What range of values of st will the short target forward

Question

Risk Management

Question 1 [Futures, Backwardation, and Contango]

The United States Oil Fund (ticker: USO) is a popular exchange-traded security that is designedto track changes in the price of oil (specifically, West Texas Intermediate or WTI). Since spot oilis expensive to store, the fund uses futures contacts to generate its returns. In simplified formthe fund uses the following strategy: At the beginning of each month, the fund enters into longpositions in one-month futures contracts, and rolls them over at maturity (i.e., it closes out thecontracts at the end of the month and takes positions in new one-month futures).

Note that at maturity, the futures contract has become a contract for immediate delivery, so hasessentially become a spot contract. Thus, at maturity, the price of the expiring futures contractmust equal the spot price. Now, the strategy involves taking a long position in a futures contractat the beginning of each month and closing it out--i.e., taking an offsetting short position inthe same futures contract--at the end of the month. Since at month's end the contract is atmaturity, this means we buy at the futures price at the beginning of the month and sell at thespot price at the end of the month.

The purpose of this question is to see if the fund's returns are affected by whether the market isin backwardation or contango. A futures market is in backwardation if the futures price is lessthan the spot price. It is in contango if the futures price is greater than the spot price.

1. First suppose the market is in contango (i.e., futures prices are greater than spot prices).Specifically, suppose the following is the series of realized spot and one-month futures pricesthat prevailed at the beginning of each month over a three month horizon:

Beginning of Month	1	2	3	4
One-month Future Price	52	59	60	62
Spot price	50	57	59	60

What is the fund's total profit (or loss) from futures trading over this three month horizon?[Note: The three-month horizon means that the fund enters into long futures positions atthe beginning of months 1,2, and 3 and closes them out respectively at the beginning ofmonths 2, 3, and 4.] Do these returns exceed or lag the actual change in the spot price ofoil over these three months?

2. Now suppose the market is instead in backwardation (futures prices are less than spotprices). Specifically suppose that the following is the series of realized spot and one-month futures prices that prevailed at the beginning of each month over a three monthhorizon:

Beginning of Month	1	2	3	4
One-month Future Price	48	55	58	58
Spot price	50	57	59	60

(Note that the spot prices are unchanged from the previous table.) What is the fund's totalprofit (or loss) from futures trading over this three month horizon? Is this greater or lessthan the actual spot price change in oil over these three months?

Question 2

This question concerns a simplified version of the target forwards contract introduced in class during the Aracruz Cellulose discussion.

You are comparing two alternatives:

- A short plain vanilla forward contract with a delivery price of K₁ = 100; and

- A short target forward with a delivery price of K₂ = 103.

Both instruments have a maturity of one month and are written on the same underlying asset.Both instruments are zero value at inception, i.e., you do not have to pay anything upfront toget into the respective contracts. If ST denotes the price of the underlying asset at maturity, the payoff from the short forward at maturity is

100 - S_T

for all ST, while the payoff from the short target forward at maturity is

103 - S_T, if S_T< 103
2 x (103 - S_T), if S_T ≥ 103

Answer the following questions:

1. For what range of values of S_T will the short target forward with delivery price 103 result in a larger payoff at maturity than the short vanilla forward contract with delivery price 100?

2. Show that the target forward with a delivery price of 103 is identical to a portfolio of twoitems: (i) a short position in a vanilla forward with a delivery price of 103, and (ii) a shortposition in a plain vanilla call option with a strike of 103. (In words, being short a targetforward is like being short an ordinary forward and, in addition, being short a call option.)That is, fill in the payoffs of each instrument in the following table and show that the totalpayoff is exactly the same as the target forward's payoff give above.

Position	Payoff if ST< 103	Payoff if ST ≥ 103
Short forward
Short call
Total

3. Assume that the one-month rate of interest is zero. What does the decomposition of the target forward in Part 2 tell you about the value of the call option embedded in the target forward? Use the following steps:

a) From the information given in the question, the plain vanilla forward with delivery price 100 and the target forward with delivery price 103 each have a value of zero.

b) Moreover, from Part 2, the value of the target forward is equal to the sum of the values of the short forward with a delivery of 103 and the short call with a strike of 103.

c) If the forward with a delivery price of 100 is worth zero, how much is the forward with a delivery price of 103 worth? That is, how much better or worse off are you from holding a forward contract that allows you to sell at 103 when the breakeven delivery price is 100?

d) Combining 1-3, how much is the short call with strike 103 worth?

Question 3

You are offered the following two one-year investment opportunities.

Investment Opportunity 1: Risk-Free Investment

The first is a standard interest-bearing deposit that will pay you the risk-free rate at the end of the year. That is, letting rf denote this risk-free rate, per $1 you invest in the deposit, you will receive

1 + r_f

at the end of the year.

Investment Opportunity 2: A Principal-Protected Note

The other is a principal-protected equity index-linked note that will pay you a fraction α of thereturns on the S&P 500 index as long as these returns are positive. Specifically, let S0 denotethe current level of the equity index, and let ST be the level of the index at the end of one year.

The net return on the equity index over the year is given by

r^~_m

= S_T/S₀ - 1

The payoffs of the equity-linked deposit are specified as follows: For every $1 invested today,

- If r~m ≥ 0, the equity-linked deposit will pay you $(1 + α r~m) at the end of the year, where α is a fraction specified in the contract.

- If r~m < 0, the equity-linked deposit will pay you $1 at the end of the year.

The objective of this question is to identify the "fair" value of α using the prices of traded options on the index. To this end, it helps to first express the payoff from the note in terms of standard option payoffs.

Expressing the Note Payoff as an Option.

In shorthand notation, if you invest $1 in the note, you receive at the end of the year the following amount:

1 + max {0, αr~m} = 1 + max {0, α (S_T/S₀) -1)}

The last term (the term max{...}) can be written equivalently as

1/S₀ x max {0, α (S_T - S₀) -1)} = α/S₀ x max {0, S_T - S₀}

In other words, this is just the payoff from α/S₀ units of an at-the-money call option on the index purchased at time 0 with a maturity date of time T (one year).

Thus, the overall payoffs (1) from the note are comprised of two parts: (a) a certainty payoff of $1 at the end of the year and (b) the uncertain payoff max{0, αr~m} which is equivalent to α/S₀ units of an at-the-money call option on the index.

To identify the "fair" value of α, proceed in the following steps.

1. As already noted, the payoff at maturity from a $1 investment in the note is a combination of positions in two common securities: (i) an investment at the risk-free rate, and (ii) an at-the-money vanilla call option on the index. Express clearly the sizes of the positions in each of these securities.

2. Let C denote the time-0 price of a one-year at-the-money vanilla call option on the index. In terms of rf and C, what is the time-0 present value of the payoffs from the note?

3. In a fairly valued note, this present value should equal the initial investment in the note. Using this, identify the fair value of α in terms of C, rf and S0.

4. Suppose you are given r_f = 5% = 0.05, C = 6.89, and S₀ = 100. What is the fair value of α?