The purpose of this question is to obtain the "trader's rule of thumb" to price an at- the-money call (that is, strike K is equal to the underlying forward price F). Consider the underlying stock pays no dividend (that is, q|=0). Recall the BS formula V = SeT N(d) Ke-T N(d2) with N(x) = du, d X == -q- In()+(rq) and d = In()(rq)T using 2T standard notations from [Hull]. (a) Simplify d and d to show that they only depend on the volatility and the time to maturity T (b) Compute N(0), N'(0) and N"(0) (c) Use Taylor's expansion at x = :0, and part (b) to write a second-order approximation expression for N(x) (d) Apply the approximation in (c) to the Black Scholes formula, show that V is ap- proximated by the formula V = c SuoTw for some constants c, u, v, w. Find C, u, v, w.