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.

[1]  J. Leo van Hemmen,et al.  Calculation of Volterra Kernels for Solutions of Nonlinear Differential Equations , 2000, SIAM J. Appl. Math..

[2]  M. Mimura,et al.  Numerical approximations to interface curves for a porous media equation , 1983 .

[3]  Daniel Sevcovic,et al.  On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile , 2005 .

[4]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

[5]  Y. Kwok.  Mathematical models of financial derivatives , 2008 .

[6]  D. Ševčovič.  Analytical and Numerical Methods for Pricing Financial Derivatives , 2011 .

[7]  Guy Barles,et al.  Option pricing with transaction costs and a nonlinear Black-Scholes equation , 1998, Finance Stochastics.

[8]  udiger Frey.  Market Illiquidity as a Source of Model Risk in Dynamic Hedging , 2000 .

[9]  Stylianos Perrakis,et al.  Option pricing and replication with transaction costs and dividends , 2000 .

[10]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[11]  H. Leland..  Option Pricing and Replication with Transactions Costs , 1985 .

[12]  K. Mikula.  A HIGHER ORDER SCHEME FOR THE CURVE SHORTENING FLOW OF PLANE CURVES , 2009 .

[13]  Paul Wilmott,et al.  The Feedback Effect of Hedging in Illiquid Markets , 2000, SIAM J. Appl. Math..

[14]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[15]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.