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University of Connecticut
Math 3615: Financial Mathematics Problems
Fall 2008
Examples – Module 10 10/2/ In these examples, ignore the effect of brokerage commissions and bid-ask spreads. Assume all options are European style (i.e., can be exercised only on the expiration date). Also assume that no dividends are paid.
- Suppose that Stock A is currently trading at 60 and the continuously compounded risk- free interest rate is 0.05. (a) Write a formula for the profit from buying Stock A today and selling it in 6 months (i.e., express profit as a function of S0.5, the stock’s price 6 months from now).
(b) If a 6-month call option on Stock A with a strike price of 60 has a premium of 6, write a formula for the profit from writing a call option on one share of Stock A.
(c) Now write an expression for the combined profit for an investor who executes both of the transactions described in (a) and (b).
(d) Is there a single security that could be purchased or sold to produce the same profit (for any possible price of Stock A at t = 6 months) as the combination of (a) and (b)? If so, what is the name of that security?
What would its price have to be in order to produce the same profit as the combination of (a) and (b)?
- You are considering purchasing a share of Stock B, which is currently trading at 100. (a) If the continuously compounded risk-free interest rate is 0.06, what is the formula for the profit you will realize if you purchase a share of Stock B now and sell it in one year?
(b) On the following page are two sets of coordinate axes, one set for a payoff diagram and one set for a profit diagram. Sketch the payoff and profit diagrams for the transactions described above (purchase Stock B at 100 and sell it one year later).
(c) Now suppose that you are unable to purchase Stock B because you do not have 100 in cash, but you would still like to have a “long” position in the stock. You are able to find another investor who is willing to enter into a forward contract with you to deliver one share of Stock B one year from today for a price that the two of you determine today. What is a fair forward price for this contract?
(d) On the same axes you used for buying and selling a share of stock, sketch the payoff and profit diagrams for this forward contract. Label the lines (“stock” and “forward”) and label any points where each line crosses an axis or changes slope.
(e) On reflection, you realize that you could create the same result as the forward contract by using a put option and a call optionswith a 100 strike price, plus a position in a Treasury bond. Given that the put option you will use sells for a premium of 10, describe these securities below, sketch and label the payoff and profit diagrams for each of them on the same axes as before, and also sketch the total payoff and profit diagrams for the combination of these three securities.
Call Option Put Option Treasury Bond
Strike Price Maturity Value
Term Term
Premium Price Purchased or Written?
Purchased or Sold Short?
- Investor 1 has a position in Stock C consisting of the following options:
- a purchased one-year call option on Stock C with a strike price of 60
- a written one-year put option on Stock C with a strike price of 50
(a) Write expressions for the payoff from each of these options as a function of S 1 , the price of Stock C one year from now. Then write an expression for the combined profit from both of these options.
(b) Investor 2 holds a long position in a one-year forward contract with a delivery price of 50. How can Investor 2 add other derivatives so that the entire holding is equivalent to Investor 1’s position. (Assume that one-year calls and puts are available with strike prices of 50 and 60.)
(c) Investor 3 holds a long position in a one-year forward contract with a delivery price of 60. How can Investor 3 add other derivatives so that the entire holding is equivalent to Investor 1’s position. (Assume that one-year calls and puts are available with strike prices of 50 and 60.)
(d) The following table shows prices of certain options mentioned above. Given that Stock C’s current price is 54, and the continuously compounded risk-free interest rate is 0.05, fill in the values of the remaining derivatives in the table.
Security Strike Price = 50 Strike Price = 60 Call 10.
Put 8.
Forward
