Delta hedging is one of the ways to reduce the risk of an option position, aimed at reducing the sensitivity of the derivatives portfolio to small fluctuations in the price of the underlying asset. Frequent delta hedging is a typical task for traders who specialize in options with expiry dates of up to one month.
Short-term options have a high gamma (at-the-money), so traders have to watch the delta very closely. Vega of short-term options is low. Therefore, trading short-term options involves active gamma trading, and as a result, active delta hedging.
For example, a long call position can be hedged by selling the underlying asset. If the delta and gamma of a call option are equal to 70% and 10% respectively, the trader must sell 7 shares for every 10 call options to make the portfolio delta neutral (delta equals to zero). Thus, if the price of the underlying asset moves insignificantly, the value of the portfolio consisting of a long 10 calls and a short position of 7 shares will remain unchanged, i.e. the portfolio has become insensitive to share price fluctuations.
Scenario 1: the price went up from $50 to $55
Since the gamma of one option was 10%, the delta of the entire combined portfolio grew from zero to 100%, i.e. by the value of the delta of one share (the delta of any share is equal to 1 or 100%). In order to hedge the delta to zero, a trader needs to short another stock.
Scenario 2: price went down from $50 to $45
Because of the gamma of the delta portfolio dropped from zero to minus 100%. In order to hedge the delta to zero, a trader needs to buy one Volkswagen share.
The positive gamma allows to hedge delta at a profit, while the negative gamma always results in losses from delta hedging. However, this does not mean that the negative gamma never needs to be hedged. Generally, an option position with a positive gamma means a negative theta, and vice versa. Therefore, a trader with a positive gamma will always lose on theta. A theta is the price that trader pays to have positive gamma. On the other hand, a negative gamma position implies a profit from receiving theta.