Option replication with transaction costs: general diffusion limits

Transaction costs preclude the construction of hedging strategies for general contingent claims. Leland introduced the concept of diffusion limits of hedging strategies in a small transaction cost limit. This paper establishes such diffusion limits for very general hedging strategies and also establishes leading order asymptotic expressions for the replication error. In addition to subsuming previously considered temporal strategies, the results in this paper yield new results, namely expressions for replication errors of stock price strategies and a variety of "renewal" strategies. Most importantly, this paper provides a unified methodology for calculating hedging strategies and replication errors in the small transaction cost limit. This is an essential component of optimization methods, when, for example one is trying to minimize replication error for a given initial portfolio value.