2013) 1663-1670.

[2] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. politic. Eco., 81 (1973) 637-659.

[3] D.J. Duffy, Finite Difference Methods in Financial Engineering, A Partial Differential Equation Approach, John Wiley and Sons, England, 2006.

[4] M. Mehdizadeh Khalsaraei, An improvement on the positivity results for 2-stage explicit Runge- Kutta methods, J. Comput. Appl. Math., 235 (2010) 137-143.

[5] M. Mehdizadeh Khalsaraei, Positivity of an explicit Runge-Kutta method, Ain. Shams. Eng. J., 6(4) (2015) 1217-1223.

[6] M. Mehdizadeh Khalsaraei and F. Khodadosti, Nonstandard finite difference schemes for differential equations, Sahand. Commun. Math. Anal, 1(2) (2014) 47-54.

[7] M. Mehdizadeh Khalsaraei and R. Shokri Jahandizi, Positivity preserving schemes for Black-Scholes equation, Res. J. Fin. Account., 6(7) (2015) 101-105.

[8] M. Mehdizadeh Khalsaraei and R. Shokri Jahandizi, A modified explicit method for the Black-Scholes equation with positivity preserving property, J. Math. Comput. Sci., 15 (2015) 299-305.

[9] M. Mehdizadeh Khalsaraei and R. Shokri Jahandizi, Positivity preserving schemes with application to finance: option pricing, Appl. Math. Engin. Manage. Tech., 3(4) (2015) 212-220.

[10] M. Milev and A. Tagliani, Discrete monitored barrier options by finite difference schemes, Math. Edu. Math., 38 (2009) 81-89.

[11] M. Milev and A. Tagliani, Low volatility options and numerical diffusion of finite difference schemes, Serdica. Math. J., 36(3) (2010) 223-236.

[12] M. Milev and A. Tagliani, Nonstandard finite difference schemes with application to finance: option pricing, Serdica. Math. J., 36(1) (2010) 75-88.

[13] M. Milev and A. Tagliani, Numerical valuation of discrete double barrier options, J. Comput. Appl. Math., 233 (2010) 2468-2480.

[14] M. Milev and A. Tagliani, Efficient implicit scheme with positivity preserving and smoothing properties, J. Comput. Appl. Math., 243 (2013) 1-9.

[15] J.M. Ortega, Matrix Theory, Plenum Press, New York, 1988.

[16] J.M. Ortega, Numerical Analysis: a second course, Academic Press, New York, 1990.

[17] R. Rannacher, Finite element solution of diffusion problems with irregular data, Numer. Math., 43 (1984) 309-327.

[18] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1985.

[19] D. Tavella and C. Randall, Pricing Financial Instruments: The Finite Difference Method, John Wiley and Sons, New York, 2000.

[20] B.A. Wade, A.Q.M. Khaliq, M. Yousuf, J. Vigo-Aguiar, and R. Deininger, On smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options, J. Comput. Appl. Math., 204 (2007) 144--158.

[21] P. Wilmott, Derivatives: The Theory and Practice of Financial Engineering, John Wiley and Sons, Chichester, 1998.

[22] G. Windisch, M-Matrices in Numerical Analysis, in: Teubner-Texte Zur Mathematik, Leipzing, 1989.