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When and at-the-money regular call or put approaches expiration, gamma tends to infinity. However, for practical purposes, there is only a finite change in delta. The problem is that if any of the options in your portfolio gets a crazy high number, this ruins the usefulness of the gamma of the whole portfolio.

I guess that that for practical applications you can choose a fixed change dx in the value of the underlying and compute gamma numerically using the pricing function f(x):

= ( (f(x+dx)-f(x)/dx - (f(x)-f(x-dx)/dx )/dx
= ( f(x+dx) - 2*(f(x) - f(x-dx)) / dx^2

The absolute value of this numerically computed gamma could be used as a ceiling for the absolute value of the closed-form formula-computed gamma.

Ideally, dx could be something like the standard deviation of daily price changes, or any number (of this order of magnitude) that the traders find easy to mentally manage.

I wrote the above just looking at the crystal ball. How do traders and financial engineers tackle this problem in practice?

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Many traders build a spreadsheet of how their delta changes across a range of market moves (-10%, -8%, ,....+8%, +10%) for example. That is a lot more useful than a single gamma number. It also means that the answers are finite and useful.

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In addition to computing Gamma for the overall portfolio, I also compute Gammas for each separate expiration date. When I look at it I see immediately that the "funny" gamma is in the subportfolio that is very close to expiration. I can then identify the option that is near ATM and treat it differently than others if I want (such as taking it out of the portfolio and managing it separately from the others, for the short time until expiration).

[But changing the way Gamma is defined I think is not a good idea].

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There is a greek for $\frac{\partial \Gamma}{\partial S}$, it's called speed.

How do traders and financial engineers tackle this problem in practice?

As a third order greek, speed is not easy to manage mentally. Traders usually do a full repricing of their whole portfolio and scenario analysis where they see their delta/gamma at different levels of spot, rather than using speed.

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