# Gamma

• Gamma is the rate of change in delta for unit change in the price of the stock
• For a portfolio of options with the same underlying asset, the gamma is the change in the value of the portfolioâ€™s delta with the change in the portfolio value:
• A large gamma suggests that delta will change rapidly as the price of the underlying stock changes
• The following table presents and compares the sign of Gamma and Delta
 Option Position Delta Gamma Call Long + + Put Long - + Stock Long 1 0 Call Short - - Put Short + - Stock Short -1 0

• Gamma reaches its maximum absolute value when a stock is trading at the money or near the money
• It reduces in value as the security moves further out of the money or further in the money.
• Calculation of Gamma
• Gamma for European options can be calculated using the following formula:

• Where symbols have their usual meaning

• Gamma Neutral Portfolio
• Suppose we have a delta neutral portfolio of options that has a positive gamma
• In order to make the portfolio gamma neutral one would need to undertake trades in options that reduce the gamma to zero
• However, these trades will lead to a new portfolio, which will not be delta neutral anymore
• Therefore as a second step one would need to trade in stocks so that the portfolioâ€™s delta neutrality
• is restored
• This will not impact the gamma neutrality of the portfolio as stocks carry a zero gamma
• Relation between Gamma, Delta and Theta
• For a delta neutral portfolio, when theta is large and negative, gamma will tend to be large and positive. The converse of this holds too. Thus, if delta is zero and theta is large and positive, gamma is likely to be large in magnitude and negative in sign