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 --> path dependent. 


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, sensitivity to asset moves can be dynamically hedged out over option's lifetime. 

P&L will come from the accumulated action of continuously re-balancing to keep the portfolio delta-neutral over time.

--> Gamma P&L  --> 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.

For small moves in the underlying --> replicating hedge is accurate. 

For larger moves in the underlying --> long options position > replicating hedge in both direction. 

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-->  chance of ending ITM can change rapidly, even for 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 gamma remains constant: 

- Option gains/loses value at the average delta.   -->   average delta = initial delta + 1⁄2 gamma x dS

- Replicating hedge gains/loses value at 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:

- it measures change in delta per unit of underlying 

- it is dependent on the absolute level on the underlying 


Moving to the concept of the dollar-gamma is more useful because:

- it allows us to directly calculate the gamma P&L for a given percentage underlying move. 

- it 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. 


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 (0.03)2 = $4,500. 


Gamma dollar gamma


Dollar gamma graphs



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