European Option Pricing with Transaction Costs

TLDR

The paper studies European option pricing in a Black–Scholes–type market with proportional transaction costs, where perfect replication is impossible and holding an option carries inherent risk. The option price is defined as the utility difference between trading with and without the option obligation, leading to two stochastic optimal control problems whose value functions are unique viscosity solutions of a fully nonlinear quasi‑variational inequality. The resulting price coincides with Black–Scholes when costs vanish, the viscosity solutions guarantee convergence of binomial discretisation schemes, and numerical experiments show that for long‑dated options the writer must charge a premium equal to the transaction cost of buying one share.

Abstract

The authors consider the problem of pricing European options in a market model similar to the Black–Scholes one, except that proportional transaction charges are levied on all sales and purchases of stock. “Perfect replication” is no longer possible, and holding an option involves an essential element of risk. A definition of the option writing price is obtained by comparing the maximum utilities available to the writer by trading in the market with and without the obligation to fulfill the terms of an option contract at the exercise time. This definition reduces to the Black–Scholes value when the transaction costs are removed. Computing the price involves solving two stochastic optimal control problems. This paper shows that the value functions of these problems are the unique viscosity solutions, with different boundary conditions, of a fully nonlinear quasi-variational inequality. This fact implies convergence of discretisation schemes based on the “binomial” approximation of the stock price. Computational results are given. In particular, the authors show that, for a long dated option, the writer must charge a premium over the Black–Scholes price that is just equal to the transaction charge for buying one share.