Singular Stochastic Control and a Modified Black-Scholes Theory Incorporating Transaction Costs

Abstract

In the presence of transaction costs, it is no longer possible to perfectly replicate the payoff of a European option by trading in the underlying stock. We develop a new option hedging strategy based on minimizing the expected cumulative hedging error and additional cost of rebalancing due to proportional transaction costs. The resulting singular stochastic control problem can be shown to be equivalent to an optimal stopping problem with relatively low computational complexity. Our results show that an optimal hedge consists of selling or buying the underlying stock whenever the holding of shares falls above or below a no-transaction band containing the option's delta. Using a self-financing argument, we then derive a writer's and a buyer's values of an option as the expected total cost of portfolio adjustments when the optimal hedge is carried out and establish bounds on these values.