Binomial model

    1. Consider the single period binomial model with d < 1 + r < u. Prove from first principles (i.e., without using the fundamental theorems of asset pricing) that there is a unique equivalent martingale measure. 2. Consider the single period binomial model. Suppose r = 0.02, u = 1.05, d = 0.9 and S0 = 50. Consider a put option on the stock with strike price K = 49. (a) Compute the equivalent martingale measure (qu, qd). (b) Find the no-arbitrage price of the claim at time 0. (c) Find the replicating portfolio. 3. Consider a single period binomial model which satisfies the no-arbitrage condition d < 1 + r < u. Consider two contingent claims C and C 0 where (cu, cd) = (1, 2) and (c 0 u , c0 d ) = (1, 4). You are given that the no-arbitrage price of C is 1.5, and the no-arbitrage price of C 0 is 2. (a) Find the interest rate r. Hint: Use the two given claims to generate other cashflows. (b) Find the risk-neutral probabilities qu and qd. 4. Write down two different sets of parameters (u, d, r, S0) of the single period binomial model which give the same equivalent martingale probabilities. Must they give the same prices for all contingent claims? Explain your reasoning. 5. Consider the trinomial model in Example 2.1 of the lecture notes: r = 0, S0 = 1, u = 3, m = 2 and d = 1/2. Characterize all contingent claims that can be replicated. That is, give a general expression of the payoffs cu, cm, cd that can be replicated. 6. Consider a single-period model with 4 states and 3 risky assets (apart from the risk-free asset). Write Ω = {ω1, ω2, ω3, ω4}. Assume that (S j 0 )0≤j≤3 = 1 1.7 2 2 and (S j 1 (ωi))1≤i≤4,0≤j≤3 =     1 1 3 1 1 2 3 3 1 2 1 1 1 1 1 3     . 1 (Here the column index j runs from 0 to 3. So the risk-free interest rate is 0.) (a) Show that the model is arbitrage-free and complete. (b) Consider the contingent claim (C(ωi))1≤i≤4 =     1 0 0 1     . Find the no-arbitrage price of C at time 0 and find a replicating portfolio. Is the replicating portfolio unique? 7. Consider the gambling strategy in Example 3.2 of the lecture notes, where the initial wealth is 0. Find the exact distribution of the wealth M10 at time t. (This means finding the probability mass function P(M10 = x) for the possible values of x.) Verify numerically that E[M10] = 0.