On Leland's Option Hedging Strategy with Transaction Costs

Nonzero transaction costs invalidate the Black-Scholes (1973) arbitrage argument based on continuous trading. Leland (1985) developed a hedging strategy which modifies the Black-Scholes hedging strategy with a volatility adjusted by the length of the rebalance interval and the rate of the proportional transaction cost. Leland claimed that the exact hedge could be achieved in the limit as the length of rebalance intervals approaches zero. Unfortunately, the main theorem (Leland 1985, P1290) is in error. Simulation results also confirm opposite findings to those in Leland (1985). Since standard delta hedging fails to exactly replicate the option in the presence of transaction costs, we study a pricing and hedging model which is similar to the delta hedging strategy with an endogenous parameter, namely the volatility, for the calculation of delta over time. With transaction costs, the optimal hedging volatility is substantially different from the stock's volatility under the criterion of minimizing the total absolute replication error weighted by the probabilities that the option is in or out of the money. This model provides a close explanation of the skewness of the implied volatilities for equity options. As a special case, option prices from our model are identical to the Black-Scholes option prices when transaction costs are ignored. Data on Samp;P500 index cash options from January to June 2002 are used to illustrate the model