Cubic B-Spline Collocation Method for Pricing Path Dependent Options
We consider an efficient pricing method based on the cubic B-spline collocation method in which the approximation to the exact solution can be expressed as a linear combination of cubic B-spline basis functions. An earlier work has proposed to use such technique to solve the generalised Black–Scholes PDE to price European options only. In this work, we extend the application of the B-spline collocation method to price path-dependent options such as American, Barrier and Asian options under the Black–Scholes model. Our numerical results show that the new scheme is a method of choice for option pricing. Indeed the scheme has the same computational complexity as the standard second order finite difference scheme since they both require the solution of tridiagonal linear systems. However our numerical experiments reveal that the B-spline collocation method is more accurate in the infinity norm when solving convectively dominated financial problems and yields second order convergent prices and hedging parameters for both continuous and discretely sampled path-dependent options.
Keywordscollocation method cubic B-splines basis functions path-dependent option pricing
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- 1.Wu, L., Kwok, Y.K.: A front-fixing difference method for the valuation of American options. Journal of Financial Engineering 6, 83–97 (1997)Google Scholar
- 7.Věceř, J.: A new PDE approach for pricing arithmetic average Asian options. The Journal of Computational Finance 4, 105–113 (2001)Google Scholar
- 8.Prenter, P.M.: Splines and Variational Methods. Wiley (1975)Google Scholar
- 9.de Boor, C.: A Practical guide to splines. Springer (1972)Google Scholar
- 12.Mittal, R.C., Jain, R.K.: Numerical solutions of nonlinear Fisher’s reaction-diffusion equation with modified cubic B-spline collocation method. Mathematical Sciences (2013)Google Scholar
- 16.Tangman, D.Y., Peer, A.A.I., Rambeerich, N., Bhuruth, M.: Fast simplified approaches to Asian option pricing. The Journal of Computational Finance 14, 3–36 (2011)Google Scholar
- 18.Fang, F., Oosterlee, C.W.: Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions. Numerische Mathematik 114, 27–62 (2009)Google Scholar
- 19.Zhang, B., Oosterlee, C.W.: Efficient Pricing of Asian Options under Lévy Processes Based on Fourier Cosine Expansions, Report submitted to TU Delft University of Technology (2011)Google Scholar