2 Two-Period Binomial Tree (Medium, 30 points) There are three total periods t = 0, 1, 2 and two assets with known prices, a stock (S) and a money market fund (B). Period 0 is today, and is not random. The stock price is So and the money market fund starts at B(0) = 1. The short-term interest rate is ro. From period 0 to period 1, there are two possible realizations. With probability Po, we could go to the up state, in which case the stock price rises to S(1) = us(0). With the remaining probability 1 Po, we could also go to the down state, in which case the stock price falls to S(1) ds(0). Here, u > 1 and d ro, and the conditional probability of going to the up-up" state is pu- If, at t = 1, we are in the "down" state (S(1) = dSo), then the stock can still rise and fall by u and d. However, the short-term interest rate falls to ra eu so that there is no arbitrage between the money market account and the stock. (a) (5 points) Draw the binomial tree. Make sure to include the stock S(t) and the money market fund B(t). (b) (10 points) Suppose we add a new (redundant) asset R with price R(t) at time t. Suppose you were given the four possible time t = 2 prices Ruu, Rud, Rdu, and Rdd- Using no-arbitrage arguments, find the two t = 1 prices R, and Rd, and the one t = 0 t = price Ro. = = (c) (5 points) Use your answer above (or any other method you like) to find the three risk- neutral probabilities of the stock price increasing: qurqd, and 9o. (Recall: qu is the risk-neutral version of Pu, so it is the risk-neutral conditional probability of the stock price increasing again after the first increase. qd is the same in the "down" state. qo is the risk neutral probability of the stock price increasing from t = 0to t = 1.) (d) (10 points) Suppose So = 100, u = 1.1, d = 0.95, ro = 0.02, ru = 0.03, ra = 0.01, po = 0.6, Pu = 0.7, and pa=0.5. Find the price Co of a European call option that expires at time t = 2 and has strike price $100. In case you haven't dealt with options before, this means that at expiration you get the difference between the stock price and the strike price if this is positive, and zero otherwise. This means that C(2) = max{S(2) K,0}. Use this to compute Cuu, Cud, Cdu, Cdd, and then use whatever method you like to figure out the t = 0 price. (e) (2 points bonus) What is the stochastic discount factor? &o = 1, but give me the remaining values &u,&d, &uu, Sud, &du, and Edd. Do this in the general case, not the numerical example for (e). 2 Two-Period Binomial Tree (Medium, 30 points) There are three total periods t = 0, 1, 2 and two assets with known prices, a stock (S) and a money market fund (B). Period 0 is today, and is not random. The stock price is So and the money market fund starts at B(0) = 1. The short-term interest rate is ro. From period 0 to period 1, there are two possible realizations. With probability Po, we could go to the up state, in which case the stock price rises to S(1) = us(0). With the remaining probability 1 Po, we could also go to the down state, in which case the stock price falls to S(1) ds(0). Here, u > 1 and d ro, and the conditional probability of going to the up-up" state is pu- If, at t = 1, we are in the "down" state (S(1) = dSo), then the stock can still rise and fall by u and d. However, the short-term interest rate falls to ra eu so that there is no arbitrage between the money market account and the stock. (a) (5 points) Draw the binomial tree. Make sure to include the stock S(t) and the money market fund B(t). (b) (10 points) Suppose we add a new (redundant) asset R with price R(t) at time t. Suppose you were given the four possible time t = 2 prices Ruu, Rud, Rdu, and Rdd- Using no-arbitrage arguments, find the two t = 1 prices R, and Rd, and the one t = 0 t = price Ro. = = (c) (5 points) Use your answer above (or any other method you like) to find the three risk- neutral probabilities of the stock price increasing: qurqd, and 9o. (Recall: qu is the risk-neutral version of Pu, so it is the risk-neutral conditional probability of the stock price increasing again after the first increase. qd is the same in the "down" state. qo is the risk neutral probability of the stock price increasing from t = 0to t = 1.) (d) (10 points) Suppose So = 100, u = 1.1, d = 0.95, ro = 0.02, ru = 0.03, ra = 0.01, po = 0.6, Pu = 0.7, and pa=0.5. Find the price Co of a European call option that expires at time t = 2 and has strike price $100. In case you haven't dealt with options before, this means that at expiration you get the difference between the stock price and the strike price if this is positive, and zero otherwise. This means that C(2) = max{S(2) K,0}. Use this to compute Cuu, Cud, Cdu, Cdd, and then use whatever method you like to figure out the t = 0 price. (e) (2 points bonus) What is the stochastic discount factor? &o = 1, but give me the remaining values &u,&d, &uu, Sud, &du, and Edd. Do this in the general case, not the numerical example for (e)