Question 1  [24 points]

Consider an economy with two dates (t=0,1) and three states at t=1. The following three assets are traded:

Asset A has a payoff of xA=(2,0,0)

Asset B has a payoff of xB=(1,3,0)

Asset C has a payoff of xC=(2,2,4)

at t=1. In other words, asset B pays off $1 in state 1, $3 in state 2, and $0 in state 3 at t=1.

(a)       Suppose the prices of asset A, B and C at t=0 are pA=1.8, pB=3.3 and pC=6.2 respectively. Explain whether there is an arbitrage. [6p]

(b)       A put option on asset C with exercise price E=3 is traded. What is the price of this option at t=0?  [3p]

(c)       Design a portfolio with payoff (0,0,2) at t=1. What is the price of this portfolio at t=0? [4p]

Now consider an (equivalent) economy with three save bonds and four dates (t=0,1,2,3).

In other words, e.g. bond B costs $3.20 at t=0 and pays off $1 at t=1, $3 at t=2, and $0 at t=3.

(d)       What is the definition of a yield curve (or term structure of interests)? [2p]

(e)       Determine the yield curve in this economy. [6p]

(f)        If there is no arbitrage, the yield curve is always flat. Is this statement true?   [3p]

Question 2  [12 points]

Consider a firm which current stock price has experienced a sharp decline from above $40 to  $20 now (at t=0). Suppose an investor believes that the stock price will rebound to $30 at t=1.

An investment advisor suggests the investor to buy the following portfolio of options.

Sell 1 call option with exercise price E=$20

Buy 2 call options with exercise price E=$30

Sell 1 call option with exercise price E=$36

(a)       Draw the payoff of this strategy at t=1.  [6p]

(b)       Should the investor follow the recommendation? Please explain.  [2p]

Suppose at t=0 the price (premium) of the call with E=$30 is $2.40 and the price of the call with E=$36 is $1.30. Suppose there is no arbitrage.

(c)       Can you determine the minimum price of the call with E=$20? If so, what is it? If not, what additional information do you need? [4p]

Question 3  [12 points]

Consider an economy with three states which occur with probabilities (0.2, 0.4, 0.4). Suppose an entrepreneur has $140 and owns a firm that has a single project which generates the state    dependent cash flows (100, 100, 300) at t=1. The investment costs are $140 at t=0. The

market portfolio generates the payoff (100, 200, 300) and has an expected return of 12%. The risk free rate is 3%. Suppose the CAPM holds.

(a)       What is the beta of this project? [4p]

(b)       Explain whether the entrepreneur should conduct the project.  [6p]

(c)       Suppose a private equity company wants to acquire the firm. At what price is the entrepreneur willing to sell the firm?   [2p]