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Therefore, in practice an option is delta hedged at discrete points in time and is called dynamic hedging. Although this might seem inconvenient, it is exactly the reason that traders make money on volatility. This is discussed in Chapter 3. 10 GAMMA Since delta changes whenever the stock price changes, it is useful to have a measure that captures this relationship. This measure is called gamma and gives the sensitivity of delta to a small change in stock price. Mathematically, gamma is the derivative of delta with respect to the stock price.
This means that in order to be delta hedged the trader will sell an additional 200 shares at € 27. 50, which means that the trader will buy back 200 shares at € 25. Just from rebalancing the delta, the trader has made 2 × 200 = € 400. To put this in a general formula; if a stock moves by y after which the option is delta hedged and the stock then moves back to its original level, the profit on one option is: y × (change in δ). This means that the profit from a move y without moving back to its original level is 1 · y × (change in δ).
For example, when both the implied volatility of the 100 % call and the implied volatility of the 120 % go down, but the implied volatility of the 100 % call goes down more than the implied volatility of the 120 % call. However, this scenario is not as clear cut as the above three scenarios. For example, if the vega of the 120 % call is larger than the vega of the 100 % call, a parallel downward shift in implied volatility for all strikes has a larger negative effect on the trader’s long 120 % call position than the positive counter effect on the trader’s short 100 % call position.