Can the Black-Scholes-Merton Model Survive under Transaction Costs? An Affirmative Answer

Abstract

We derive a reservation purchase price for a call option price under proportional transaction costs. The price is derived in discrete time for a general distribution of the returns of the underlying asset, as in Constantinides and Perrakis (CP, 2002, 2007). We then consider a lognormal diffusion model of these returns, and we formulate a discrete time trading version that converges to diffusion as the time partition becomes progressively more dense. Given the existence of a partition-independent and tight upper bound already derived in CP (2002), we focus on the lower bound. We show that the CP approach results in a lower bound for European call options that converges to a non-trivial and tight limit that is a function of the transaction cost parameter. This limit defines a reservation purchase price under realistic trading conditions for the call options and becomes equal to the exact Black-Scholes-Merton value if the transaction cost parameter is set equal to zero. We also develop a novel numerical algorithm that computes the CP lower bound for any discrete time partition and converges to the theoretical continuous time limit in a relatively small number of iterations. Last, we extend the lower bound results to American index and index futures options.