Midterm Exam Possible Solutions

K Foster, Options & Futures, Eco 275, CCNY, Spring 2010

1. (10 points) You're going to buy corn futures; each full-sized contract is 5000 bushels, while mini-contracts are 1000 bushels (delivered to a few locations in Illinois). Currently the contract for July 2010 delivery (on the 14^th of that month) trades at 375 (cents per bushel).
1. How much does a single full-sized contract cost? A mini-contract?

A single full-size contract costs 5000*3.75 = $18,750; a mini-contract costs $3750.

2. Given interest rates of 2.5%, what range of prices would not present arbitrage opportunities?

Spot prices and futures prices should be linked; F0 = S0e^rT so the spot price should be 3.7188, otherwise there are arbitrage opportunities.

3. Initial margin is $1350 so assume you buy one contract with only that margin amount. The maintenance margin is $1000. How large of a price decline would trigger a margin call?

Each penny of price decline costs $50 for the whole contract so the price could decline only 7 cents before there is a margin call.

2. (20 points) You have a portfolio with calls and puts on oil contracts; all of these options are European. The current oil price is $80.68/barrel. You are long 10 calls with strike of 82 and short 8 calls with strike of 84. (Contract size is 1000 barrels in each; each option brings the right but not the obligation to buy 1000 barrels.) You own 5 puts with strike price of 81 and 5 puts with strike price of 80. You are short 7 puts with strike prices of 83.
1. Draw the payoff graph for your portfolio.

2. If you bought 10,000 barrels (or 10 contracts) at a price of 80.50/barrel, how would the payoff graph change?

3. (25 points) You are considering whether to buy an at-the-money European put, expiring in a month, on a stock that is currently worth 50. After one month assume the stock will be worth 55 or 45. The risk-free rate is 2%.
1. What are the risk-neutral probabilities of the stock rising or falling?

The risk-neutral probabilities, p and (1 – p), must satisfy so substitute in r=0.02 and T=1/12 and solve: so p=.50834 and (1 – p) = .49166.

2. What is the delta for the put?

Use Hull's Δ-method to find the number of shares to combine with a short position in the call to get a riskless portfolio. The put is worth 0 if the stock goes to 55 or is worth 5 if the stock goes to 45. So the portfolio is worth 55Δ if the stock goes up or 45Δ – 5 if the stock goes down. Set these equal to get Δ=-.5.

3. What is the fair value of the put?

Use the risk-neutral probabilities to find the put value is = 2.4542.

Or use Hull's Δ-method to find that the riskless portfolio, which has current value = -27.4542, is composed of Δ units of stock and one short put, 50Δ – put, so the put price is 27.4542 – 25 = 2.4542 – the same answer as with the probabilities.

4. (20 points) The Greek government has seen prices of its bonds fall dramatically. Consider two (fictitious) bonds; both pay 100 semi-annually. One is an on-the-run 5 year bond with just 1.5 years remaining (3 remaining payments); the other matures in 3 years and also pays 100 semi-annually.
1. As of September 2009, the first bond traded at a price of 292.65. The second bond traded at a price of 662.61. Assuming discrete semi-annual compounding, what was the implied zero rate for 1.5 years?

Find the value of r that solves , which is R=2.5% (you can solve this by guessing a couple rates – the graph gives clues about the likely value).

2. What was the forward rate from 1.5 to 3 years?

Now find the value of the rate that solves -- my typo had 662.61 not 562.61 (which would give 3.75%). From the correct number you'd find a forward rate of 5%.

3. By March 2010 (6 months late), Greece's fiscal crisis had begun worrying the financial markets so bond prices plunged. Now the same first bond from part (a), with just 2 payments left now, trades at a price of 192.74; the second bond with 2.5 years remaining trades at 463.27. What is now the implied zero rate for 1 year?

Now solve so R=5%.

4. What is the forward rate from 1 to 2.5 years?

Solve so R=5.2% so the forward rate is 5.4%.

5. Actual Greek bonds saw the yield curve compact, as short rates rose substantially while longer yields rose by less; the graph below (data from Greek Central Bank) shows yields on 3 – 30 year bonds. Explain why this might be so.

Answers will vary; somehow explain theories of yield curve slope.

6. How has the duration of these bonds changed?

You can calculate up to eight separate durations/modified durations; the table summarizes them:

	maturity 1.5 or 1 yr		maturity 3 or 2.5 yr
	R old	R new	R old	R new
duration	0.996	0.747	1.723	1.474
modified duration	0.996	0.747	1.723	1.474