Pricing European options with proportional transaction costs and stochastic volatility using a penalty approach and a finite volume scheme

Abstract In this paper we propose a combination of a penalty method and a finite volume scheme for a four-dimensional time-dependent Hamilton–Jacobi–Bellman (HJB) equation arising from pricing European options with proportional transaction costs and stochastic volatility. The HJB equation is first approximated by a nonlinear partial differential equation containing penalty terms. A finite volume method along with an upwind technique is then developed for the spatial discretization of the nonlinear penalty equation. We show that the coefficient matrix of the discretized system is an M -matrix. An iterative method is proposed for solving the nonlinear algebraic system and a convergence theory is established for the iterative method. Numerical experiments are performed using a non-trivial model pricing problem and the numerical results demonstrate the usefulness of the proposed method.

[1]  Valeri I. Zakamouline.  European option pricing and hedging with both fixed and proportional transaction costs , 2006 .

[2]  P. Boyle,et al.  Option Replication in Discrete Time with Transaction Costs , 1992 .

[3]  Kok Lay Teo,et al.  Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation , 2006 .

[4]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[5]  Valeri Zakamouline.  American option pricing and exercising with transaction costs , 2005 .

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

[7]  European option pricing with transaction costs and stochastic volatility: an asymptotic analysis , 2012, 1211.4396.

[8]  Song Wang,et al.  THE VISCOSITY APPROXIMATION TO THE HAMILTON-JACOBI-BELLMAN EQUATION IN OPTIMAL FEEDBACK CONTROL: UPPER BOUNDS FOR EXTENDED DOMAINS , 2009 .

[9]  Wen Li,et al.  A numerical method for pricing European options with proportional transaction costs , 2014, J. Glob. Optim..

[10]  Michael Monoyios.  Option pricing with transaction costs using a Markov chain approximation , 2004 .

[11]  Song Wang,et al.  Pricing american options under proportional transaction costs using a penalty approach and a finite difference scheme , 2013 .

[12]  Stewart Hodges,et al.  Optimal delta-hedging under transactions costs , 1997 .

[13]  Wen Chen,et al.  A finite difference method for pricing European and American options under a geometric Lévy process , 2014 .

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

[15]  Song Wang.  A penalty approach to a discretized double obstacle problem with derivative constraints , 2015, J. Glob. Optim..

[16]  Song Wang,et al.  Penalty Approach to the HJB Equation Arising in European Stock Option Pricing with Proportional Transaction Costs , 2009 .

[17]  Shuhua Zhang,et al.  A superconvergent fitted finite volume method for Black–Scholes equations governing European and American option valuation , 2015 .

[18]  Song Wang.  A penalty method for a finite-dimensional obstacle problem with derivative constraints , 2014, Optim. Lett..

[19]  Stephen Figlewski.  Options Arbitrage in Imperfect Markets , 1989 .

[20]  Utility Based Option Evaluation with Proportional Transaction Costs , 2000 .

[21]  Song Wang,et al.  A Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities , 2006, Computing.

[22]  Song Wang,et al.  A novel fitted finite volume method for the Black-Scholes equation governing option pricing , 2004 .

[23]  Anders Damgaard.  Computation of Reservation Prices of Options with Proportional Transaction Costs , 2000 .

[24]  Song Wang,et al.  Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs , 2015, Appl. Math. Comput..

[25]  K. Toft..  On the Mean-Variance Tradeoff in Option Replication with Transactions Costs , 1996 .

[26]  D. Liu.  European Option Pricing with Transaction Costs , 2000 .

[27]  C. Sgarra,et al.  American option valuation in a stochastic volatility model with transaction costs , 2015 .

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

[29]  R. Uppal,et al.  Optimal Replication of Options with Transactions Costs and Trading Restrictions , 1993, Journal of Financial and Quantitative Analysis.