Section A: Answer only ONE question from this section
Question 1 [20 marks]
For each of the statements below, state whether it is True or False, justifying your answer:
a. In a market in which one of the state price vectors is ψ = h [−1 0 1 i]′ , there is definitely arbitrage.
b. According to the consumption CAPM, investors’ endowments play a role in determining prices.
c. Consider a market with two assets. Given some prices, it is always the case that we can graphically find an investor’s optimal combination of the two assets as the point of tangency between the investor’s budget constraint and his indifference curves.
d. Consider using the binomial model for pricing a derivative. The price we calculate for the derivative will be exactly the same, no matter what value we choose for the probability p of the high return in the model calibration.
Question 2 [20 marks]
A commercial bank wants to sell a new derivative security. If ST is the stock price at expiration, this new derivative pays S γ T , for some γ > 0. Suppose that stock returns follow a geometric Brownian motion with annual mean rate of return equal to 10% and annual volatility σ = 10% a year. The risk-free rate of return is r = 5% per year, the contract expires in two years, and the initial stock price is 5. You have been hired as an analyst to price the power contract with γ = 0.5. Proceed in steps:
a. Explain, in words, how a derivative is priced in continuous-time finance using the risk-neutral method.
b. Write the SDE for the stock price using BP , the Brownian motion under the objective probability measure.
c. Rewrite the SDE for the stock price using BQ, the Brownian motion under the risk-neutral measure. State which theorem you are using here. Also explain in words how the change of measure works.
d. Use Ito’s lemma to find the stochastic differential equation for ln St under the risk-neutral measure. Integrate the SDE for ln St to find the risk-neutral distribution of ln ST as of time 0, where T is the time to expiration. e. Express the price of the derivative (at t = 0) as a risk-neutral expectation. Evaluate this expectation.
Section B: Answer ALL questions from this section
Question 1 [20 marks]
a. Is the market complete?
b. Can you calculate the risk-free return in this market? If yes, what is it?
c. Find state prices consistent with A and S and the Law of One Price, and then determine whether there is arbitrage.
d. If there is arbitrage, propose an arbitrage portfolio and explain what type of arbitrage it is. If there is no arbitrage, state what the risk-neutral probabilities are.
Question 2 [20 marks]
For each of the statements below, state whether it is True or False, justifying your answer:
a. In a one-period model with dates today and tomorrow, a riskless asset is one whose payoff tomorrow is known with certainty.
b. Consider the consumption CAPM model we have discussed in class. The risk-free return does not depend on investors’ time preferences.
c. Consider the economy with two investors and two assets, as we discussed in class, and denote with p1 p2 the ratio of the prices for asset 1 and 2, respectively. If, at this price ratio, there is excess supply for asset 1, then the equilibrium price ratio is higher.
d. The annual log true return of a stock is i.i.d. normally distributed with mean and variance 0.06 and 0.24, respectively. You want to write a 2-period binomial model to price a derivative that expires in 4 months and whose payoffs depend on the price of this stock. In this model, the high per-period return for the stock (i.e., Ru in the notation used in class) is 1.324.
Question 3 [20 marks]
Consider an individual with preferences represented by u (x1, x2) = x1 + √ x2, where x1 and x2 represent consumption of good 1 and good 2, respectively. Assume that:
• consumption (of either good) cannot be negative,
• the ratio of the two goods’ prices is p1 p2 = 2, and
• the individual is initially endowed with 0 units of good 1 and 0.5 units of good 2. What is the individual’s optimal allocation? Solve analytically, showing steps in detail, and also demonstrate graphically.
Question 4 [20 marks]
Consider a three-period model, with t = 0 representing the first date and t = 3 the last date. There are two assets: the risk-free bank account and the stock. The risk-free rate of return, in each period, is constant and equal to r = 0.
The one-period stock returns are independent and identically distributed, and can take two values, Ru = 2 or Rd = 0.5, with objective probabilities pu = 1/ 2 and pd = 1 2/ . The initial stock price is 10. Consider a “down-and-out” barrier call option that matures at the end of the third period (i.e., T = 3), and has strike price K = 10 and barrier B = 6. Such an option pays
and otherwise CT = 0.
a. Construct the stock price tree.
b. Consider an arbitrary payoff at t = 3. Is it possible to price this payoff?
c. Calculate the risk-neutral probabilities in each one-period sub-model.
d. Write the barrier call option’s payoffs at the terminal nodes of the tree at t = 3.
e. Price the barrier call option.
f. At t = 0, find the amount deposited in (or borrowed from) the bank, and the number of shares required for the self-financing portfolio that perfectly replicates the payoff of the barrier call option at maturity.
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