Please, in no particular order read the Wikipedia entry about futures contracts and listen to the fun and educational episode of Planet money about credit risk in cafeteria futures. Both are linked below:
Wikipedia: Futures
Planet money: Delicious cake futures
(10 points) Provide the definitions of the forward contract and the futures contract. Then, highlight at least twosimilarities and at least two differences between the two.
The files “dow.csv” and “dow-futures.csv” contain index prices and the futures prices for delivery on a particular date in the future.
(5 points) Import the data into a data frame (or two?). Draw the time-plot of the evolution of the closing index prices and the futures prices on the same coordinate system. You do not need to put the calendar days on the horizontal axis, but you do need to label your axes and give your time-plot a title indicating the dates. Make sure that you plot the two trajectories in different colors indicating in the text which color corresponds to which company.
Here is a link to a straightforward tutorial which will help you accomplish the above neatly:
(10 points) What do you notice when you compare the timeplots? Which of the formulae for calculating futures prices you encountered on the Wikipedia entry come in handy?
The file “apple-calls.csv” contains call prices observed on February 14th for European call options on the stocks of Apple Inc with expiry on May 19th and with varying strike prices.
(5 points) Import the data into a data frame. Create a line plot of the mid price (defined as the midpoint between the bid and the ask price) as it depends on the strike.
(5 points) What do you notice about the monotonicity and convexity of the above curve?
(15 points) Prove that the same conclusions regarding the monotonicity and convexity of the call prices with respect to the strike must hold in general under the no arbitrage assumption.
(5 points) Using put-call parity, find the formula for the interest rate if all other ingredients are known.
(5 points) Based on the above, what can you say about the interest rates \(r\) you expect to obtain for varying values of the strike price? Bear in mind the standing assumptions we made in the beginning of the semester.
(10 points) Based on the data set “apple-parity.csv”, calculate the continuously compounded, risk-free interest rate for all the strike prices given. Be careful about how you use the bid and ask/offer prices of the options. Set \(T=0.25\). Plot the values of the interest rate you obtain as they depend on the strike prices.
(5 points) What do you notice about the above plot? Does it or does it not agree with your prediction from the second question in the problem? Substantiate your answer.
To what do you attribute the discrepancies you observed in the previous problem? Think about it a bit and note your ideas. Then, consult the paper by Brenner and Galai uploaded into Canvas (under Files in the folder articles). Provide at least four substantiated reasons.
Note that this question is worth a quarter of your score for this project! A few lines of your musings will give you hardly any credit at all.