Stratégies Options - Gamma Hedging Principles

We saw that delta is not always the best way to hedge a derivative position.
Delta moves and can lead to under-hedged portfolios, thus to large losses.

→ Jumps happen and short option position can be very tricky to delta-hedge for example.
→ Option Delta is a varying quantity. In order to stay with the same delta exposure, adjustments are needed.

In order to avoid to re-balance continuously the portfolio, it's worth to gamma-hedge.

I - Principle

If we want to have an impact on the way delta varies, we have to deal with options.
Spot delta remains 100%, there is no gamma over there. The natural way to find gamma is to trade another option.

II - Example

Options O1 et O2 are quoted with deltas ∆1 = 0.47 et ∆2 = 0.23 and gammas Г1 = 0.15 and Г2 = 0.06.

The portfolio is long 47 shares and short 100 options with delta of 47%.
This portfolio is delta neutral.
Global delta is + 47 - ( 100 * 0.47 ) = 47 - 47 = 0
What we want is a steady delta.

- A Problem :
If the spot moves up, delta ∆1 would rise and won't be full hedged by the long position on asset.
- If the spot goes up by $1 new delta would be ∆1’= ∆1 + Г1 = 0.47 + 0.15 = 0.62.
Hence 15 shares are needed to be bought in order to remain neutral.

If the spot dropped down, delta ∆1 would decrease and the position would be over-hedged.
- If the spot goes down by $1, lnew delta ∆1’’= ∆1 - Г1 = 0.47 - 0.15 = 0.32. thus 15 shares are needed to be sold inorder to stay delta neutral.

- A Solution :
We already know that :
Г1 = 0.15
Г2 = 0.06

If one just trade 100*0.15 / 0.06 = 250 options O2, it would be enough to neutralize the whole delta.
The portfolio would be then :
- long 47 shares
- short 100 options O1
- long 250 options O2

In fact, it's enough to trade Г1 / Г2 options to be gamma hedged.

Where
Г1 O1's option gamma
Г2 O2's option gamma
Portfolio is then ' gamma-hedged'

- A Consequence :
The whole delta is now far from zero.
An adjustment is needed.

O2 delta was 0.23. Buying 250 O2 Options leads to an 'over-delta' of 250*0.23 = 57.5 shares
To remain neutral, it's needed to short 57 shares.
We were long 47 shares. By shorting 57 share, the portfolio is now net short 10 shares.

Final portfolio is :
- short 10 shares
- short 100 options O1
- long 250 options O2

This is a delta-gamma hedged portfolio.

Next : Gamma Hedging : Illustration
Previous : Gamma Hedging : A First Attempt

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Delta Hedging : A First Attempt
Gamma Hedging : A First Attempt

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