Solve the following:

Let Em , , be the annually compounded yields on m and n year zero-coupon bonds respectively, with n > m. (i) Explain what is meant by the annually compounded forward interest rate for the period (m, n), and show algebraically how you would calculate this forward rate. [3] (ii) Show that the continuously compounded forward interest rate for the time interval (m, n) is given by no, - mom 1- m where om,6, are the continuously compounded forces of interest for the zero- coupon bond yields of term m and n respectively. [4] (iii) If a 14-year zero-coupon bond yields 4.9% p.a. and a 15-year zero-coupon bond yields 5.0% p.a., both annually compounded, calculate the one-year forward interest rate for year 15. Comment on the level of this forward rate relative to the zero-coupon rates.2 (i) Outline the use and significance of Ito's lemma for the valuation of derivatives in a Black-Scholes-Merton economic world. [2] (ii) Use Ito's lemma to determine the differential equation for the value of the following derivatives under the stock price process ds = usdt + osdz, where = is a standard Brownian motion, and u and o are constants: (a) European Call option, strike price &, period to maturity T. (b) American Call option, strike price &, period to maturity 7, where the payoff is the excess (if positive) of the square of the stock price above the strike price. In each case, state precisely the boundary conditions which apply, without attempting to solve the equations. [11] (iii) Compare in general terms the instantaneous stock hedges for the options in (ii). In particular, compare the way in which they vary with the underlying stock price. [2] (iv) Describe how your answers to (ii) and (iii) would change if instead the stock price followed the process ds = usd + One Pisdz for some constants Go and B.(i) (a) State the three necessary and sufficient conditions for the continuous process W, to be a P-Brownian Motion. (b) State the requirements for a probability measure Q to be equivalent to P. [3] (ii) Let W, be a continuous P-Brownian Motion process. Consider the "change of measure" process , = Ep dQ dP IF, , where Q is a measure equivalent to P and _ _= exp (-yW7 -by'7 ) for some time horizon de T, constant y and given filtration F, . [Ep[...] represents expectation with respect to measure P.] Show that W, =W, + yr is a continuous Q-Brownian Motion process. [11] (iii) Comment on the significance of the result in (ii). [2] (Hint: You may use the fact that a random variable X is a normal Mu,) under measure P if and only if Ep (exp(0X)) = exp(Ou +1202c) for all real 0.) [Total 16]4 Rivendale is a non-dividend paying equity, with current price 100. You are considering some strategies based upon buying and selling options on Rivendale. All these options expire three months from now. (i) Draw approximate aggregate Profit/Loss diagrams for the following two option strategies: (a) Buy 1 European at-the-money call option. (b) Sell 2 European put options strike 80. Buy 1 European put option strike 90. Buy 1 European call option strike 110. Sell 2 European call options strike 120. Diagrams are required for each strategy as at now, two months from now and three months from now. [8] (ii) Draw an approximate diagram for the Gamma of the option strategy in (i) (a) above, as at now, two months from now and nearly three months from now. [4] (Note: Sketch all the time curves for a single strategy on the same graph, plotted against the Rivendale stock price using a suitable scale. You should allow for the estimated purchase price of the underlying options where appropriate. Any curves may of course be drawn freehand.) [Total 12]