Consider a single-period trinomial model with state space = {B,M,G}, a bank account offer- ing zero...

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Consider a single-period trinomial model with state space = {B,M,G}, a bank account offer- ing zero interest rate (r = 0), as well as a stock with current price So = 100, while R(B) = 1/10, R(M) = 0 and R(G) = 1/5. 6. Sam has log-utility and P{M} = 1/2; no other information for Sam's P{B} and P{G} is given. Show that utility indifference prices for Sam are given by expectations of payoffs under the risk-neutral probability with Q{B} = 1/3 and Q{M} = 1/2. 7. Explicitly solve Max's expected log-utility maximization problem max E [log X1], where X0 > 0, X1 [7 marks] in terms of optimal fraction * of investment in the stock, where the (only) assumption made for Max's subjective probability P is that P{B} = (3/2)P{G}. [7 marks] = - = = 8. Extend now the model to a multi-period trinomial, with zero bank interest, and each time period having independent returns (Rt; t = 1,2,...), and such that P[Rt -1/10] = (3/2)P[Rt : 1/5], P[R = 0] = 1 P[Rt = 1/10] P[R = 1/5]. In other words, stock returns are i.i.d. with the similar probabilities as in part 7 above, from which we keep the same set-up and notation. Show that the portfolio * also maximizes the long-run growth lim((1/T) log XT) over all portfolios , where X is the time-T wealth of investing a constant proportion of current wealth at the stock in each period. [6 marks] Consider a single-period trinomial model with state space = {B,M,G}, a bank account offer- ing zero interest rate (r = 0), as well as a stock with current price So = 100, while R(B) = 1/10, R(M) = 0 and R(G) = 1/5. 6. Sam has log-utility and P{M} = 1/2; no other information for Sam's P{B} and P{G} is given. Show that utility indifference prices for Sam are given by expectations of payoffs under the risk-neutral probability with Q{B} = 1/3 and Q{M} = 1/2. 7. Explicitly solve Max's expected log-utility maximization problem max E [log X1], where X0 > 0, X1 [7 marks] in terms of optimal fraction * of investment in the stock, where the (only) assumption made for Max's subjective probability P is that P{B} = (3/2)P{G}. [7 marks] = - = = 8. Extend now the model to a multi-period trinomial, with zero bank interest, and each time period having independent returns (Rt; t = 1,2,...), and such that P[Rt -1/10] = (3/2)P[Rt : 1/5], P[R = 0] = 1 P[Rt = 1/10] P[R = 1/5]. In other words, stock returns are i.i.d. with the similar probabilities as in part 7 above, from which we keep the same set-up and notation. Show that the portfolio * also maximizes the long-run growth lim((1/T) log XT) over all portfolios , where X is the time-T wealth of investing a constant proportion of current wealth at the stock in each period. [6 marks]