A Convexity Approach to Option Pricing with Transaction Costs in Discrete Models

We study the option pricing problems under transaction costs in a discrete financial model with a riskless bond and one risky asset. For general models with a transaction fee, a perfectly replicating portfolio (if it exists) may not be optimal, i.e., a perfectly replicating portfolio may cost more than certain portfolios which super replicate the contingent claim. In this paper, we shall define two adapted convex set-valued processes H-t (2.1) and R-t (2.2) associated with a contingent claim and demonstrate how its (seller's) price and optimal portfolio processes may depend on them (Theorem 2.6). A sufficient condition (Theorem 2.12) is given to ensure the uniqueness of optimal portfolio process. As an application, we investigate when a perfect replication (if it exists) is optimal for a contingent claim and prove in a unified way some known results regarding the optimality of perfect replication using Ht and Rt. Finally, we shall demonstrate our methods in trinomial models and define two classes of contingent claims called AS and DS which include long and some short European call options respectively and under which we are able to determine their prices and the exact shapes of the H-t and R-t.