1. Theoretical vs. Actual Futures Price with market data: This problem asks you to get recent data in order to compare theoretical versus actual futures prices. (For this problem, don't worry about differences between compounding conventions - you can assume all rates are continuously compounded)
Using the sources below, find the S&P 500 index futures settlement price expiring in March 2016.
https://www.cmegroup.com/trading/equity-index/us-index/sandp-500_quotes_settlements_futures.html
You can find the maturity (settlement) date here:
https://www.cmegroup.com/trading/equity-index/us-index/sandp-500_product_calendar_futures.html
Note the trading day for the price. Look up the closing price of S&P 500 index on the same trading day (e.g., Google Finance, Yahoo Finance).
Find an approximate S&P dividend yield and 6-month riskless interest rates. Sources for these are
- https://www.indexarb.com/dividendAnalysis.html
- https://www.bloomberg.com/markets/rates-bonds/government-bonds/us/
Using this information and the formula for the theoretical (fair) index futures price, fill in the following table. Compare your answer to the data from the futures data website.
| Maturity Date |
S&P Index Level |
Years to Maturity |
6-month rate |
Dividend yield |
Theoretical Futures |
Actual Futures |
2. Value of the Index Futures contract: We learned to value futures using formulas that adjusted for either discrete income (e.g., quarterly dividends) or continuous income (e.g., dividend yield). In this problem you will do both types of valuation. Use the information sources in Problem #1 to get the data needed to value the S&P 500 index futures.
a) Assume that you took a LONG position in the March 2016 contract in early August at a price of 2000. What is the current value of this future contract if you treat the dividend income as a yield?
b) Recalculate the current value of the futures contract using the dollar amount of dividends. For the estimated dividends you can use the Divisor Adjusted Dividends from https://www.indexarb.com/dividendAnalysis.html. Assume the Divisor Adjusted Dividend is the Present value of the dividends over the life of the contract (i.e., that it represents I in the futures pricing/valuation formulas).
c) How much do the values in parts a) and b) differ?
3. Stock Index Arbitrage: Suppose we observe the monthly spot price of the S&P 500 Index and corresponding futures price over an eight month period for a stock index futures contract maturing in month 8. Here are the prices in index points:
| Month |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
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|
|
| Spot Price |
1539.21 |
1552.95 |
1587.44 |
1603.48 |
1659.12 |
1653.47 |
1693.59 |
1710.66 |
|
|
|
|
|
|
|
|
|
| Futures Price |
1544.76 |
1547.45 |
1577.21 |
1584.09 |
1640.33 |
1670.81 |
1691.65 |
1710.66 |
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| Theoretical Futures Price |
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a) Assume that the risk-free rate is 0.13% per month and the dividend yield is .25% per month. Fill out the table for the theoretical Futures Price each month.
b) Plot the observed Spot Price, observed Futures Price, and the Theoretical Futures Price over the eight months to maturity.
c) Assume that the roundtrip transaction cost for doing an index arbitrage trade is 1.8 index points. Analyze the monthly data above to identify any months it may have been possible to have made an arbitrage profit. Create a new figure that plots the observed Future Price, Theoretical Futures Price, and includes arbitrage bands so that is easy to visualize any months in which arbitrage would be possible.