Calculating gamma of gamma

S

stevenpaul

Hello everyone,

I was wondering if it is possible to calculate the gamma of the gamma--that is, the change in gamma relative to the change in the underlying's price--using the basic Greeks (i.e. delta, gamma, theta, etc.). Would you quantitative finance guys around here kindly post how this is done (if it is indeed possible) in prose? I'll probably misinterpret an equation. We don't ever get gamma of gamma data in option analytics programs, but the main Greeks are readily available. It would be great if we could set up a formula in Excel to calculate gamma of gamma based on the readily available values of the main Greeks.

Thank you very much!
 
What you're looking for is called "Speed", a third order derivative. Here's the formula:
690a4f3b0988d17f564c67b9d0f4b6e7.png


( Gamma / UL price ) * ( d1 / v*sqrt(T) + 1 )


Let me know if you need an example or explaining of the formula
 
Quote from TskTsk:

What you're looking for is called "Speed", a third order derivative. Here's the formula:
690a4f3b0988d17f564c67b9d0f4b6e7.png


( Gamma / UL price ) * ( d1 / v*sqrt(T) + 1 )


Let me know if you need an example or explaining of the formula
More...

Thank you Tsk. I'm unclear on d1, v, and T. Can you name those variables? Do you know of a shortcut for computing gamma of gamma (which, as you point out is also known as Speed) using the values of the other, more common Greeks? I guess a shortcut isn't necessary, as long as an Excel formula can handle the heavy lifting. If you, TskTsk, or anyone here is up for a fun challenge, any chance we could get an actual Excel formula? It might enrich the forum to have the formula on record here.

Thanks again.
 
v*sqrt(T) is time normalized volatility. In other words, a volatility of 0.25 gives a time-normalized volatility of 0.25*sqrt( Days to expiry / 365)

d1 is given by this formula:
eb21c0dcca19d35edd7f48e7dcdc6598.png



As for spreadsheets, I do have a sheet with BSM + all of the higher order greeks (Zomma, Vomma, Vanna, Charm, Speed and all that). I got it here: http://www.elitetrader.com/vb/attachment.php?s=&postid=3452907

Be aware that some of the formulas are not correct on that sheet. The basic formulas for Vega, Theta, Delta, Gamma, Rho are working fine. Speed is working fine as well. However some of the others are not completely accurate. I have the same sheet and have corrected the ones not working, however the sheet is now also heavily customized, and I'd rather not upload the whole thing, so I'd have to extract the formulas into a separate sheet and upload, which could take some time...
 
Here are some important points surrounding gamma of gamma, in prose:

The more time remaining, the more "stable" the gamma is, and the less quickly it changes as the underlying moves. As you approach expiration, gamma changes wildly as the underlying moves.

Eyeball your position, and note which strikes are net long and which are short. Doesn't matter whether the options are puts or calls - a 120 put and a 120 call are identical as far as gamma is concerned. So for this calculation, the following 3 positions are identical: long 50 120 calls and short 30 120 puts, long 70 120 puts and short 50 120 calls, long 20 120 calls and zero position in the 120 puts.

As the underlying moves toward a strike where you are net long options, your overall position gamma will increase - slowly and gently with a lot of time remaining until expiration, radically as you approach expiration. As the underlying moves toward a strike where you are net short options, overall position gamma will decrease.

As your gamma increases, so will your vega. That's why a comfortable position is to be long strikes below the market (if we're talking about, say, S&P500 options) and short strikes above the market. I'm assuming delta neutrality.

With such a position, every time the market goes down you get long vega and IV goes up, so you make money. Every time the market goes up you get short vega and IV goes down, so you make money again. That's why the lower strikes are more expensive than the higher strikes (in terms of IV). If, say, the S&P500 was at 1300, and you could buy the 1250 puts and calls for the same IV at which you could sell the 1350 puts and calls, then you would just buy 1250s, sell 1350s, get delta neutral with the underlying, and make money all day. In short order you'd be richer than Bill Gates. The market isn't THAT inefficient. That's a major reason why the lower strikes cost more than higher strikes.

It's good this was brought up. Change in gamma as the underlying moves is vitally important in option trading, but it is little talked-about and unfortunately poorly understood.
 
Good stuff, yes speed is important closer to expiration. I've been burnt on it before, if you got like 2 hours to expiration and a major move happens past your gamma strikes, you're gonna build or lose deltas extremely quickly, so it's a bitch to manage.
 
Thank you very much, dmo and TskTsk. I found a wonderful spreadsheet done up by Espen Haug which provides all the various Greeks with graphs for each of them over price and time. It is on a CD-Rom included in his __The Complete Guide to Option Pricing Formulas__. It's under copyright, otherwise I'd love to include it here. Still, I appreciate the sheet you linked us to, Tsk.

The aforementioned sheet offers Greeks from several models. If interested in forex option trading, would you use the Garman and Kohlhagen forex option model of 1983 over the Black-Scholes?

Are there other higher order Greeks that you think are as important to take into consideration as Speed?

Do you imagine that neutralizing Speed would be helpful to long gamma traders seeking to stay in trades and continue scalping, rather than having to restraddle after big moves?
 
Quote from stevenpaul:

Thank you very much, dmo and TskTsk. I found a wonderful spreadsheet done up by Espen Haug which provides all the various Greeks with graphs for each of them over price and time. It is on a CD-Rom included in his __The Complete Guide to Option Pricing Formulas__. It's under copyright, otherwise I'd love to include it here. Still, I appreciate the sheet you linked us to, Tsk.

The aforementioned sheet offers Greeks from several models. If interested in forex option trading, would you use the Garman and Kohlhagen forex option model of 1983 over the Black-Scholes?

Are there other higher order Greeks that you think are as important to take into consideration as Speed?

Do you imagine that neutralizing Speed would be helpful to long gamma traders seeking to stay in trades and continue scalping, rather than having to restraddle after big moves?
More...

To calculate speed of your position at any particular moment is not enough - more important is to have a sense of what will happen to your position gamma as the underlying moves over a range. For that you should plot position gamma over a range of underlying prices using Hoadley, perhaps the Haug CD, or whatever other tool you like. Then you have a complete understanding of your position's character.

Minimizing speed was a big concern when I was on the floor and trying very hard to stay gamma neutral. The best way to do that was to be net long and short options at alternating strikes. For example, long options at the 101 strike, short at the 102 strike, long at the 103 strike, short at the 104 strike, etc. With a lot of time remaining, such a position will keep your gammas steady as a rock as the underlying moves. With little time remaining though, strike-to-strike premium relationships break down and your gamma will change wildly with such a position as the underlying moves. If you're not a market-maker though, then staying gamma neutral is probably not your concern.

If what you're doing is just basically being long gamma and scalping gammas, then of course speed will be lower if your long gamma is spread across strikes (ie long the 101s, the 102s, the 103s, the 104s) than if you're just long at one strike. But that's probably a less important consideration than the cost of those gammas. In other words, I'd probably be more concerned about buying cheap gammas at liquid strikes. Minimizing speed by being long a series of strikes would be lower on my priority list. As a practical matter it may make more sense to buy more straddles after a big move because you're sticking with trading the at-the-moneys which are now more liquid than they were when they were far away from the money.
 
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