Introduction

- Gamma measures the change in delta due to the change in underlying price.

- Gamma represents the second-order sensitivity of the option to a movement in the underlying asset’s price.

- The higher the gamma, the more convex the theoretical payout. 

- Gamma is not a measure of value --> low/high gamma does not mean cheap/expensive --> IV = measure of option's value. 

 

The price of an option as a function of the underlying price is non-linear.

Gamma allows for a second-order correction to Delta to account for this convexity.

This convexity in the underlying price is what gives the option value. 

To see the second-order effect in pricing --> we will use models that assume some form of randomness in asset’s price.

 

BS Gamma for both calls and puts (since they have the same shape)--> Gamma

Dollar Gamma = Cash Gamma = Gamma * S2

 

 

Delta-hedged ptf --> ptf has been hedged by trading in the underlying asset against small mvmts in these assets. 

As a second-order effect --> Gamma becomes significant when large move in the underlying occurs. 

To hedge this Gamma --> need to trade other European options (convex instruments) so that Gammas cancel out. 

Lower Gamma --> lower need for large and frequent rebalancing of Delta. 

 

Gamma of a ptf of options = sum individual gammas of the options 

 

Gamma is important as it enables the traders to derive the profit on an option for any given stock move.

- Positive gamma --> one needs to sell stocks if S goes up and buy stocks if S goes down to be delta hedged.

- Negative gamma --> one needs to sell stocks if S goes down and buy stocks if S goes up to be delta hedged.

 

No free lunch

Gamma does not come for free.

An option holder pays for the right of buying low and selling high by means of the theta, the time decay of an option. 

The holder of an option needs to earn back the daily loss of the option by taking advantage of the underlying's moves.

The seller of an option makes money on the theta and loses it by rebalancing delta by buying high and sellling low. 

 

As we move into exotic structures, we will see that these may have quite different Gamma profiles to the European options.

Their gamma can change sign.

 

 

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We have seen that the delta for the call and put have the same shape, both increasing with the stock value. 

So the gamma for both instruments is identical and positive at all times. 

 

We have seen that most of the change in the delta occurs around the ATM point. 

So the gamma should be large around this point.

 

We have seen that delta get smoother for larger time to maturity. 

So we can expect a smoother gamma for a longer maturity. 

 

The graph of the gamma resembles the bell-shaped curve of the Normal distribution. 

 

Any trader that has sold options can tell you how tricky it is to manage the book near maturity when S ~ K.

The gamma gets so big that the hedging needs to be done very fast. 

As he is the option seller, he is gamma short, with a huge gamma position. 

If the market keeps moving around K, every rebalance will cost him money and the amount of theta for is limited.  

In some cases, in highly volatile markets, it is sometimes better to either fully hedge at once or fully unwind. 

If you are wrong, it will cost you, but rebalancing can be even more costly (slippage or bid/offer spreads). 

 

 

 

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