1. Consider a European call option on a non-dividend-paying underlying with maturity date T, for which the strike price is equal to the forward price of the underlying. We say that the option is at-the-money-forward, and denote its price (at t = 0) by CAMF. (a) [12 marks] Show that, under Black-Scholes assumptions, CAMF = S (20(6/2) 1], = where S is the price of the underlying at t = 0, 0(-) is the standard normal cdf, and := OVT. (b) [7 marks] Use a first-order approximation of the formula for CAMF in part (a) around = 0 to obtain the ratio CAMF/S as a linear function of . Hint: Recall that the first-order Taylor series approximation of a function f around xo is given by f(x) = f(x0) + f'(x0)(x xo). (c) [6 marks] An at-the-money-forward European call expiring in 3 months is worth 4% of the under- lying. Using the approximation derived above, calculate its implied (annual) volatility