Help in solving these questions.

The solution to the Black-Scholes equation for the price I' (assuming a risk-free force of interest / ) of a European put option maturing & years from now with strike price K on a stock that pays dividends at force q whose current spot price is S is: V = Ke "(-dz ) - Se-""((-d ) where d , dy = log(S / K)+ (r -91/20 ju ev (i) Show that the hedge ratio A = is given by A = -e ""@(-dj). [8] as a2 v (ii) Hence find a formula for I= [2] as2 [Total 10]An option is described as "at-the-money-forward" if the price of the underlying asset equals the strike price discounted at the risk-free rate over the remaining life of the option. (i) Show that, according to the Black-Scholes model, the price of an at-the-money- forward 7-year call option on a non-dividend-paying share is approximately a proportion 0.40 VT of the price of the underlying asset, where o is the annual volatility of the underlying asset. [5] Hint: For a differentiable function: f(x) = f(0)+xf'(0) (ii) Hence estimate the price of a 3-month at-the-money-forward call option on shares worth Elm with a volatility of 20% per annum. [2] [Total 7]