Gamma of an option is the second partial derivative of the theoretical value of an option wrt the underlying. It should be the rate of change of Delta wrt to a small change on the underlying. However many textbook (e.g. Trading option greeks, Passarelli) says that gamma is conventionally Stated in terms of Delta per dollar move.
Let’s suppose we want to use the B&S Model for a call option on non divident paying stock. Let’s also suppose that:
S = 52 (the underlying)
K = 50 (the strike)
tau = 0.25 (the time to maturity)
r = 0.12 (the risk free rate)
sigma = 0.3 ( the volatility of the underlying)
Then we have: Call = 5.057387 Delta = 0.7041836 Gamma = 0.04429147
I want to estimate How the option value will be if the stock change from 52 to 53 ( in this situation the B&S model would give as exact answer Call = 5.783055).
As the first approximation (Delta) i would do: Call = 5.057387 + (53 -52)0.7041836 = 5.761571 (which is not equal to 5.783055) Then of i want to be more precise, i could use gamma as well: The new Delta should be 0.7041836 + 0.04429147 (gamma stated ad Delta per dollar move) or 0.7041836(1+0.04429147), i.e. Rate of change of Delta. Why?