3. Let S 2. Consider the following option-pricing problem. There are K = 3 securities: (i) a stock, with initial price q > 0 and payoff r = (991, b91), where g>b> 0 are the "good" and "bad" gross returns respectively; (ii) a riskless bond, with initial price q2 > 0 and payoff r2 = (Rq2, Rq2), where R1 is the riskless return; (iii) a put option on the stock with initial price q3 and payoff r3 = (r13, 23), where rs3 = max{K - rs1, 0}, s = 1,2, and K > 0 is the exercise price of the option. (The put option gives its holder the right, but not the obligation, to sell the stock for K per share after the state is revealed.) (a) Derive necessary and sufficient conditions on g, b and R for the absence of arbitrage involving only the stock and bond. (b) Assuming no-arbitrage, for the three securities, calculate the put option price q3 explicitly in terms of 91,9, b, R, and K. Find the risk- neutral probabilities and 72 and show that q3 = R-E" [3], where E denotes expectation w.r.t. (T1, T2).