PLEASE WRITE ALL THE CALCULATIONS AND FORMULAS

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=8? [3p]

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 $90,000 at date t=1 and sales of $100,000 in state 1, $120,000 in state 2, and $200,000 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 $42 per share. Will this IPO be successful, i.e. will there be a positive demand for the shares?