Optimal Delta Hedging for Options

The “practitioner Black-Scholes delta” for hedging options is a delta calculated from the Black-Scholes-Merton model (or one of its extensions) with the volatility parameter set equal to the implied volatility. As has been pointed out by a number of researchers, this delta does not minimize the variance of changes in the value of a trader's position. This is because there is a non-zero correlation between movements in the price of the underlying asset and implied volatility movements. The minimum variance delta takes account of both price changes and the expected change in implied volatility conditional on a price change. This paper determines empirically a model for the minimum variance delta. We test the model using data on options on the S&P 500 and show that it is an improvement over stochastic volatility models, even when the latter are calibrated afresh each day for each option maturity. We present results for options on the S&P 100, the Dow Jones, individual stocks, and commodity and interest-rate ETFs