Estimating Gamma for Calls or Puts

But just as Delta is an estimate, so is Gamma. It should be applied only to better comprehend the relationship found between Delta and the underlying security. In other words, Gamma denotes the momentum or acceleration in Delta’s movement. Options traders consider Gamma a representation of risk because it measures Delta in this manner. High Gamma is seen as higher risk and low Gamma is seen as lower risk. But these conclusions should be drawn with caution. As with all estimates, it is not realistic to act on the assumed probability of outcomes, but to use Gamma and other “Greeks” as portions of a broader analysis of risks.

Price changes in the underlying and premium changes in the option are clear indications of varying levels of risk, and many options traders try to match these price movements to individual risk tolerance. To accomplish this, a trader needs to appreciate how Gamma performs and what it reveals. Gamma develops as a “volatility risk premium” factor and may be more pronounced whenever volatility increases over the option’s lifetime. It is also true that ignoring an option’s gamma can lead to incorrect inference on the magnitude of the volatility risk premium … the S&P options are used as a test case to demonstrate the impact of ignoring gamma on the estimation of the market priced of volatility risk. The findings show that the more prices fluctuate, the greater the variability in the estimation of volatility risk premium when gamma is ignored. [Doran, James S. (2007). The influence of tracking error on volatility risk premium estimation. The Journal of Risk 9 (3), 1-36]

This means that growing momentum of price movement (volatility) makes conclusions less reliable when Gamma is not considered, and this is the point that should not be overlooked. If a trader follows Delta but ignores Gamma, the momentum factor will not appear as readily, and a false sense of certainty can result.

A related factor is moneyness of the option. When the option is at or very close to the money, Gamma is likely to have maximum value. As the option  moves deeper in or out of the money, Gamma will become increasingly lower. Complicating this even more, proximity to expiration is also a factor. When the option’s expiration date approaches, Gamma tends to move higher for ATM options and lower for ITM and OTM contracts. For these reasons, relying on Delta alone is dangerous. Reviewing both Delta and Gamma (often called the Delta of the Delta), you get a more accurate picture of how volatility affects premium and changes overall risks.

As an overview of Gamma behavior, you may simplify these relationships by observing that low volatility causes higher ATM Gamma and lower ITM and OTM Gamma, with the lowest point being zero Gamma. In this simplified version, Gamma directly summarizes Delta behavior (and Delta directly summarizes option volatility and behavior).

Gamma, to the extent that it reveals behavior in implied volatility of the option, still should be recognized as being influenced by the underlying’s historical volatility. Just as Delta directly reveals this historical volatility and its influence of option pricing, Gamma indirectly does the same thing by showing how Delta movement occurs.

The Gamma trend is most readily seen in the responsiveness of option premium to changes in time value. The longer the time to expiration, the less reaction will be seen in the option premium. A phenomenon witnessed by all options traders; and the closer the time to expiration, the more responsive premium becomes. The combined behavior of Delta and Gamma can be used to spot volatility trends with the expiration timeline in mind.

The formula for Gamma is developed to summarize the second derivation of the option’s value, [Chriss, Neil A. (1997). Black-Scholes and Beyond. New York: McGraw-Hill, pp. 311-312]  or
 

Γ =  ∂²V  ÷  ∂²S

Γ = Gamma

∂²V = second derivation of the option

∂²S = second derivation of the underlying

A simplified calculation is to calculate the difference in Delta from one period to another, divided by changes in the underlying. However, this may be inaccurate because underlying price movement is not the same for all issues. A one-point price move for a $100 stock is much more substantial than a one-point move for a stock currently valued at $50 or another at $100. Using this method for calculating Gamma is only entirely reliable when used to compare Delta and Gamma for two underlying stocks with identical prices. This is not practical, however. It presents the same problem as that of how chart scaling is done. A “move” in price for one stock cannot be compared to that of another and, likewise, simplified Gamma calculations for stocks at different prices can be substantially distorted if prices are not at least close to one another.

Options traders seeking reliable Delta and Gamma outcomes also must be aware of the influence of unusually strong price movement. For example, if an earnings surprise moves the underlying 10 points when typical daily movement is one to two points, what does it mean for Gamma? Because this is not typical in terms of degree of movement, this is where Delta, Gamma and all other Greeks can easily become distorted.

It makes sense to think of Gamma and other Greeks as circumstantial evidence of risk changes. You still need for concrete evidence and may view Delta and Gamma as confirming indicators of ever-changing volatility in the option premium. Remember, though, that even non-conclusive results can be revealing and useful. Or , putting this another way, “Some circumstantial evidence is very strong, as when you find a trout in the milk.” [Thoreau, Henry David, Journal, November 1850]

Michael C. Thomsett is a widely published author with over 80 business and investing books, including the best-selling Getting Started in Options, coming out in its 10th edition later this year. He also wrote the recently released The Mathematics of Options. Thomsett is a frequent speaker at trade shows and blogs on his website at Thomsett Guide as well as on Seeking Alpha, LinkedIn, Twitter and Facebook.

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