Professor Juhani Linnainmaa
Tuck School of Business
Futures and Options Markets
Instructions: This final exam is due on Tuesday, March 12, at 5PM. For all prob- lems, you can assume that you can trade an asset in any quantity unless otherwise specified. For example, one can buy 0.31 shares. There are 6 questions for a total of 60 points. This test has 9 pages.
Resources you can use: This is a take-home exam and is to be completed in- dividually without collaboration or assistance from others. The following is a non- exhaustive list of examples of allowed and non-allowed resources. This exam is covered under the Tuck Honor Code. If you have any doubts about any specific resources, err on dis-allowed, and feel free to ask me.
1. Any material from class, your group project answers, and the textbook.
2. Material on the internet to the extent that you are NOT collaborating or receiving assistance from other people.
-Examples: Downloading data from Bloomberg, reading articles on In- vestopedia or Wikipedia, or looking up a definition.
1. Any material that is collaboration or direct assistance from others.
-Examples: Consulting someone about the problem (or a “hypothetical” problem that is just like the problem) in any way. This includes, but is not limited to, online chat/phone/video, e-mail, or forums.
2. Other groups’ project answers, consulting a previous year’s exam, or someone’s answers to that exam.
3. Consulting your own group members, classmates, or any person even outside the FOM. This includes asking for clarification from your group-mates about their work on the group projects.
4. Plagiarizing material is strictly prohibited. Obviously!
What to turn in:
One Excel spreadsheet which contains, for each problem, a worksheet tab with all relevant analyses and qualitative responses. If you used any software that is not Excel, you need to include all program files, although you should only need Excel.
For each calculation, you must show all your work clearly. A correct answer with insufficient work, or work that I cannot easily decipher, may not receive full credit.
Some questions may ask for your thoughts in addition to calculations. A correct calculation with insufficient reasoning or explanation may not receive full credit.
Each question is worth 10 points, but some questions are harder than others. Please make your best attempt at every question—there is partial credit and there is no penalty for wrong answers.
Consider two stocks, A and B, both non-dividend-payers today and for the foresee- able future. The price of A is $200 per share today, and the price of B is $75 per share. The continuously compounded risk-free rate is 2% per year.
A) Consider an 8-month put option on the S&P 500 struck at K = 2100. The current price is S = 2050, the risk-free rate is 3%, and the dividend yield on the index is 2%. Use a four-period binomial tree (with CRR values of u and d) to price the option using a volatility of σ = 15%. Value the option supposing it is American. Be sure to indicate when early exercise is optimal, if ever. (4 points.)
B) Consider the marriage problem. In the version from class, if you choose not to marry the last person, you receive zero. Suppose instead the following: if you choose not to marry the last person, your happiness from being eternally single is 0.75. With three people to go, what is your optimal strategy? (3 points.)
C) Consider a digital call option (sometimes known as a binary call option) which entitles the option holder to $1 if the stock price equals or exceeds the strike price at expiration, and nothing otherwise. A digital put option entitles the option holder to $1 if the stock price falls below the strike at expiration.
A non-dividend-paying stock currently trades at $52. A six-month digital call option with strike 50 trades at $0.6536; a digital put option of the same strike and maturity trades at $0.3355. A six-month zero-coupon risk-free bond with face value of $100 trades at $99.50.
Determine whether there is an arbitrage opportunity. If so, demonstrate the arbitrage and calculate the arbitrage profit. If not, show that prices are con- sistent with no-arbitrage. If there is insufficient data to determine whether there is an arbitrage, indicate why. (3 points.)
A common perception is that futures are a promising investment when the futures curve is downward-sloping.
A) Discuss the merits of this logic for equity index futures. (2 points.)
B) Discuss the merits of this logic for commodity futures. Include in your dis- cussion why you may reach a different conclusion than in part A above. (3 points.)
C)Discuss the merits of this logic for VIX futures, and explain whether this logic has held using the data from Project2 CFE VIX Futures2.xlsx. Include in your discussion why you may reach a different conclusion than in parts A and B above. (5 points.)
A)Use the hypothetical data provided in Final DeltaHedge.xlsx to compute the daily market-making profits to a short 40-strike call plus delta-hedge which is rebalanced each day. Then compute the present-value of these profits and explain why the hedged position gained or lost money. Then compute the delta, gamma, and theta of the option Greeks for each day, and explain at which points the market-maker faced the greatest risk. Assume the stock does not pay dividends. (5 points.)
B) With 40 days to go, graph the 1-day holding period profit of the market maker’s delta-hedged position against possible stock prices. At what level of stock price movement does the market maker lose money over the next day? (5 points.)
A common piece of “wisdom” is that stocks are safer in the long run. Your friend is a strong proponent. She reasons that if you have a long enough horizon, the law of large numbers works in your favor, and you will safely earn the average return on stocks. She goes on to argue that this makes stocks a better investment than bonds in the long run, as stocks will safely earn higher returns.
You and your friend agree to discuss this issue within the context of the Black- Scholes model. You consider option prices struck on the S&P 500 total return index (this allows you to assume δ = 0 throughout, since the index capitalizes dividends). For the sake of discussion, you both agree that the Black-Scholes assumptions are satisfied: e.g., stock prices are log-normally distributed with constant volatility, interest rates are constant, and so on. You also both agree that the current stock price is fairly-valued.
A) Compute Black-Scholes prices for put options struck at-the-money, where the current index level is S = 1000, the risk-free rate is 0%, and the implied volatility is 18%, for maturities of 1 year, 3 years, 5 years, 10 years, 20 years, 50 years, and 100 years. Do these prices support the idea that stocks are safe in the long run? (2 points.)
B)Your friend is quite good at statistics. She figures that, if the market risk premium is 5% (ann. c.c.) and prices are lognormally distributed, the following table describes: 1) the probabilities that ATM puts expire in the money, 2) the expected payoffs if they do expire in the money—E(K ST ST < K)—and
3) the expected payoffs of the option:
Your friend argues the stock market is thus safe because expected payoffs from long-horizon puts struck ATM are small. You’re not sure these calculations are right, but you take them at face value. Point out why (and if) your friend’s logic is mistaken. (3 points.)
C) Your friend goes on to argue that using option prices to determine whether stocks are safer than bonds is non-sense because options “do not account for the 5% expected return on the market—the expected return could be 10% and the option price would be the same.” Comment on whether this is true and explain whether this invalidates your argument. (1 point.)
D) Finally, your friend says your argument is not realistic because you have as- sumed a risk-free rate of zero. She proposes that you input rf = 4%.
Calculate option prices with this risk-free rate for all the T above. You will find that the price of the 100-year ATM put option is lower than the 1-year ATM put price. Ha ha! Your friend claims victory.
Explain the intuition in Black-Scholes for why this occurs. Then, go on to explain why the new numbers, while correctly calculated, do not support her argument. Propose an alternative calculation that corrects the flaw in her logic. (4 points.)
Use the data provided in Project 2 to evaluate the performance of a strategy that purchases call options with 30 days to expiration, and holds them through expiration.
For 96, 100, 102% K/S moneyness levels, compute the monthly alphas using the SPXT as the benchmark market return, and compare the monthly Sharpe ratio of the strategy versus the Sharpe ratio of the market. Comment on what these measures may or may not suggest about the performance of the strategy. (10 points.)