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

37 Pages Posted: 14 Mar 2008

See all articles by Michal Czerwonko

Michal Czerwonko

Concordia University

Stylianos Perrakis

Concordia University, Quebec - John Molson School of Business

Date Written: February 20, 2008

Abstract

We examine the stochastic dominance bounds for call options in the presence of proportional transaction costs, developed in a discrete time and for a discrete or continuous state model of the returns of the underlying asset by Constantinides and Perrakis (CP, 2002, 2007). We 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, for which the results of that study were not available in a useful formulation. We then show that the CP lower bound for European call options 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. The limit is a Black-Scholes type expression that becomes equal to the exact Black-Scholes value if the transaction cost parameter is set equal to zero, thus providing the only known generalization of the Black-Scholes model that produces useful results under transaction costs. 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 options.

Keywords: option pricing, option bounds,incomplete markets, stochastic dominance, transaction costs, diffusion processes

JEL Classification: G12, G13

Suggested Citation

Czerwonko, Michal and Perrakis, Stylianos, Can the Black-Scholes Model Survive under Transaction Costs? An Affirmative Answer (February 20, 2008). Available at SSRN: https://ssrn.com/abstract=1105579 or http://dx.doi.org/10.2139/ssrn.1105579

Michal Czerwonko (Contact Author)

Concordia University ( email )

1455 De Maisonneuve Blvd. W.
Montreal, Quebec H3G 1M8
Canada

Stylianos Perrakis

Concordia University, Quebec - John Molson School of Business ( email )

1455 de Maisonneuve Blvd. W.
Montreal, Quebec H3G 1M8
Canada

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