Optimal hedging of options with small but arbitrary transaction cost structure

In this paper we consider the problem of hedging options in the presence of cost in trading the underlying asset. This work is an asymptotic analysis of a stochastic control problem, as in Hodges & Neuberger [1] and Davis, Panas & Zariphopoulou [2]. We derive a simple expression for the 'hedging bandwidth' around the Black-Scholes delta; this is the region in which it is optimal not to rehedge. The effect of the costs on the value of the option, and on the width of this hedging band is of a significantly greater order of magnitude than the costs themselves. When costs are proportional to volume traded, rehedging should be done to the edge of this band, when there are fixed costs present, trading should be done to an optimal point in the interior of the no-transaction region.