Question

1. [Valuing a Put Option] Suppose a stock is currently trading for $60, and in one period will either go up by 20% or fall by 10%. If the one-period risk-free rate is 3%, what is the price of a European put option that exprires in one period and has an exercise price of $60?

2. Suppose the uncertainty involves two possible states (0- = 1,2) with equal probability 0.5. The investment A produces a state-contingent rate of return 3% at the state 1 and 5% at the state2. The investment B produces a state-contingent rate of return 2% at the state 1 and 8% at the state 2

a. Based on the state-by-state dominance, which investment project would you prefer?

b. Based on the mean-variance criterion, which investment project would you like better?

c. Which investment project would you rather have? Tell me your criterion

3. Consider the following portfolio of two risky assets: the asset 1 with return r1 and the asset 2 with return r2. We invest x dollars in the asset 1 and (1-x) dollars in the asset 2, where 0<=x<=1.

a. Calculate the expected value of the portfolio E[rp]

b. Calculate the variance of the portfolio, Var(rp)

c. Based on your findings on the part b. what kind of assets you should choose when constructing the portfolio.

d. CAPM assets that all investors will hold the optimal portfolio. What is the optimal portfolio in this context?

4. Why do economists use a utility function to present an economic agent's preference? Is this utility-based approach plausible?

5. How do we measure risks?

Answer 1

You have asked 5 unrelated and different questions in the same post. Some of these questions have further sub parts. I will address the first question in entirety. Please post the balance questions separately.

1. [Valuing a Put Option] Suppose a stock is currently trading for $60, and in one period will either go up by 20% or fall by 10%. If the one-period risk-free rate is 3%, what is the price of a European put option that expires in one period and has an exercise price of $60?

S_o = $ 60; u = (1 + 20%) = 1.20; d = 1 - 10% = 0.9; r = 3%, T = 1 year, S_u = u x S₀ = 1.2 x 60 = $ 72, S_d = d x S₀ = 0.9 x 60 = $ 54, Strike price, K = $ 60

Payoff from Put option in up state, P_u = max (K - S_u, 0) = max (60 - 72, 0) = 0

Payoff from Put option in down state, P_d = max (K - S_d, 0) = max (60 - 54, 0) = 6

Risk neutral probability of up state, p = (e^rT - d) / (u - d) = (e^{3% x 1} - 0.90) / (1.20 - 0.90) =  0.4348

Hence, price of Put option today = e^-rT x [p x P_u + (1 - p) x P_d] = e^{-3% x 1} x [0.4348 x 0 + (1 - 0.4348) x 6] = $ 3.29