A Boundary Element Formulation for the Pricing of Barrier Options

Affiliation(s)

Institute of Applied Mathematics, National Cheng-Kung University, Taiwan.

Department of Finance, National Dong Hwa University, Taiwan..

Institute of Applied Mathematics, National Cheng-Kung University, Taiwan.

Department of Finance, National Dong Hwa University, Taiwan..

ABSTRACT

In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a boundary value problem of heat equation with a moving boundary. The boundary integral representation and integral equation are derived. A boundary element method is designed to solve the integral equation. Special quadrature rules for the singular integral are used. A numerical example is also demonstrated. This boundary element formulation is correct.

In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a boundary value problem of heat equation with a moving boundary. The boundary integral representation and integral equation are derived. A boundary element method is designed to solve the integral equation. Special quadrature rules for the singular integral are used. A numerical example is also demonstrated. This boundary element formulation is correct.

KEYWORDS

Boundary Element Method; Black-Scholes Equation; Moving Boundary; Option Pricing; Barrier Option

Boundary Element Method; Black-Scholes Equation; Moving Boundary; Option Pricing; Barrier Option

Cite this paper

Shen, S. and Hsiao, Y. (2013) A Boundary Element Formulation for the Pricing of Barrier Options.*Open Journal of Modelling and Simulation*, **1**, 30-35. doi: 10.4236/ojmsi.2013.13006.

Shen, S. and Hsiao, Y. (2013) A Boundary Element Formulation for the Pricing of Barrier Options.

References

[1] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062

[2] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143

[3] J. C. Cox, S. A. Ross and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of Financial Economics, Vol. 7, 1979, pp. 229-264. doi:10.1016/0304-405X(79)90015-1

[4] N. Kunitomo and M. Ikeda, “Pricing Options with Curved Boundaries,” Mathematical Finance, Vol. 2, No. 4, 1992, pp. 275-298. doi:10.1111/j.1467-9965.1992.tb00033.x

[5] H. Geman and M. Yor, “Pricing and Hedging Double-Barrier Options: A Probabilistic Approach,” Mathematical Finance, Vol. 6, No. 4, 1996, pp. 365-378. doi:10.1111/j.1467-9965.1996.tb00122.x

[6] A. Pelsser, “Pricing Double Barrier Options Using Laplace Transforms,” Finance and Stochastics, Vol. 4, No. 1, 2000, pp. 95-104. doi:10.1007/s007800050005

[7] R. Zvan, K. R. Vetzal and P. A. Forsyth, “PDE Methods for Pricing Barrier Options,” Journal of Economics Dynamics & Control, Vol. 24, No. 11, 2000, pp. 1563-1590. doi:10.1016/S0165-1889(00)00002-6

[8] S. Sanfelici, “Galerkin Infinite Element Approximation for Pricing Barrier Options and Options with Discontinuous Payoff,” Decisions in Economics and Finance, Vol. 27, No. 2, 2004, pp. 125-151. doi:10.1007/s10203-004-0046-1

[9] A. M. L. Wang, Y. H. Liu and Y. L. Hsiao, “Barrier Option Pricing: A Hybrid Method Approach,” Quantitative Finance, Vol. 9, No. 3, 2009, pp. 341-352. doi:10.1080/14697680802595593

[10] S. Y. Shen and A. M. L. Wang, “On Stop-Loss Strategies for Stock Investments,” Applied Mathematics and Computation, Vol. 119, 2001, pp. 317-337. doi:10.1016/S0096-3003(99)00229-5

[1] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062

[2] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143

[3] J. C. Cox, S. A. Ross and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of Financial Economics, Vol. 7, 1979, pp. 229-264. doi:10.1016/0304-405X(79)90015-1

[4] N. Kunitomo and M. Ikeda, “Pricing Options with Curved Boundaries,” Mathematical Finance, Vol. 2, No. 4, 1992, pp. 275-298. doi:10.1111/j.1467-9965.1992.tb00033.x

[5] H. Geman and M. Yor, “Pricing and Hedging Double-Barrier Options: A Probabilistic Approach,” Mathematical Finance, Vol. 6, No. 4, 1996, pp. 365-378. doi:10.1111/j.1467-9965.1996.tb00122.x

[6] A. Pelsser, “Pricing Double Barrier Options Using Laplace Transforms,” Finance and Stochastics, Vol. 4, No. 1, 2000, pp. 95-104. doi:10.1007/s007800050005

[7] R. Zvan, K. R. Vetzal and P. A. Forsyth, “PDE Methods for Pricing Barrier Options,” Journal of Economics Dynamics & Control, Vol. 24, No. 11, 2000, pp. 1563-1590. doi:10.1016/S0165-1889(00)00002-6

[8] S. Sanfelici, “Galerkin Infinite Element Approximation for Pricing Barrier Options and Options with Discontinuous Payoff,” Decisions in Economics and Finance, Vol. 27, No. 2, 2004, pp. 125-151. doi:10.1007/s10203-004-0046-1

[9] A. M. L. Wang, Y. H. Liu and Y. L. Hsiao, “Barrier Option Pricing: A Hybrid Method Approach,” Quantitative Finance, Vol. 9, No. 3, 2009, pp. 341-352. doi:10.1080/14697680802595593

[10] S. Y. Shen and A. M. L. Wang, “On Stop-Loss Strategies for Stock Investments,” Applied Mathematics and Computation, Vol. 119, 2001, pp. 317-337. doi:10.1016/S0096-3003(99)00229-5