Could you help me with Question 2 and Question 3 in the exercise?

Thank you very much.

Tri Vi Dang Corporate Finance Columbia University Spring 2016 Exercise Set 2 Due Tuesday, 2/16 This problem set consists of four questions. You can obtain a maximum of 40 points. Question 1 [16 points] Consider an economy with three assets and two dates (t=0,1) and three states at t=1. Let 1 2 3 X 2 1 4 1 3 1 1.4 p 1.8 p 3 be the matrix of asset payoffs at t=1 and p the vector of asset prices at t=0. Suppose p3=2. (a) Is there an arbitrage? [3p] (b) If yes, find an arbitrage portfolio. [3p] Suppose p3=1.2. (c) Does an arbitrage portfolio exist? [2p] (d) Can you create a portfolio with payoff of (60, 70, 120) at t=1 and what is the t=0 price of such a portfolio? [4p] (e) Determine the (implicit) risk free rate in this economy. [4p] 1 Question 2 [8 points] Consider an economy with two dates (t=0,1) and at t=1 there are three states. The following three stocks are traded: x1=(10,0,30) x2=(0,20,40) x3=(20,20,0) The t=0 prices of these stocks are given as follows (p1, p2, p3)=(12, 14, 8). (a) Is there an arbitrage? [2p] Suppose an investment firm sells options. (b) What is the t=0 price (premium) of a call option on stock 2 with exercise price E=10? [3p] (c) What is the t=0 price (premium) of a put option on stock 1 with exercise price E=8? [3p] Question 3 [6 points] Consider the economy (above) again where the following set of stocks is traded: x1=(10,0,30) x2=(0,20,40) x3=(20,20,0) for the prices (p1, p2, p3)=(12, 14, 8). Suppose a start-up company wants to go public. The firm has total costs of $60,000 at date t=1 and sales of $80,000 in state 1, $100,000 in state 2, and $120,000 in state 3. The firm wants to issue 1,000 IPO shares. (A share is endowed with a cash flow right of 0.1% of the total profits of the firm.) The underwriter suggests an IPO price of $32 per share. Will this IPO be successful, i.e. will there be a positive demand for the shares? 2 Question 4 [10 points] (a) Financial engineering deals with the design of new assets. Draw the payoff (at t=1) of the following bull butterfly spread: Purchase 1 call with exercise price a Sell 2 calls with exercise price (a+b)/2 Purchase 1 call with exercise price b as a function of the underlying stock price S at t=1 where a=20 and b=40. [4p] (b) An individual agent thinks that there is a high probability that the Dow Jones will have a payoff (or points) between a=12000 and b=17000 at t=1. Design a digital option (see Figure 1) as a sequence of calls on the Dow that converges to a pure bet on getting $1 on the interval [12000, 17000], i.e. if the Dow lies between S[12000, 17000] at t=1, then the portfolio of calls pays off exactly $1. The payoff is 0 otherwise. [6p] Figure 1 (Digital option) payoff 1 a=12000 b=17000 S Hint: You have to modify the sell strategies of a bull butterfly spread to obtain a payoff as given in Figure 2 and then adjust n and appropriately. Figure 2 payoff of portfolio 1=n a a+ b 3 b S