Note that to first \(\operatorname{order} N(d)=\frac{1}{2}+d / \sqrt{2 \pi}\). Use this to derive the value of a call option when the stock price is at the present value of the strike price; that is, \(S=K e^{-r T}\). Specifically, show that \(C \simeq .4 S \sigma \sqrt{T}\). Also show that \(\Delta \simeq \frac{1}{2}+.2 \sigma \sqrt{T}\). Use these approximations to estimate the value of the call option of Example 15.2.

Data from Example 15.2

Let us calculate the value of the same option that was a 5-month call option on a stock

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with a current price of $62 and volatility of 20% per year. The strike price is $60 and the interest rate is 10%. Using S = 62, K=60, = .20, and r = .10, we find In(62/60)+.12 x 5/12 d = = .641287 20/5/12 d2 d 2/5/12 = .512188. The corresponding values for the cumulative normal distribution are found by the approximation in Exercise 1 to be N(d) 739332, N(d2) = .695740. Hence the value for the call option is C=62 x .739332-60 x .95918 x.695740 = $5.798. This is close to the value of $5.85 found by the binomial lattice method.