High order method for Black-Scholes PDE. (English) Zbl 1409.91276
Summary: In this paper, the Black-Scholes PDE is solved numerically by using the high order numerical method. Fourth-order central scheme and fourth-order compact scheme in space are performed, respectively. The comparison of these two kinds of difference schemes shows that under the same computational accuracy, the compact scheme has simpler stencil, less computation and higher efficiency. The fourth-order backward differentiation formula (BDF4) in time is then applied. However, the overall convergence order of the scheme is less than \(\mathcal O(h^4+\delta^4)\). The reason is, in option pricing, terminal conditions (also called pay-off function) is not able to be differentiated at the strike price and this problem will spread to the initial time, causing a second-order convergence solution. To tackle this problem, in this paper, the grid refinement method is performed, as a result, the overall rate of convergence could revert to fourth-order. The numerical experiments show that the method in this paper has high precision and high efficiency, thus it can be used as a practical guide for option pricing in financial markets.
MSC:
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
Keywords:
option pricing; Black-Scholes formula; compact difference scheme; backward differentiation formula; grid refinement methodReferences:
| [1] | Tavella, D.; Randall, C., Pricing Financial Instruments: The Finite Difference Method (2000), Wiley: Wiley New York |
| [2] | McCartin, B. J.; Labadie, S. M., Accurate and efficient pricing of vanilla stock options via the Crandall-Douglas scheme, Appl. Math. Comput., 143, 39-60 (2003) · Zbl 1053.91062 |
| [3] | C.W. Oosterlee, C.C.W. Leentvaar, X.Z. Huang, (2005) Accurate American Option Pricing by Grid Stretching and High Order Finite Differences equation, Working papers, DIAM, Delft University of Technology, the Netherlands.; C.W. Oosterlee, C.C.W. Leentvaar, X.Z. Huang, (2005) Accurate American Option Pricing by Grid Stretching and High Order Finite Differences equation, Working papers, DIAM, Delft University of Technology, the Netherlands. |
| [4] | Tangman, D. Y.; Gopaul, A.; Bhuruth, M., Numerical pricing of options using high-order compact finite difference schemes, J. Comput. Appl. Math., 218, 270-280 (2008) · Zbl 1146.91338 |
| [5] | Jain, M. K.; Jain, R. K.; Mohanty, R. K., A fourth-order difference method for the one-dimensional general quasilinear parabolic differential equation, Numer. Methods Partial Differential Equations, 6, 311-319 (1990) · Zbl 0715.65067 |
| [6] | Spotz, W. F.; Carey, G. F., A high-order compact formulation for the 3D Poisson equation, Numer. Methods Partial Differential Equations, 12, 235-243 (1996) · Zbl 0866.65066 |
| [7] | Shukla, R. K.; Zhong, X., Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation, J. Comput. Phys., 204, 404-429 (2005) · Zbl 1067.65088 |
| [8] | Patel, Kuldip Singh; Mehra, Mani, High-Order Compact Finite Difference Method for Black-Scholes PDE, (Agrawal, P. N.; etal., Mathematical Analysis and Its Applications (2015)), 393-403 · Zbl 1337.65119 |
| [9] | Guo, J.; Wang, W., An unconditionally stable, positivity-preserving splitting scheme for nonlinear Black-Scholes equation with transaction costs, Sci. World J., 2014, 1, 525207 (2014), 2014, (2014-5-7) |
| [10] | Guo, J.; Wang, W., On the numerical solution of nonlinear option pricing equation in illiquid markets, Comput. Math. Appl., 69, 2, 117-133 (2015) · Zbl 1360.91151 |
| [11] | Edeki, S. O.; Owoloko, E. A.; Ugbebor, O. O., The modified Black-Scholes model via constant elasticity of variance for stock options valuation, (Progress in Applied Mathematics in Science & Engineering (2016), AIP Publishing LLC), 83-90 · Zbl 1527.91163 |
| [12] | Black, F.; Scholes, M. S., The Pricing of Options and Corporate Liabilities, J. Polit. Econ., 81, 637-654 (1973) · Zbl 1092.91524 |
| [13] | Süli, E.; Endre; Mayers, D., An Introduction To Numerical Analysis (2003), Cambridge University Press · Zbl 1033.65001 |
| [14] | Leentvaar, C. C.W., Numerical Solution of the Black-Scholes Equation with a Small Number of Grid Points (2003), Delfty University of Technology, (Master’s thesis) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
