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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



Comments (10)

  • 1. Shez | 10/05/2021
Hi Max,
Thanks for the insightful post. Just one question regarding the gamma pnl for short futures. Is the delta not 1 (I know technically it is exp[-r(T-t)] ) for long futures, thus making the short position -dS or -exp[-r(T-t)]dS. In this case the overall terms in the gamma pnl would not cancel. Similarly what if you shorted stock instead of shorting futures, in which case it would surely be -dS.
  • 2. Fred | 08/10/2020
Hi, would you recommend any reference/book for option knowledge like this (is there a book that covers the topic you wrote for instance?)? I would like to learn options in more depth. Thanks!
  • maxxman100 | 03/01/2021
Honestly, I suggest using the new website which contains a book and an option pricer that can be used alongside the book. I will try to add one chapter per week in the book. You can find the new website at . The book is under the 'Derivatives Academy' section and the pricer is under the 'Derivatives Pricer' section. Let me know if you have any questions. You can contact me at
  • maxxman100 | 09/10/2020
Hi Fred, Can you write me an email on with your background and goals so that I can provide you with the right material you are looking for? Regards, Maxime
  • 3. Rohit | 14/07/2020
Hi Maxx,
Appreciate your notes on these topics. You mentioned you would post notes on a new website?
Please post the link if it is something you are ok to share.
  • maxxman100 | 15/07/2020
Hi Rohith, Yes, I have been busy with another project that I have recently released (trainings on financial markets in Brussels). I have had to postpone the other website. I will surely have to wait for 2021 for the new website. Hopefully, I should be able to integrate in the website some exotic option calculators that I have personally developed for my students.
  • 4. Mike | 23/11/2019
Can I say
Overall position delta is equal to overall position gamma multiply by chg in underlying divided by 2
  • maxxman100 | 20/01/2020
Hi Mike, I need to set up notifications when a comment is written here I am sorry. I do not come here often. I am still working on another website with more advanced features :) To answer your question: No! Gamma does not tell you anything on your overall delta position. Hedging delta does not impact your gamma. Gamma tells you how your delta position moves when the underlying moves. An option price is convex with respect to the underlying price. Delta hedging by trading the underlying stock is purely linear and you will be hedged only for small moves of the underlying (for small moves, linear approximation of the convex curve is fine). However, for large moves, your linear approximation is not fine anymore as your delta changed (the slope changed as the option price is convex (positive second derivative)). The larger the gamma, the more delta changes and the more frequent you should rebalance your delta hedge. I hope I answered your question properly. Br, Maxx
  • 5. Andy | 27/11/2018

i am confused, in the dollar gamma example, should one not square the 3%? why are you multiplying by 0.032?

Thank you
  • maxxman100 | 03/04/2019
Hi Andy, I am sorry for the late reply. I did not even know that it was public. I was just writing in here as a back up and use it as content for a lesson on structured products and market finance. To answer your question, there is nothing to be confused about. It is actually 0.03^2. You can easily see it. If you do 50*100.000*(0.03^2), you will surely find 4.500$. I will create a new website and sort/improve the content. I will gladly keep you updated if you want. Thanks spotting it though and hope it helped! Regards, Maxx

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The NEW website is OUT! 

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