- Volume 51 Issue 4
DOI QR Code
AN ADAPTIVE FINITE DIFFERENCE METHOD USING FAR-FIELD BOUNDARY CONDITIONS FOR THE BLACK-SCHOLES EQUATION
- Received : 2013.08.26
- Published : 2014.07.31
We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.
Black-Scholes equation;finite difference method;far-field boundary conditions;adaptive grid;Peclet condition
- M. Brennan and E. Schwartz, Finite difference methods and jump processes arising in the pricing of contin-gent claims: a synthesis, J. Financ. Quant. Anal. 13 (1978), no. 3, 461-474. https://doi.org/10.2307/2330152
- G. W. Buetow and J. S. Sochacki, The trade-off between alternative finite difference techniques used to price derivative securities, Appl. Math. Comput. 115 (2000), no. 2-3, 177-190. https://doi.org/10.1016/S0096-3003(99)00141-1
- C. Christara and D. M. Dang, Adaptive and high-order methods for valuing American options, J. Comput. Financ. 14 (2011), no. 4, 74-113.
- D. J. Duffy, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, John Wiley & Sons, New York, 2006.
- S. Figlewski and B. Gao, The adaptive mesh model: a new approach to efficient option pricing, J. Financ. Econ. 53 (1999), no. 3, 313-351. https://doi.org/10.1016/S0304-405X(99)00024-0
- R. Geske and K. Shastri, Valuation by approximation: a comparison of alternative option valuation techniques, J. Financ. Quant. Anal. 20 (1985), no. 1, 45-71. https://doi.org/10.2307/2330677
- D. Jeong, Mathematical model and numerical simulation in computational finance, Ph.D. Thesis, Dep. Mathematics, Korea Univ., Korea, 2012.
- R. Kangro and R. Nicolaides, Far field boundary conditions for Black-Scholes equations, SIAM J. Numer. Anal. 38 (2000), no. 4, 1357-1368. https://doi.org/10.1137/S0036142999355921
- B. J. Kim, C. Ahn, and H. J. Choe, Direct computation for American put option and free boundary using finite difference method, Jpn. J. Ind. Appl. Math. 30 (2013), no. 1, 21-37. https://doi.org/10.1007/s13160-012-0094-9
- G. Linde, J. Persson, and L. von Sydow, A highly accurate adaptive finite difference solver for the Black-Scholes equation, Int. J. Comput. Math. 86 (2009), no. 12, 2104-2121. https://doi.org/10.1080/00207160802140023
- P. Lotstedt, S. Soderberg, A. Ramage, and L. Hemmingsson-Franden, Implicit solution of hyperbolic equations with space-time adaptivity, BIT Numer. Math. 42 (2002), no. 1, 134-158. https://doi.org/10.1023/A:1021978304268
- P. Lotstedt, J. Persson, L. von Sydow, and J. Tysk, Space-time adaptive finite difference method for European multi-asset options, Comput. Math. Appl. 53 (2007), no. 8, 1159-1180. https://doi.org/10.1016/j.camwa.2006.09.014
- R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci. 4 (1973), no. 1, 141-183. https://doi.org/10.2307/3003143
- J. Persson and L. von Sydow, Pricing European multi-asset options using a space-time adaptive FD-method, Comput. Vis. Sci. 10 (2007), no. 4, 173-183. https://doi.org/10.1007/s00791-007-0072-y
- J. Persson and L. von Sydow, Pricing American options using a space-time adaptive finite difference method, Math. Comput. Simulation 80 (2010), no. 9, 1922-1935. https://doi.org/10.1016/j.matcom.2010.02.008
- O. Pironneau and F. Hecht, Mesh adaption for the Black and Scholes equations, East-West J. Numer. Math. 8 (2000), no. 1, 25-35.
- E. Schwartz, The valuation of warrants: Implementing a new approach, J. Financ. Econ. 4 (1977), no. 1, 79-93. https://doi.org/10.1016/0304-405X(77)90037-X
- D. Tavella and C. Randall, Pricing Financial Instruments: The Finite Difference Method, John Wiley & Sons, New York, 2000.
- J. Topper, Financial Engineering with Finite Elements, John Wiley & Sons, New York, 2005.
- P. Wilmott, J. Dewynne, and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993.
- R. Windcliff, P. A. Forsyth, and R. A. Vetzal, Analysis of the stability of the linear boundary condition for the Black-Scholes equation, J. Comput. Financ. 8 (2004), 65-92.
- S. Zhao and G. W. Wei, Option valuation by using discrete singular convolution, Appl. Math. Comput. 167 (2005), no. 1, 383-418. https://doi.org/10.1016/j.amc.2004.07.010
- R. Zvan, P. A. Forsyth, and K. R. Vetzal, Robust numerical methods for PDE models of Asian options, J. Comput. Financ. 1 (1998), 39-78.
- Y. Achdou and O. Pironneau, Computational Methods for Option Pricing, SIAM, Philadelphia, 2005.
- F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ. 81 (1973), no. 3, 637-659. https://doi.org/10.1086/260062
- M. Brennan and E. Schwartz, The valuation of American put options, J. Financ. 32 (1977), no. 2, 449-462. https://doi.org/10.1111/j.1540-6261.1977.tb03284.x
- Second order accuracy finite difference methods for space-fractional partial differential equations vol.320, 2017, https://doi.org/10.1016/j.cam.2017.01.013
- PATH AVERAGED OPTION VALUE CRITERIA FOR SELECTING BETTER OPTIONS vol.20, pp.2, 2016, https://doi.org/10.12941/jksiam.2016.20.163
- Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations vol.2015, 2015, https://doi.org/10.1155/2015/359028
- Accurate and Efficient Computations of the Greeks for Options Near Expiry Using the Black-Scholes Equations vol.2016, 2016, https://doi.org/10.1155/2016/1586786