Delta Hedging, Gamma and Dollar Gamma

In this section we outline how vanilla options can be used to trade volatility.

Options are exposed to a wide range of factors:

- performance of underlying

- time-to-maturity

- volatility

- rates

- dividends

- ... 

 

The first-order exposure to moves in the underlying can be hedged out by the familiar delta-hedging process.

This leaves the exposure to volatility, paid for in time-decay as the most important sensitivity.

However this is not a pure volatility exposure, but is path dependent, varying over time and as the underlying moves towards and away from the option strike.

 

Suppose we hold a call option. To be delta-neutral, we can sell an amount of the underlying equivalent to the delta of the option. 

For ATM option, delta is about 0.5.

For other options, delta is more dependent on the volatility level. 

Delta of ITM option tends towards 1 as it becomes more ITM; and does so more quickly at lower levels of volatility. 

Delta of OTM option tends to 0 as it becomes more OTM; and does so more quickly for low volatility. 

 

Gamma P&L

By frequently re-adjusting this delta-hedge (in theory continuously, in practice usually daily) the sensitivity to direction of the underlying can be dynamically hedged out over the lifetime of the option. 

P&L will come from the accumulated action of continuously re-balancing the holding of the underlying, in order to keep the entire portfolio delta-neutral over time. This is referred to as the gamma P&L, and is paid for in the option premium which is marked to market as lost theta. 

 

Delta graphs

 

 

How is this gamma P&L actually made?

Essentially the gamma measures the convexity of the option.

This convexity always works in favour of long options positions --> although a replicating hedge is accurate for small moves in the underlying, for larger morves the long option will outperform the replicating hedge in both directions. 

For a delta-hedged option, the gamma P&L will be the outperformance of the option over the replicating hedge.

 

Gamma P&L is largest for short-maturity ATM options --> where the chance of the option ending ITM can change rapidly, even for relatively small moves. 

The more the delta changes, the more the replicating delta-hedge will underperform the long options position. 

 

How much will it underperform?

The actual amount depends on the difference between the initial and final delta of the option, which is gamma x dS. 

Assuming the gamma remains constant, the option gains or loses value at the average delta (average delta = initial delta + 1⁄2 gamma x dS). 

The replicating hedge gains or loses value at the initial delta. 

Therefore, the difference in performance between the option and the replicating hedge = 1/2 * Gamma * dS2

The gamma P&L from a move in the underlying is proportional to the gamma of the option and the square of the move.

 For instance the gamma P&L from a 2% move will be four times that of a 1% move. 

 

Gamma p l

 

Gamma calculation

 

Dollar Gamma = cash P&L from delta-hedging process

Gamma is a useful concept, but since it measures change in delta per unit of underlying, it is dependent on the absolute level on the underlying.

Example: gamma of an option on a stock worth €10 will be double the gamma of the equivalent option on a stock worth €20 (with same characteristics).

Moving to the concept of the dollar-gamma is more useful. 

It allows us to directly calculate the gamma P&L for a given percentage underlying move and makes it much easier to compare gamma exposures across different underlyings.

Dollar delta = cash equivalent exposure of the underlying. 

Dollar gamma = change in the dollar delta for a 1% move in the underlying. 

This dollar gamma is therefore equal to the normal gamma multiplied by the square of the value of the underlying and divided by 100. 

 

Dollar delta dollar gamma

 

For a return R -->  gamma P&L = 50 $Γ x R2.

Example: if we hold a position which is long $100,000 of dollar-gamma and the underlying moves by 3%, the P&L will be 50 x $100,000 x 0.032 = $4,500. 

 

Gamma dollar gamma

 

Dollar gamma graphs

 

 

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