Pricing Call and Put Options: In the text we mentioned contracts called “call” and “put” options as examples of somewhat more sophisticated ways in which one can take a short or long position in the market.
A. Suppose, as in exercise 20.7, that the current price of oil is $50 a barrel. There are two types of contracts one can buy: The owner of contract 1 has the right to sell 200 barrels of oil at the current price of $50 a barrel 1 year from now. The owner of contract 2 has the right to buy 200 barrels of oil at the current price of $50 a barrel. Assume in this exercise that the annual interest rate is 5%.
(a) Suppose, as in exercise 20.7, that Larry thinks the price of oil will rise while Darryl thinks it will fall. Consider Larry first and suppose he feels quite certain that oil will sell for $75 a barrel 1 year from now. What’s the most he is willing to pay to buy contract 2? What is the most he is willing to pay to buy contract 1?
(b) Next consider Darryl who is quite certain that oil will be trading at $25 a barrel 1 year from now. What is the most that he is willing to pay for the two contracts?
(c) Which contract allows you to take a short position and which allows you to take a long position in the oil futures market?
(d) Suppose that contract 1 currently sells for $6,000. What does that tell us about the market’s collective prediction about the price of a barrel of oil 1 year from now?
(e) Suppose instead that contract 2 currently sells for $6,000. What does that tell us about the market’s prediction of oil prices 1 year from now?
B. In part A, we considered only a single call or put option at a time. In reality, a much larger variety of such futures contracts can exists at any given time.
(a) Suppose that a call option gives the owner the right to buy 200 barrels of oil at $50 one year from now. You observe that this futures contract is selling for $3,000 in the market. What is the market’s prediction about the price of oil one year from now? (Assume again an interest rate of 0.05.)
(b) Suppose someone else has just posted another call option contract for sale—this one entitles the owner to buy 200 barrels of oil at a price of $43 one year from now. How much do you predict this contract will sell for?
(c) Then a put option is posted for sale —this one allows the owner to sell 200 barrels of oil at a price of $71 one year from now. What do you think this option will be priced at by the market?
(d) Let PC (p̅,p,q,r,n) be the price of a call option to buy q barrels of oil n years from now at price p̅ when the market interest rate is r and the market expectation of the actual price of oil
n years from now is p. What is the equation that defines PC ?
(e) Let PP (p̅,p,q,r,n) be the price of a put option to sell q barrels of oil n years from now at price p when the market interest rate is r and the market expectation of the actual price of oil n years from now is p. What is the equation that defines PP ?
(f) The price p at which oil actually sells at any given time is called the spot price. Illustrate what you have just found in a graph with the spot price p on the horizontal axis and dollars on the vertical. First, graph the relationship of PC to p (holding fixed p̅, q, n and r ). Label intercepts and slopes. Then graph the same for PP . Where must these intersect? Explain.
(g) Illustrate the same thing in a second graph, except this time put the call or put price p on the horizontal axis. Where do the PC and PP lines now intersect? Explain.