Suppose that the current price of a stock that doesn’t pay dividends is S = 40, and the stock volatility is \sigma = 20%. You are long an European Call expiring in 1 year. You want to delta-hedge your position. You don’t remember the strike price for the option, but know that the Call gamma is \Gamma = .0331.

Using the formula for Call gamma, and since the stock doesn’t pay dividends,

\Gamma = 1/(S * \sigma * \sqrt{T} * \sqrt{2\pi}) exp (-.5 d_1^2)

solving for d_1, and choosing d_1 > 0 gives d_1 = .90536, so that

\Delta = N(d_1) = .81736

however, this line of reasoning is wrong. the delta is supposed to be .6388.

(the solution for this problem involves first calculating the missing parameters and then using Black-Scholes PDE to solve for delta)

does anyone know why i can’t use \Gamma to infer d_1 and hence the delta of the European call option on a non-dividend paying stock (using \Delta = N(d_1))? thank you.