GREEKS: GAMMA

​

Option dealers use Delta hedges to temporarily counteract the effects of changes in the underlying.

 

As the underlying market price moves the hedge ratio of an option changes, and the position must be re - balanced.

 

For a short Call Option, as the underlying increases in price the dealer must buy more than previously estimated, and as it decreases, it must sell some of its hedge, at a reduced price.

 

This effect is known as Gamma.

​

A short gamma position is said to have negative convexity - referring to a cost curve which is always above its tangent.

 

The measure of the required rebalancing of a Delta hedge is known as Gamma.

​

In Mathematical terms the Delta is the first derivative of the Option Price to changes in the underlying, and Gamma is the second derivative.

​

​

​

RE-BALANCING OF DELTA HEDGES AND THE COST OF GAMMA 

​

For a dealer delta hedging an options book, Gamma is the measure of the expected re-balancing of the dealers net position.

​

A dealer short an option that is Delta hedged, monetizes a loss every time the position is re-balanced.

​

The examples below display how a dealer delta hedging has a small loss in every re-balancing step, and how different paths have similar re-balancing costs.

​

The dealer starts on the left with a short call position and 50% delta hedge. As the share price goes up, the delta of the option increases - so he must now buy 25% more shares at a higher price. If the share price goes down again, he must sell the shares he bought at a higher price at the new lower price.

 

In the first example the price keeps going down. The option expires out of the money, but the dealer still has a loss, resulting from the re-balancing of his delta hedges.

 

The second example adds another path - showing how the cost of re-balancing is similar even if the stock price takes a different path. 

​

​