Computation of Reservation Prices of Options with 
 Proportional Transaction Costs

Abstract

In this paper we analyze the numerical scheme applied in Damgaard (2000a) to compute reservation prices of European options. In addition, we suggest and implement a procedure for computing reservation purchase prices of American options. In the paper we consider a continuous time economy with proportional transaction costs where the investors are assumed to have finite time horizons and HARA utility functions defined over terminal wealth. In the European case we show that the value functions are unique viscosity solutions of their respective HJB equations. We suggest and implement a discretization scheme for computing reservation prices of European options. This corresponds to solving a highly non-linear pde in time with three state variables. Convergence proofs for the employed discretization schemes are provided, and numerical examples are given. For the case of American call options we give an example showing that the presence of transaction costs implies under some circumstances that it is optimal to exercise an American call option written on a non-dividend paying security before maturity.