Semi-Dynamic Hedging with Transaction Costs

The process of forming a replicating trading strategy that mimics the value of an option is of great interest in mathematical finance. One process of forming a replicating trading strategy involves the use of the European call option solution to the Black-Scholes partial differential equation. While this process does a fair job of providing direction as to how to rebalance a replicating trading strategy, it makes the assumption that readjustments can be made in continuous time. This is not realistic, since it is physically impossible to adjust continuously but also because of transaction costs. Once transaction costs are considered, the idea of using a self-financing strategy, or a strategy in which a portfolio is formed and no funds are required after the initial investment, is no longer realistic. Various ways of discretizing this continuous process will be examined in order to find an optimal strategy that both accurately approximates the value of an option and yet remains as close to self-financing as possible. Uniform partitioning will be examined but will most likely not be the sole basis of our optimal strategy. We demonstrate that, with the proper set of conditions, a strategy for readjusting a replicating portfolio can be found such that our goals are properly met. Once this strategy has been developed, the assumption of the absence of transaction costs will be eliminated, and a more realistic way of forming a replicating trading strategy that helps hedge away risk will have been formed.