4. Arbitrage (30 points) Suppose you are in the business of making bets about the weather a month from now. For simplicity: there are three possible outcomes. It could be sunny, it could be rainy, or it could be windy. For this problem, let time 0 denote today, and time 1 denote one month from now. In betting on the weather, you can make four investments: You can save money in a money market account (which accrues interest at the short-term interest rate). A time 0 value of B(0) = 1 grows to B(1): = 1.25 by time 1, regardless of what the weather ends up being. You can invest in a security S, which has value S(1) and S(1) = 0 otherwise. It has initial price So today. 1 if it is sunny a month from now, You can invest in a security W, which has value W(1) now, and W(1) = 0 otherwise. It has initial price Wo today. 1 if it is windy a month from You can invest in a security NW (NW for not windy), which has value NW (1) = 1 if it is sunny or rainy a month from now, and NW (1) = 0 if it is windy. It has initial price NWo today. Suppose that the probability of it being sunny is 50%, the probability of it being rainy is 49%, and the probability of it being windy is 1%. You know the following time-0 prices: So = 0.36 NW = 0.77 (a) (3 points) Draw the (trinomial) tree associated with the setup of this problem. Make sure you denote all four of the asset prices in each state. (b) (5 points) Suppose you were offered the windy security at a price Wo = 0.04. Explain what the arbitrage opportunity is. Hint: can you replicate the money market account payoff using the three other securities? (c) (5 points) Suppose there are no arbitrage opportunities. What must the price Wo be? For the rest of this problem, suppose that the windy contract is offered at this no-arbitrage price Wo computed in (c). If you were unable to find Wo, leave your other answers in terms of Wo. (d) (12 points) Find the risk-neutral probabilities for each of the possible outcomes (sunny, rainy, or windy). Denote the three probabilities by qs, QR, and qw, respectively. Hint: as you only just learned these, remember that the risk-neutral probability measure Q must price all of our securities as discounted expected cash flows. That is for VB, S, W, NW. V(0) B(0) EQ B(1) (e) (5 points) Use your risk-neutral probabilities to price a contract that pays out 3 if it is sunny, 5 if there is windy, and 2 if it is rainy. (You can do this problem even if you couldn't answer (d).)