Gamma of an Option – Table of Contents
What is Gamma of an Option in Finance?
The term “gamma of an Option” refers to the range of the change in the delta of an option in response to the unit change in the price of the underlying asset of the option. Gamma can be expressed as the second derivative of the premium of the option with respect to the price of the underlying asset. It can also be expressed as the first derivative of the delta of the option with respect to the price of the underlying asset.
The formula for gamma function can be derived by using a number of variables which includes asset’s dividend yield (applicable for dividend-paying stocks), spot price, strike price, standard deviation, option’s time to expiration and the risk-free rate of return.
Mathematically, the gamma function formula of an underlying asset is represented as,
where,
- d1 = [ln (S / K) + (r + ơ2/2) * t] / [ơ * √t]
- d = Dividend yield of the asset
- t = Time to the expiration of the option
- S = Spot price of the underlying asset
- ơ = Standard deviation of the underlying asset
- K = Strike price of the underlying asset
- r = Risk-free rate of return
For non-dividend paying stocks, the gamma function formula can be expressed as,
Explanation of the Gamma Option in Finance
The formula for gamma in finance can be derived by using the following steps:
Step 1: Firstly, determine the spot price of the underlying asset from the active market, say the stock market for an actively traded stock. It is represented by S.
Step 2: Next, determine the strike price of the underlying asset from the details of the option. It is denoted by K.
Step 3: Next, check whether the stock is paying any dividend and if it is paying then note the same. It is denoted by d.
Step 4: Next, determine the maturity of the option or time to expiration and it is denoted by t. It will be available as details pertaining to option.
Step 5: Next, determine the standard deviation of the underlying asset and it is denoted by ơ.
Step 6: Next, determine the risk-free rate of return or asset return with zero risks for the investor. Usually, the return of government bonds is considered as the risk-free rate. It is denoted by r.
Step 7: Finally, the formula for the gamma function of the underlying asset is derived by using asset’s dividend yield, spot price, strike price, standard deviation, option’s time to expiration and a risk-free rate of return as shown below.
Example of Gamma Option Finance Formula (with Excel Template)
Let us take the example of a call option with the following data.
Also, calculate the gamma at spot price
- $123.00 (out of money)
- $135.00 (at the money)
- $139.00 (in the money)
(i) At S = $123.00,
d1 = [ln (S / K) + (r + ơ2/2) * t] / [ơ * √t]
= [ln ($123.00 / $135.00) + (1.00% + (30.00%)2/2) * (3 / 12)] / [30.00% * √(3 / 12)]
= -0.3784
Therefore, the gamma function calculation of the option can be calculated as,
Option’s gamma S=$123.00
= e-[d12/2 + d*t] / [(S*ơ) * √(2ℼ*t)]
= e-[0.22352 /2+ (3.77% * 3/12)] / [($123.00 * 30.00%) * √(2π * 3/12)]
= 0.0193
(ii) At S = $135.00,
d1 = ln (S / K) + (r + ơ2/2) * t] / [ơ * √t]
= [ln ($135.00 / $135.00) + (1.00% + (30.00%)2/2) * (3 / 12)] / [30.00% * √(3 / 12)]
= 0.2288
Therefore, the gamma function calculation of the option can be calculated as,
Option’s gamma S=$135.00
= e-[d12/2 + d*t] / [(S*ơ) * √(2ℼ*t)]
= e-[ 0.22352 /2+ (3.77% * 3/12)] / [($135.00 * 30.00%) * √(2π * 3/12)]
= 0.0195
(iii) At S = $139.00,
d1 = [ln (S / K) + (r + ơ2/2) * t] / [ơ * √t]
= [ln ($139.00 / $135.00) + (1.00% + (30.00%)2/2) * (3 / 12)] / [30.00% * √(3 / 12)]
= 0.2235
Therefore, the gamma function calculation of the option can be calculated as,
Option’s gamma S=$139.00
= e-[d12/2 + d*t] / [(S*ơ) * √(2ℼ*t)]
= e-[ 0.22352 /2+ (3.77% * 3/12)] / [($139.00 * 30.00%) * √(2π* 3/12)]
= 0.0185
For detail calculation of gamma function formula refer the given excel sheet below.
Relevance and Uses
It is important to understand the concept of gamma function because it helps in the correction of convexity problems seen in the case of hedging strategies. One of its application is the delta hedge strategy which seeks a reduction of gamma in order to hedge over a wider price range. However, the reduction of gamma results in a reduction of alpha too.
Further, the delta of an option is useful for a shorter time period, while gamma helps a trader over a longer horizon as the underlying price changes. It is to be noted that the value of gamma approaches zero as the option goes either deeper into the money or deeper out of the money. The gamma of an option is the highest when the price is at the money. All the long position options have a positive gamma, while all the short options have negative gamma.
You can download this Gamma Function Formula Excel Template from here – Gamma Function Formula Excel Template
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