Gamma of an Option

What is the 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 optionOptionOptions are financial contracts which allow the buyer a right, but not an obligation to execute the contract. The right is to buy or sell an asset on a specific date at a specific price which is predetermined at the contract date.read more 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 include asset dividend yield (applicable for dividend-paying stocks), spot price, strike price, standard deviation, option’s Time to expiration, and the risk-free rate of returnRisk-free Rate Of ReturnA risk-free rate is the minimum rate of return expected on investment with zero risks by the investor. It is the government bonds of well-developed countries, either US treasury bonds or German government bonds. Although, it does not exist because every investment has a certain amount of risk.read more.

Mathematically, the gamma function formula of an underlying asset is represented as,

You are free to use this image on your website, templates etc, Please provide us with an attribution linkHow to Provide Attribution?Article Link to be Hyperlinked
For eg:
Source: Gamma of an Option (wallstreetmojo.com)

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 priceStrike PriceExercise price or strike price refers to the price at which the underlying stock is purchased or sold by the persons trading in the options of calls & puts available in the derivative trading. Thus, the exercise price is a term used in the derivative market.read more 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

Example of Gamma Option Finance Formula (with Excel Template)

Let us take the example of a call optionThe Example Of A Call OptionCall Options are derivative contracts that enable the buyer of the option to exercise his right to buying particular security at a pre-specified price popularly known as strike price on the date of the expiry of such a derivative contract. It is important to note that the call option is a right, not an obligation.read more 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,

d_{1 =} 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 a detailed calculation of gamma, function refer the given excel sheet above.

Relevance and Uses

It is important to understand the concept of gamma function because it helps in the correction of convexityConvexityConvexity of a bond is a measure that shows the relationship between bond price and yield, and it helps risk management tools to measure and manage a portfolio's exposure to interest rate risk and loss of expectation.read more problems seen in the case of hedging strategies. One of its applications 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 long positionsLong PositionsLong position denotes buying of a stock, currency or commodity in the hope that the future price will get higher from the present price. The security can be bought in the cash market or in the derivative market. The course of action suggests that the investor or the trader is expecting an upward movement of the stock from is prevailing levels.read more 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

This has been a guide to Gamma of an Option and its definition. Here we discuss Gamma Formula in Finance along with calculation and examples in excel and downloadable excel template. You can learn more about financing from the following articles –

• Definition of In the Money
• Formula of Alpha
• Call Options vs. Put Options
• Futures vs. Options