Option Pricing with Transaction Costs Using a Markov Chain Approximation

An e cient algorithm is developed to price European options in the presence of proportional transaction costs, using the optimal portfolio framework of Davis (1997). A fair option price is determined by requiring that an in nitesimal diversion of funds into the purchase or sale of options has a neutral e ect on achievable utility. This results in a general option pricing formula, in which option prices are computed from the solution of the investor's basic portfolio selection problem, without the need to solve a more complex optimisation problem involving the insertion of the option payo into the terminal value function. Option prices are computed numerically using a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. Comparisons with approximately replicating strategies are made. The method results in a uniquely speci ed option price for every initial holding of stock, and the price lies within bounds which are tight even as transaction costs become large. A general de nition of an option hedging strategy for a utility maximising investor is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed. JEL classi cation: C61; G11; G13