Finite Volume Schemes for Solving Nonlinear Partial Differential Equations in Financial Mathematics
In order to estimate a fair value of financial derivatives, various generalizations of the classical linear Black–Scholes parabolic equation have been made by adjusting the constant volatility to be a function of the option price itself. We present a second order numerical scheme, based on the finite volume method discretization, for solving the so–called Gamma equation of the Risk Adjusted Pricing Methodology (RAPM) model. Our new approach is based on combination of the fully implicit and explicit schemes where we solve the system of nonlinear equations by iterative application of the semi–implicit approach. Presented numerical experiments show its second order accuracy for the RAPM model as well as for the test with exact Barenblatt solution of the porous–medium equation which has a similar character as the Gamma equation.
Keywordsfinite volume method second-order scheme financial mathematics
Unable to display preview. Download preview PDF.
This work was supported by grants APVV–0351–07 and VEGA 1/0733/10.
- 1.Avellaneda, M., Levy, A., and ParŠas, A.: Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance 2, (1995) 73Ű-88Google Scholar
- 2.Balažovjech M., Mikula K.: A Higher Order Scheme for the Curve Shortening Flow of Plane Curves. Proceedings of ALGORITMY, STU Bratislava, (2009) 165–175Google Scholar
- 3.Barles, G., and Soner, H. M.: Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stoch. 2, 4 (1998) 369Ű-397Google Scholar
- 4.Black, F., and Scholes, M.: The pricing of options and corporate liabilities. The Journal of Political Economy 81, (1973) 637Ű-654Google Scholar
- 5.Jandačka, M., Ševčovič, D.: On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. J. Appl. Math. 3, (2005) 235Ű-258Google Scholar
- 6.Kratka, M.: No mystery behind the smile. Risk 9, (1998) 67Ű-71Google Scholar
- 7.Kwok, Y. K.: Mathematical models of financial derivatives. Springer-Verlag, Singapore (1998)Google Scholar
- 8.Leland, H. E.: Option pricing and replication with transaction costs. Journal of Finance 40, (1985) 1283Ű-1301Google Scholar
- 9.Merton, R.: Theory of rational option pricing. The Bell Journal of Economics and Management Science, (1973) 141Ű-183Google Scholar
- 10.Mimura, M., Tomoeda, K.: Numerical approximations to interface curves for a porous media equation. Hiroshima Math. J. 13, Hiroshima University, (1983) 273–294Google Scholar
- 11.Schönbucher, P. J., and Wilmott, P.: The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math. 61, 1 (2000) 232-Ű272Google Scholar
- 12.Ševčovič, D., Stehlíkova, B., Mikula, M.: Analytical and numerical methods for pricing financial derivatives. Nova Science Publishers, Hauppauge NY (2011)Google Scholar