Quickly calculate portfolio delta and gamma

Options trading skills tested :

Practical understanding of delta and gamma of an options portfolio.

How to set up the training exercise

Load a Custom Play Volcube simulation and make a few quotes and execute some trades against the broker in reasonable size. To begin with, leave auto-hedge on or delta hedge your option trades yourself. Later, try this exercise with an un-hedged option position.

A vega position

A delta-hedged short call spread position

Open up the thumbail which shows a Volcube simulation where we have sold 250 call spreads (that have been delta-hedged). You may want to print the image or at least have it in a place where you can read this and see the numbers in the image. We will use this as an example of how to perform this options training drill.

In this exercise, we are going to look at the portfolio delta in the Risk Detail at different underlying price and use this to estimate our gamma. Then we will reverse the process and use our gamma to estimate our change in delta.

Look at the delta number in the Risk Detail versus the spot price of 100.00. It is zero. This means our portfolio is currently fully delta-hedged versus this spot price. Now take a look at the cell to the left. This is our portfolio delta with the spot trading at 98.50. We are long 907 deltas. We know we are long because the number is green. Short or negative numbers are in red. So, the spot has fallen by $1.50 and our portfolio delta has increased to 907. What does this tell us about our portolio gamma over this move in the spot? Well, a quick calculation (and this is the whole point of the exercise) tells us we must be short approximately 600 gamma. Why short gamma? Because as the spot has fallen (a negative change in price), our delta has become more positive; that always indicates short gamma. Why 600? Because the gamma number tells us how much our delta changes for a 1 point move in the underlying. Since the underlying moved by 1.5 points ($1.50) and our delta changed by roughly 900, we must be short roughly 600 gamma.

Now we can double-check our reasoning by looking at the actual gamma numbers in the Risk Detail. Versus 100.00, we are short 532 gamma. Versus 98.50, we are short 669 gamma. So we can say that as the spot product dropped in price from 100 to 98.50, we were, on average, short roughly 600 gamma (the average of short 532 and short 669). This ties in nicely with our calculation above.

Now let’s work the other way; from our gamma numbers to our delta. Versus 100.00, we are short 532 gamma. As the spot rallies, versus 101.50, we are short 356 gamma. In other words, we are less short. (This is due to the nature of our short call spread position in this example). So we can say that we are approximately short 444 gamma between 100 and 101.50. Therefore, we would expect that from being delta neutral against 100.00, we will be short roughly 666 delta versus 101.50. This is because the change in delta is simply the change in the spot price times the portfolio gamma. And sure enough, versus 101.50, our portfolio delta is short 671. The discrepancy is due to the fact that our average is just an approximation and the actual rate of change of gamma is not constant. Nevertheless, it is obviously a good estimate.

The point of this exercise is to make sure we understand the nature of the relationship between gamma and delta. Although the maths is fairly trivial, it is still important to practise making the calculations yourself. Professional option traders will be able to make such calculations in a split second. There are two aspects to this exercise; learning how long or short gamma leads to short or long delta positions for rallies/falls in the spot price and learning about the magnitude of these changes. Try making the calculation for yourself at other points on the Risk Detail slide. For more advanced Volcubers, try starting with a non-zero at-the-money delta.

Trinomial trees and option Greek calculations

One other interesting note. For option pricing models that rely on trinomial (or other trees), there are often no closed-form equations giving easy readings of the gamma or delta or other Greeks. In that case, the method used to estimate the Greeks is precisely that used in this training exercise. For example, to estimate delta, we could value an option versus one spot price and then value it versus another. The delta is then implied by the change in the option value. This is exactly the same idea as above. We estimated the gamma of the portfolio by checking the delta at one spot price, and then again at another. By definition, the gamma (sign and magnitude) can be implied by the difference in the delta.





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These articles are for informational purposes only. They should not be regarded as an offer to sell or as a solicitation or an offer to buy any financial or derivative products.

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