There are two securities in the market: a risky asset (stock) whose price at time t is S(t) and a bond whose value at time t is B(t). Trading only takes place now at time t = 0 and payoffs are revealed at maturity T. The bond costs $1 now and pays $(1 + R) at maturity, where R is the one-period discretely compounded riskless return. The asset costs So now and can take M possible values at maturity T. The actual probability of the stock price S(T) taking values s; > 0 is p > 0, for j = 1,..., M and j=1 Pj = 1. For simplicity we assume that s s2 0 and maturity T is a contingent claim written on asset S with the following maturity payoff function g(.) g(S( [S(T) K, if S(T) K, K S(T), if S(T) < K. a) A portfolio containing stocks and bonds is constructed to replicate the payoff of a straddle. Write down the conditions this portfolio must satisfy. b) Is it always possible to find such a portfolio? c) Using the put-call parity show that a straddle can be perfectly replicated using call options, stocks and bonds. Now assume that M = 2, So= $100, $ = $95, $2 = $105, R = 3%, P = 50%, K = $99, T = 1 d) Compute the replicated portfolio for the straddle. Compute the value of the staddle at time period t = 0, under the assumption of no-arbitrage. e) Calculate 91 and 92 and compare them to P1 and P2. Use 91 and q2 to compute the value of the straddle at time t = = 0. Compare this value to the value obtained using the no-arbitrage principle. (Total for Question 26: 25 marks) 3 marks 2 marks 5 marks 6 marks 9 marks