1. Outcomes one through five in a single-period framework correspond to elements in the

following probability vectors that exist in ℙ and in ℚ spaces:

ℙ = [0, .1, .21, .29, .4] T

ℚ = [0, .4, .3, .2, .1] T

Thus, for example, the probability of outcome 1 is zero under both ℙ and ℚ.

a. Are ℙ and ℚ equivalent probability measures?

b. If the current riskless return rate equals 10%, what is the current value of a put option on a

stock with the following payoff vector under these same 5-outcome risk-neutral measures

with ℚ: [20, 30, 40, 50, 60] T ? You should assume that the put has an exercise price equal

to 35.

c. Suppose that a futures contract trades on the stock in part b. What is the current futures

price on this contract?

d. Suppose that there is a call with an exercise price of 35 trading on the stock from part b.

What is the expected risk-neutral value of this call contingent on it being exercised?

e. Consider the stock for which the payoff vector is given in part b. If one were to use the

riskless one-year bond as the numeraire for pricing purposes, what would be the current

stock price under its equivalent martingale measure based on the equivalent probability

measure ℚ? (Make sure that you denominate your final numerical answer in terms of

either the correct number of dollars or riskless bonds.)

2. Let {r t , t ≥0} (the return on a stock) be an arithmetic Brownian motion.

a. Suppose that r t is made up of two components, an instantaneous drift with expected value µ

= .05 and a variance σ 2 = .25. What is the probability that r 5 is between .3 and .5?

b. Suppose that the price of a stock follows a geometric Brownian motion process. Suppose

that the stock’s initial value S 0 = 1, its instantaneous drift r t has an expected value µ = .05 per

year and an annual variance σ 2 = .25. What is the probability that the stock is worth more than 2 in five years P[S 5 > 2]?

c. Is the return process for this stock a martingale?

3. A put and a call are selling for $5 each on a share of stock currently worth $50. Both the put

and call expire in one year and have exercise prices equal to $50. The market for stocks and

options are perfectly efficient, with no-arbitrage pricing evident.

a. What is the riskless return rate in this economy?

b. How would your answer to part a of this question change if investors were strongly risk

averse?

c. In this same economy, suppose that the spot price of gold is $1,800 per ounce. What is the

futures price of an ounce of gold assuming that the Expectations Hypothesis for futures pricing

holds? Ignore part d of this question to answer this part c.

d. Now, assume a single exception to perfect market efficiency, with the annual cost of storing

gold being $2 per ounce. However, the riskless interest or return rate is still consistent with the

correct answer for part a. Would this futures market for gold more likely be in contango,

backwardation, both or neither?

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