Dynamic Hedging Portfolios for Derivative Securities in the Presence of  Large Transaction Costs
Marco Avellaneda and Antonio Paras
Applied Mathematical Finance 1994

We introduce  a  new  class of strategies for  hedging  derivative  securities  in the presence of transaction costs assuming continuous-time prices for the underlying asset.  We  do not  assume necessarily that the payoff is convex as in Leland  (1985)  or that transaction costs are small compared to the price  changes  between  successive portfolio adjustments as in Hoggard, Whalley and Wilmott (1993). The type of hedging strategy to be used depends on the  magnitude of the Leland number A=sqrt(2/pi)*(k/sigma*sqrt(dt)), where k is the round-trip transaction cost, sigma is the volatility of the underlying asset, and dt is the time-lag between transactions. If A<1 it is possible to implement modified Black Scholes delta-hedging strategies, but not otherwise. We propose new hedging strategies that can be used with A>=1 to control effectively hedging risk and transaction costs. These strategies are associated with the solution of a nonlinear obstacle problem for a diffusion equation with volatility sigma_A=sigma*sqrt(1+A). In these strategies, there are periods in which rehedging takes place after each interval dt and other periods in which a static strategy is required. The solution to the obstacle problem is simple to calculate, and closed-form solutions exist for many problems of practical interest.