ROBUST AND ACCURATE METHOD FOR THE BLACK-SCHOLES EQUATIONS WITH PAYOFF-CONSISTENT EXTRAPOLATION

Title & Authors
ROBUST AND ACCURATE METHOD FOR THE BLACK-SCHOLES EQUATIONS WITH PAYOFF-CONSISTENT EXTRAPOLATION
CHOI, YONGHO; JEONG, DARAE; KIM, JUNSEOK; KIM, YOUNG ROCK; LEE, SEUNGGYU; SEO, SEUNGSUK; YOO, MINHYUN;

Abstract
We present a robust and accurate boundary condition for pricing financial options that is a hybrid combination of the payoff-consistent extrapolation and the Dirichlet boundary conditions. The payoff-consistent extrapolation is an extrapolation which is based on the payoff profile. We apply the new hybrid boundary condition to the multi-dimensional Black-Scholes equations with a high correlation. Correlation terms in mixed derivatives make it more difficult to get stable numerical solutions. However, the proposed new boundary treatments guarantee the stability of the numerical solution with high correlation. To verify the excellence of the new boundary condition, we have several numerical tests such as higher dimensional problem and exotic option with nonlinear payoff. The numerical results demonstrate the robustness and accuracy of the proposed numerical scheme.
Keywords
multi-dimensional Black-Scholes equations;operator splitting method;extrapolation;linear boundary condition;high correlation;
Language
English
Cited by
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References
1.
Z. Cen, A. Le, and A. Xu, Finite difference scheme with a moving mesh for pricing Asian options, Appl. Math. Comput. 219 (2013), no. 16, 8667-8675.

2.
D. J. Duffy, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, John Wiley and Sons, 2006.

3.
A. Esser, General valuation principles for arbitrary payoffs and applications to power options under stochastic volatility, Financ. Markets and Portfolio Manage. 17 (2003), no. 3, 351-372.

4.
A. Golbabai, L. V. Ballestra, and D. Ahmadian, A Highly Accurate Finite Element Method to Price Discrete Double Barrier Options, Comput. Econ. (2013), 1-21.

5.
E. G. Haug, The Complete Guide to Option Pricing Formulas, McGraw-Hill, New York, 1998.

6.
R. C. Heynen and H. M. Kat, Pricing and hedging power options, Financ. Eng. JPN. Markets 3 (1996), no. 3, 253-261.

7.
S. Ikonen and J. Toivanen, Operator splitting methods for American option pricing, Appl. Math. Lett. 17 (2004), no. 7, 809-814.

8.
K. J. In't Hout and S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Numer. Anal. Model 7 (2010), no. 2, 303-320.

9.
A. Q. M. Khaliq, D. A. Voss, and K. Kazmi, Adaptive ${\theta}$-methods for pricing American options, J. Comput. Appl. Math. 222 (2008), no. 1, 210-227.

10.
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.

11.
MathWorks, Inc.,MATLAB: the language of technical computing, http://www.mathworks.com/, The MathWorks, Natick, MA., 1998.

12.
C. Reisinger and G. Wittum, On multigrid for anisotropic equations and variational inequalities Pricing multi-dimensional European and American options, Comput. Vis. Sci. 7 (2004), no. 3-4, 189-197.

13.
A. Tagliani and M. Milev, Laplace Transform and finite difference methods for the Black-Scholes equation, Appl. Math. Comput. 220 (2013), 649-658.

14.
P. G. Zhang, Exotic Options: a Guide to Second Generation Options, World Scientific, Singapore, 1998.

15.
N. Zheng and J. F. Yin, On the convergence of projected triangular decomposition methods for pricing American options with stochastic volatility, Appl. Math. Comput. 223 (2013), 411-422.

16.
R. Zvan, K. R. Vetzal, and P. A. Forsyth, PDE methods for pricing barrier options, J. Econom. Dynam. Control 24 (2000), no. 11, 1563-1590.