Delta Hedging
Risk Management Practices

Optimal Delta Hedging for Options ─ A Non-Technical Summary

Related Project: Risk Management & Market Liquidity 

Delta hedging is an important way in which traders manage the risk in a portfolio of options.  

Optimal Delta Hedging for Options

John Hull and Alan White

Joseph L. Rotman School of Management
University of Toronto

 The Global Risk Institute funded this research along with the preparation of this paper. 


Delta hedging is an important way in which traders manage the risk in a portfolio of options. The delta of an option measures the sensitivity of its price to the price of the underlying asset. For example, a delta of 0.3 for a stock option indicates that a small change in the price of the stock would be accompanied by a change in the price of the option that is 30% as much. The sale of 1000 options on the stock could therefore be hedged by buying 300 shares of the stock.

The calculation of delta depends on the option pricing model being used. The most popular model is the Black-Scholes model, but one of the weaknesses of this model is that the volatility of the underlying asset is assumed to be constant. In practice, for stocks, stock indices, and many other assets, volatility tends to be negatively correlated with price. When the price of the asset increases volatility tends to decrease; when the price of the asset decreases volatility tends to increase. This phenomenon leads to a decrease delta. For example, if the delta calculated from the Black-Scholes model (the “Black-Scholes delta”) is 0.3, the delta calculated when expected movements in volatility are taken into account might be 0.25. The latter is referred to as the “minimum variance delta”.

Given that delta hedging is relatively straightforward, it is important that traders get as much mileage as possible from it. Switching from the Black-Scholes delta to the minimum variance delta is therefore a desirable objective. Indeed it has two advantages. First, it lowers the variance of daily changes in the value of the hedged position. Second, it lowers the exposure to volatility changes because part of this exposure is handled by the position that is taken in the underlying asset.

One popular approach to calculating the minimum variance delta involves implementing a more advanced model than Black-Scholes where the volatility is stochastic. Hull and White (2016) develop another approach where the sensitivity of expected volatility changes to asset price changes is a quadratic function of the Black-Scholes delta divided by the product of the asset price times the square root of the time to maturity. The model involves three parameters which are estimated at the beginning of each month using data from the previous 36 months. These estimates are then used for the whole of the month.

White and Hulls model was tested using millions of price quotations on options on the S&P 500, the S&P 100, the Dow Jones Industrial Average of 30 stocks (DJIA), the individual stocks underlying the DJIA and five ETFs. In the case of the S&P 500 data, the performance of the developed model was compared with two different stochastic volatility models that are popular with practitioners; namely the SABR model and the local volatility model.  It was found that Hull and White’s model outperformed the traditional models, particularly for actively traded options. It performed similarly well for options on other stock indices. Weaker performance was observed for options on individual stocks and options on ETFs due to the increased presence of idiosyncratic noise in the price quotes for these options.

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