Gamma is defined as the rate of change of delta with respect to a given change in the underlying price. In the last post we looked at delta hedging a short call position and looked at the profit and loss incurred by the trader if the underlying price started to move.
The example was designed to be intraday (to remove the impact of time) and assumed no change in implied volatility. As a result, the trader’s profit and loss was solely determined by his hedging activity. The trader initially delta-hedged their exposure by buying 56 shares. We then moved the price up by an arbitrary 10% and the delta increased to 0.68, requiring the purchase of an additional 12 shares at the now higher spot price. The spot price then fell back to its starting point, causing delta to fall back to 0.56 and leading to the sale of the recently purchased 12 shares at the lower price. A case of buy high, sell low = make a loss.
Had the price movement been in the opposite direction (i.e. a fall followed by a rise) the end result would still have been a loss for the trader. The trader would have reduced their delta hedge by selling at a lower price and then increased it buy buying back at a higher price.
Here we can see the impact of gamma. A 10% change in the price caused delta to move from 0.56 to 0.68. Hence gamma for a 10% change in the price is 0.12. A very wise FX options trader once told me “Gamma is meaningless unless it is associated with a range of values” – this is an important point. A common way in which gamma is reported is for a 1% change in the underlying price. Notice that this is different than delta, which is designed for “small” changes in the underlying price.
Notice that if the gamma value was low, the difference between two delta values for a given change in price would be relatively small. The change in the delta hedge would be smaller and, in our example, the resulting loss would have been lower. On the other hand a higher gamma would require a larger adjustment to the delta hedge resulting in a larger profit and loss impact.
From this we can now consider some of the other definitions of gamma:
- The speed with which our profit and changes as the spot price changes
- Exposure to significant price changes
- Exposure to actual volatility
Tune in again for more on gamma!