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Posted: November 29th, 2022
Finance 622 – Derivatives
Homework #1
1. Meh Platforms (Ticker = FB) is trading at $111.44 per share. Suppose the risk-free interest rate is 4.13% at all maturities (annualized, continuously compounded). Meh is not expected to pay dividends during the next year. You are considering a forward contract maturing on January 22, two months from today.
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a. What is the forward price for delivery of FB shares that will preclude arbitrage on the forward contract above?
b. Suppose the forward price for 2 month delivery of FB stock is, instead of the amount you calculated in a, equal to $120.00 per share. Is there an arbitrage profit available, what portfolio would you construct to take advantage of this and how much would your profit be? Assume no transactions costs.
c. Next suppose the forward price for 2 month delivery of FB stock is $105.00 per share. Is there an arbitrage profit available, what portfolio would you construct to take advantage of this and how much would your profit be? Assume no transactions costs.
d. Suppose that, at an earlier date, you had entered into a short forward contract to deliver FB shares on January 19 at a delivery price of $116. Based on the forward price you calculated in part a, what is the value now of the forward contract you entered into earlier?
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e. (This part of the problem is for when you have studied the option strategies chapter.) There is a two month call option on FB with strike price of 115.00 trading at 6.54; a two month put option with strike price of 115.00 is trading at 9.35. Form a synthetic long forward contract with a delivery price of 115.00 using the two options. How much does this synthetic long forward contract with delivery price of $115.00 cost you today?
f. You could eliminate the risk of the earlier position you took out (short forward at delivery price of $115.00 from part d for delivery two months hence) by either going long a forward contract at the current forward price (your solution in part d) or by going long the synthetic forward contract you constructed in part e. Which of these methods would be better for you (lower cost)? You don’t have to construct all the positions again, just explain.
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2. European Options on XYZ stock (stock currently trading at $42.50 per share) with delivery in three months have the following prices:
Options Option Prices
Call, K = 35 8.50
Call, K = 40 4.50
Call K = 45 2.00
Put, K = 35 0.75
Put, K = 40 1.75
Put, K = 45 4.25
a. Construct profit diagrams for a holding a long call (K = 45) to maturity, purchased at the above price.
b. Construct profit diagrams for shorting a put (K=40) to maturity.
c. Construct a profit diagram for writing a covered call (K=45) against a long position in the common held to the maturity date of the call.
d. Construct the profit diagram for any bull spread using puts.
e. Construct the profit diagram for any bear spread.
f. Construct the profit diagram for a long butterfly spread using calls.
3. EXTRA CREDIT. Put together a spreadsheet that will allow you to calculate payoff and profit for any number of positions in the underlying stock and options (and diagrams). The options should all expire on the same date and be on the same underlying asset. I’m not looking for an unbelievably flexible tool, just a simple spreadsheet.
For example, you could have a column with the range of stock prices that are relevant to the question, with the spread between adjacent stock prices being set to a useful amount for the stock in question. For the above problem you could use stock prices from 0 to 80 at five dollar increments. Then have a separate table that gives the current stock price and the option prices for all the relevant options in the problem. In that table, like the above, enter the option prices. Next to the column of stock prices enter a column of profits on long stock position assuming a current stock price. Next to that column add a column with the profits to a long position for each call and each put (the calculation in the column should subtract the upfront cost of the option from the payoff on the option in question). This would be six columns given the above information.
Then construct any combinations you like. For instance suppose you want a long butterfly spread of puts: you would calculate 1 times the profit on the 35 put plus 1 times the profit on the 45 put minus 2 times the profit on the 40 put. You can figure out what any profit diagrams will look like at your leisure. (Just send me the spreadsheet to my email, tthompson17@luc.edu.)
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