Difference between Delta and Gamma Hedging

Delta hedging and Gamma hedging are two related concepts but there are differences between the two.

In both the cases a portfolio manager is trying to hedge a portfolio of stocks and options from changes in the stock prices by dynamically buying and selling stocks and options.

Delta is the first derivative of the option price with respect to the stock price, while gamma is the second derivative. So, we can define Delta as the change in the price of the option with the change in the stock price. Gamma, being the second derivative, is the change in the option’s Delta due to the change in the option’s price.

A portfolio manager will first want to make its portfolio delta-neutral, buy buying/selling stocks or options. However, as the stock price changes, the delta will change. Another objective for the portfolio manager will be to have low gamma or to hedge gamma. A low gamma indicates that the Delta will not change much when the stock prices change, so not much change will be required in the positions. A high gamma on the other hand means that the delta will change frequently and the portfolio manager will have to rebalance the portfolio frequently, buy continuously buying and selling stocks and options.

Delta and gamma can also be represented using the Taylor series, where Delta represents the first term of the Taylor series while Gamma represents the second term of the Taylor series. This is why Delta only represents a linear approximation of changes in option prices due to changes in stock prices. In reality, the changes in option prices are not linear, and a delta-neutral portfolio will only provide a linear hedge and will still have gamma. The approximation can be improved by hedging the portfolio for both delta and gamma which also considers the convexity.

So, you improve your hedge by hedging both Delta and Gamma. Delta hedging provides protection against small changes in stock prices, while gamma hedging also protects against larger changes in stock prices. You can improve your hedge even more by using Vega hedging, which protects portfolio from changes in volatility.