Economics of derivatives

1.a. Find the value of a one-month European call with exercise price $60 if the stock price ST on the expiration date T is a random variable on a risk neutral probability space Q with its uniform density function given by:

1 𝑞(𝑆𝑇)= {60

, 40 ≤ 𝑆𝑇 ≤ 100

0 , 𝑒𝑙𝑠𝑒𝑤h𝑒𝑟𝑒

Assume that the per-annum continuous time risk free rate of interest is 3.5%.

b. Whatisthecurrentvalueofa$1,000facevaluecouponrisklessbondunderthe

circumstances presented in part a above?

c. Suppose that a stock C is always worth twice as much as the stock referred to in part a

above. What would be the value of a call on this stock if its exercise price were $120?

2. Outcomesonethroughfiveinasingle-periodframeworkcorrespondtoelementsinthe 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?Youshouldassumethattheputhasanexercisepriceequal 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.)

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

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

= .05 and a variance σ2 = .25. What is the probability that r5 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 S0 = 1, its instantaneous drift rt has an expected value μ = .05 per year and an annual variance σ2 = .25. What is the probability that the stock is worth more than 2 in five years P[S5> 2]?

c. Isthereturnprocessforthisstockamartingale?

4. 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. Inthissameeconomy,supposethatthespotpriceofgoldis$1,800perounce.Whatisthe 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,assumeasingleexceptiontoperfectmarketefficiency,withtheannualcostofstoring 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?

5. Inadiscreet,two-period,perfectlyefficientmarket,astockissellingfor$1pershare.Ineach of the two one-year periods in this economy, the stock’s price will either double or drop in price by 40%. All riskless bonds will yield 10% each year; that is, the yield curve is flat. You may assume that markets are efficient and allow for no-arbitrage pricing. What is the time-zero (now) no-arbitrage market value of a 2-year call in this economy if its exercise price equals 2?