(a) Let C=C(S,t) be a European Vanilla Call Option which depends on the underlying asset S and time t. The risk-free interest rate is r and the volatility of S is . The Payoff function of this Call Option is (S)=max(SE,0). Thus, the value of the option can be calculated as C(S,t)=er(Tt)0(S)p(S,t;S,T)dS where the probability density function p(S,t;S+,tr) for a risk neutral random walk is given by p(S,t;Sr,t)=Sr2(tt)1exp(22(tt)(log(S/S)(r212)(tt))2). By direct integration, show, detailing all steps, that the value of a Europen Vanilla Call option with Exercise Price E and Expiry Date T is given by C(S,t)=SN(d1)Eer(Tt)N(d2) where N is the cumulative distribution function for a standardized normal random variable given by N(x)=21xex2/2dy. Here d1=Ttlog(S/E)+(r+2/2)(Tt) and d2=Ttlog(S/E)+(r2/2)(Tt). You can use the result N(x)=1N(x) without proof. You may wish to use the substitution x=log(S/S) and then complete the square of the associated quadratic expressiou. (b) Briefly explain how the hedging parameter =Q can be calculated using the standard Binomial Method in a risk neutral world. (c) How would holders/writers value European Call options using the binomial method if they were not planning on Delta Hedging? Discuss this from the view point of both the bolder and the writer. (d) This question inwolves Barrier Options. (i) Describe all of the features of an Up and Out Barrier Option. (ii) Describe how to determine the value of an Up and Out Barrier Option on a European Call when the Barrier is X and the Exercise