Option Gamma Made Easy
Gamma is among the most basic option Greeks, but tends to be much harder to understand than delta or vega. Recall, delta is the change in an option value when the underlying stock moves by one unit. Likewise, vega is the change in your option value when implied volatility moves by one unit. How do we practically interpret gamma?
In this writing, I will assume the reader knows basic Calculus and the basics of Black Scholes theory. If you find yourself in need of review, John Hull’s chapter, “The Greek Letters,” in his Options and Futures book will serve as a great refresher.
Unitary Gamma Basics
We define the second derivative of an option price with respect to the spot price as the unitary gamma. The concept on unitary delta is similarly defined for the first derivative of an option. The gamma captures the curvature of option price variations. In other words, it represents how unitary delta evolves.
Say you are a trader tasked with hedging a short call option. Assuming you delta hedge by purchasing the underlying stock, you will find a small mismatch between your hedge and the option value if the underlying spot price changes. The option and the hedge diverge when the underlying spot price moves far in one direction. Gamma actually captures the difference between the option and the hedge as the spot moves. The impact can be seen in Figure 1. When the spot goes up, the trader will need to increase the size of the delta hedge to minimize risk. This increased size roughly corresponds to the gamma for the example below.
A second important feature of gamma is that it peaks when the option is at the money (ATM). When the option is ATM, there is high uncertainty if it will expire in the money. Hence, the exposure will be volatile given changes to the spot price. Meanwhile, if an option is far in the money, it is almost certain it will expire in the money. In this case, exposure is more stable. See Figure 2 for the dynamics of gamma as the spot price changes.
Dollar Gamma and its Interpretation
Before building an interpretation of gamma, let’s build the concept of dollar delta. Dollar delta is defined as the total exposure of the option to the underlying spot price. It is how much the trader loses when the spot price goes to zero. Dollar delta is defined mathematically as Q*Δ*S where Q is the number of options, Δ is the delta of the option, and S is the underlying spot price.
Gamma has a similar concept called Dollar Gamma. Figure 3 shows the definition of dollar gamma where V is the value of the option. The interpretation is roughly the change in the option’s exposure when the spot changes by one unit. However, this interpretation is very approximate and may not be informative in particular scenarios. While dollar gamma does not have a clean interpretation, it is useful for computing the spot PnL of an option from the Greeks. Figure 4 shows the second order spot PnL of an option computed with the dollar gamma and represents the mathematics behind the blue curve in Figure 1. Generally, the second order approximation is relatively accurate.
To get a better sense of how the gamma impacts the dollar delta, we will take a Taylor Series of the dollar delta in Figure 5. I will assume the quantity, Q, is 1 to make the logic more simple. Δ(S) and Γ(S) represent the unit delta and gamma evaluated at price S respectively. The “$” symbol indicates dollar delta/gamma.
The logic shows how to obtain a first order approximation of the change in dollar delta when the spot increases by one unit. Line 2 drops higher order terms while line 3 evaluates the derivative with the product rule. We notice the dollar gamma does not even fully explain the first order approximation.
If an option is ATM, then the dollar gamma term will tend to dominate changes in dollar delta. This occurs as the unit delta is around 0.5 for ATM options while gamma is maximized ATM. However, this will not be the case if an option is far in the money as unit gamma rapidly declines while delta converges to 1.
To make the dollar gamma more practical, Figure 6 defines a new metric. We will call it the adjusted dollar gamma. Unlike the dollar gamma, the adjusted dollar gamma can be accurately interpreted as the approximate change in dollar delta when the spot changes by one unit.
With the adjusted dollar gamma, we now have good a idea of how exposure will change with spot. In the process of obtaining this metric, we saw that the dollar delta represents the portfolios exposure. Also, we introduced a concept similar to the dollar delta called dollar gamma. While dollar gamma is useful for computing spot PnL, its practical interpretation is somewhat limited. This lead us to derive the adjusted dollar gamma which has a more practical interpretation.
