Explain the concept of a replicating portfolio and its use in derivative pricing.

(i) (a) State what is meant collectively by "the Greeks" of an option. (b) Define delta for a derivative. [2] (ii) Assume that the assumptions underlying the Black-Scholes model hold. Derive a formula for the delta of a European call option for a non-dividend paying stock. Evaluate the delta of an at-the-money call, when the risk free rate is zero, time to expiry is one year, and volatility is 20% p.a. [7] [Hint: The Black-Scholes formula for the price of a European call option is given by f = so(d ) -ke "o(d,) where d, = [n(s/k) + (r + 0-/2ju] / ov/u ; da = di - vu, and f. s, k, r, a and u are the price of the option, the price of the stock, the strike price, the risk-free rate, the volatility and time to expiry respectively.] (ili) One investor, A, holds f1 million of equity. Another, B, holds El million in cash, and the call option evaluated in (ii), over f1 million of the same equity. The equity market rises instantly by 20%. Explain which of these investors now owns greater value. (3](i) Compare the properties of the Vasicek and the Cox-Ingersoll-Ross models for interest rates. [6] (ii) Under the Vasicek model the price model (given term /, short rate R, long rate L and parameters o and B) is P(R, 1) = exp[-D(I)R - (1 - DOO))L -(B/2)DO-] where D() = [1 - exp(-ar)]/a. Derive expressions for: (a) the spot yield (b) the forward yield the limit of the spot and forward yields as / tends to zero 6]