January 23, 2014

Last week we looked at delta — one of the most important Greek letters for options traders. Delta is significant because it captures the expected change in the option price deriving from a unit change in the underlying price. So, a delta of 0.75 means that if the underlying increases by \$1, the option will increase by \$0.75. Delta helps us understand the sensitivity of an option’s price to the underlying’s price.

But sometimes delta is not enough to characterize the expected change in an option. Remember, the price of an option isn’t a linear function of its underlying price but rather a curve with a changing slope. Delta tells us about the expected change in the option deriving from a change in the underlying, whether the change is small or large. Knowing how fast delta changes, which is tied to the change in slope, can be useful. That’s where another Greek letter comes into play — gamma.

The chart above shows that delta, which is given by the slope of the curve, is not constant. To determine the rate at which it changes, you need the first derivative of delta, or the second derivative of the option price. This reality is captured by gamma, which measures the change in delta due to a change in the underlying price.

Gamma values vary with the characteristics of the option. A call option is gamma-positive, whereas a put option is gamma-negative. As noted in our article, Looking at Options Greeks: Delta, the more in-the-money an option is, the higher delta is, so delta increases with the underlying price for a call option. Gamma must then be positive for call options. The same reasoning is applied to put options such that Gamma is negative for these. The lowest possible absolute value for both is zero, which indicates a situation in which delta isn’t affected by a change in the underlying price. This happens when the value of an option is deep in-the-money or deep out-of-the-money, and when time to maturity is huge. In these situations, the option is relatively insensitive to changes in the underlying, thus gamma is very low.

The following figure shows delta and gamma values for calls and puts on Microsoft for two different maturities. You can make the following calculations to estimate a change in delta. As an example, let’s take the call with strike price \$38 with a time to maturity of 28 days. Delta is 0.24 for that option, which means that if Microsoft’s price increases by \$1, the option will increase \$0.24. Gamma is 0.1860, which means after the \$1 change, the new delta will be 0.24 x (1+0.1860), or .285. The next time Microsoft changes by \$1, the option will change by \$0.28.

Next week we’ll take a look at vega, which measures the sensitivity of an option’s price to volatility. This is another important metric to take into account because the difference between an option value and its intrinsic value is explained by volatility. An abrupt change in volatility can impart a huge profit or loss.

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