What impacts Gamma?

1. Moneyness

Gamma tells us how much Delta will move if the underlying moves.

Gamma generally has its peak value close to ATM and decreases as the option goes deeper ITM or OTM. 

Options that are deeply ITM/OTM have gamma close to 0. 

 

2. Time to maturity

As the time to expiration draws nearer, gamma of ATM options increases while gamma of ITM/OTM options decreases. 

 

So Gamma of a European option is high when the underlying trades near the strike and there is little time left to maturity.

Near these points --> need for more frequent Delta hedging and thus inflict more hedging costs upon the trader.

Intuitively --> delta on the day of expiry will change from roughly 0% if S just < K to roughly 100% if S just > K. 

So a small change in spot --> large change in delta --> gamma is very high. 

 

3. Volatility: Gamma weakened by Volatility

We have seen that volatility attacked the delta --> not surprising it weakened the gamma too. 

Higher volatility --> less pronounced S-shape delta curve. 

Higher volatility --> less pronounced/wider more stable bell-curved gamma. 

 

- ATM region --> higher volatility lowers the gamma.

- ITM/OTM regions --> higher volatility increases the gamma.

 

We can think of this effect in terms of the time value of European options.

 

For low levels of vol: 

- deep ITM/OTM options have little time value and can only gain time value if asset moves closer to strike. 

For high levels of vol: 

- both ITM/OTM options have time value --> Gamma near strike should not be too different from away from strike. 

- gammas tend to be more stable across all strike prices. 

 

In fact, the gamma and vega play a similar role to the options price. 

Gamma is crucial to understanding how much money the hedge is making/losing. 

It was directly related to the realised volatility. 

The vega is in a way the forward measure for this. 

 

Example: 

Trader is long gamma through buying a put option. 

If market right and balanced --> paid a fair price --> money he can expect to make will be paid for through premium. 

However, if the market suddenly realises that the option was underpriced. 

Expectation becomes that stock will be more volatile --> IV will increase --> option more expensive in the market. 

The trader has a choice: 

- can keep on delta hedging and profit from this highly volatile market. 

- can cash in and sell the option at a higher price. 

 

How much money can he expect to make over lifetime of the option if he continues hedging himself? 

That amount can be shown to be identical to cashing in right now. 

Underlying this statement, there is the assumption that the IV turns out to be the correct one. 

This actually gives the trader an interesting dilemma. 

Because the market will keep changing his mind over time about the best IV that has to be used. 

Option traders will take positions when they think the IV is low/high. 

The money they will pick up by trading is directly related to the vega of these options. 

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