An adaptive finite difference method using far-field boundary conditions for the Black--Scholes equation
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
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.
Abstract
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.
참고문헌 (26)
- Y. Achdou, "Computational Methods for Option Pricing" (2005), SIAM
- F. Black, "The pricing of options and corporate liabilities", J. Political Econ, 81(3), pp.637~659 (1973)
- M. Brennan, "The valuation of American put options", J. Financ, 32(2), pp.449~462 (1977)
- M. Brennan, "Finite difference methods and jump processes arising in the pricing of contin-gent claims: a synthesis", J. Financ. Quant. Anal, 13(3), pp.461~474 (1978)
- G. W. Buetow, "The trade-off between alternative finite difference techniques used to price derivative securities", Appl. Math. Comput, 115(2-3), pp.177~190 (2000)
- C. Christara, "Adaptive and high-order methods for valuing American options", J. Comput. Financ, 14(4), pp.74~113 (2011)
- D. J. Duffy, "Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach" (2006), John Wiley & Sons
- S. Figlewski, "The adaptive mesh model: a new approach to efficient option pricing", J. Financ. Econ, 53(3), pp.313~351 (1999)
- R. Geske, "Valuation by approximation: a comparison of alternative option valuation techniques", J. Financ. Quant. Anal, 20(1), pp.45~71 (1985)
- D. Jeong, "Mathematical model and numerical simulation in computational finance", Ph.D, Dep. Mathematics, pp.2012
- R. Kangro, "Far field boundary conditions for Black–Scholes equations", SIAM J. Numer. Anal, 38(4), pp.1357~1368 (2000)
- B. J. Kim, "Direct computation for American put option and free boundary using finite difference method", Jpn. J. Ind. Appl. Math, 30(1), pp.21~37 (2013)
- G. Linde, "A highly accurate adaptive finite difference solver for the Black–Scholes equation", Int. J. Comput. Math, 86(12), pp.2104~2121 (2009)
- P. L¨otstedt, "Implicit solution of hyperbolic equations with space-time adaptivity", BIT Numer. Math, 42(1), pp.134~158 (2002)
- P. L¨otstedt, "Space-time adaptive finite difference method for European multi-asset options", Comput. Math. Appl, 53(8), pp.1159~1180 (2007)
- R. C. Merton, "Theory of rational option pricing", Bell J. Econ. Manag. Sci, 4(1), pp.141~183 (1973)
- J. Persson, "Pricing European multi-asset options using a space-time adaptive FD-method", Comput. Vis. Sci, 10(4), pp.173~183 (2007)
- J. Persson, "Pricing American options using a space-time adaptive finite difference method", Math. Comput. Simulation, 80(9), pp.1922~1935 (2010)
- O. Pironneau, "Mesh adaption for the Black and Scholes equations", East-West J. Numer. Math, 8(1), pp.25~35 (2000)
- E. Schwartz, "The valuation of warrants: Implementing a new approach", J. Financ. Econ, 4(1), pp.79~93 (1977)
- D. Tavella, "Pricing Financial Instruments: The Finite Difference Method" (2000), John Wiley & Sons
- J. Topper, "Financial Engineering with Finite Elements" (2005), John Wiley & Sons
- P. Wilmott, "Option Pricing: Mathematical Models and Computation" (1993), Oxford Financial Press
- R. Windcliff, "Analysis of the stability of the linear boundary condition for the Black–Scholes equation", J. Comput. Financ, 8, pp.65~92 (2004)
- S. Zhao, "Option valuation by using discrete singular convolution", Appl. Math. Comput, 167(1), pp.383~418 (2005)
- R. Zvan, "Robust numerical methods for PDE models of Asian options", J. Comput. Financ, 1, pp.39~78 (1998)