Option pricing with transaction costs and a nonlinear Black-Scholes equation

Abstract. In a market with transaction costs, generally, there is no nontrivial portfolio that dominates a contingent claim. Therefore, in such a market, preferences have to be introduced in order to evaluate the prices of options. The main goal of this article is to quantify this dependence on preferences in the specific example of a European call option. This is achieved by using the utility function approach of Hodges and Neuberger together with an asymptotic analysis of partial differential equations. We are led to a nonlinear Black-Scholes equation with an adjusted volatility which is a function of the second derivative of the price itself. In this model, our attitude towards risk is summarized in one free parameter a which appears in the nonlinear Black-Scholes equation : we provide an upper bound for the probability of missing the hedge in terms of a and the magnitude of the proportional transaction cost which shows the connections between this parameter a and the risk.

[1]  William Greider,et al.  One World, Ready or Not , 1997 .

[2]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[3]  H. Pagès,et al.  DERIVATIVE ASSET PRICING WITH TRANSACTION COSTS , 1992 .

[4]  H. Soner,et al.  There is no nontrivial hedging portfolio for option pricing with transaction costs , 1995 .

[5]  G. Barles,et al.  Exit Time Problems in Optimal Control and Vanishing Viscosity Method , 1988 .

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

[7]  J. Cockcroft.  Investment in Science , 1962, Nature.

[8]  A. Marco,et al.  Dynamic hedging portfolios for derivative securities in the presence of large transaction costs , 1994 .

[9]  T. Björk.  Arbitrage Theory in Continuous Time , 2019 .

[10]  G. Barles,et al.  Discontinuous solutions of deterministic optimal stopping time problems , 1987 .

[11]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[12]  Jakša Cvitanić,et al.  Hedging Contingent Claims with Constrained Portfolios , 1993 .

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

[14]  Mark Broadie,et al.  Optimal Replication of Contingent Claims under Portfolio Constraints , 1998 .

[15]  G. Constantinides.  Capital Market Equilibrium with Transaction Costs , 1986, Journal of Political Economy.

[16]  A. R. Norman,et al.  Portfolio Selection with Transaction Costs , 1990, Math. Oper. Res..

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

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

[19]  Michael T. Heath,et al.  Scientific Computing: An Introductory Survey , 1996 .

[20]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[21]  S. Kusuoka.  Limit Theorem on Option Replication Cost with Transaction Costs , 1995 .

[22]  Thaleia Zariphopoulou,et al.  Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences , 1999, Finance Stochastics.

[23]  Thaleia Zariphopoulou.  Investment-consumption models with transaction fees and Markov-chain parameters , 1992 .

[24]  I. Karatzas,et al.  On the pricing of contingent claims under constraints , 1996 .

[25]  T. Zariphopoulou.  Consumption-Investment Models with Constraints , 1994 .

[26]  H. Soner,et al.  Optimal Investment and Consumption with Transaction Costs , 1994 .

[27]  Mark H. A. Davis,et al.  European option pricing with transaction costs , 1993 .