INTRODUCTION TO THE MATHEMATICS OF FINANCE: ADDITIONAL EXERCISES

Problems indicated with a * are more challenging.

Chapter 1.
A1. A forward contract is an agreement to buy or sell an asset at a certain future time for a certain price (called the forward price). At the time a forward contract is entered into, no money changes hands. The investor who agrees to buy the asset at the future time is said to hold the long position in the contract and the investor who agrees to sell the asset is said to hold a short position in the contract. Suppose that a stock is currently selling for \$20 per share. A (long position in a) forward contract is available to buy 100 shares of the stock 3 months from now for \$20.20 per share. Suppose that a bank is offering interest at the rate of 5% per annum (continuously compounded) on a 3-month deposit. Describe a strategy for creating an arbitrage profit and establish the amount of the profit.
A2. A combination option called a strangle is obtained by taking a long position in a (European) call and a (European) put option with the same expiration date but differing strike prices, all based on the same underlying asset. An investor who buys the strangle is betting that there will be a large movement in the price of the underlying, but is uncertain whether it will involve an increase or a decrease in the price. Find a formula for the payoff for a strangle where the put has a strike price of K1 and the call has a strike price of K2 and K1 < K2. Draw a graph of this payoff as a function of the final price of the underlying asset. (Make sure to label your axes on the graph.)

Chapter 2.
A1. Consider the CRR model described in Exercise 2 of Chapter 2 as the model for a stock and a bond. A forward contract is to be offered under which the holder of a long position in the forward contract will buy 100 shares of stock at time T=2 for a fixed price F. (Here F is the total amount to be paid for the 100 shares). Remember that no money changes hands at time zero when a forward contract is written. What value should F take in order that there is no arbitrage opportunity for the investor who holds a long or a short position in the contract? Explain your reasoning fully (in particular, identify an associated European contingent claim and derive the value of F using arbitrage pricing for the contingent claim).

Chapter 3.
A1*. Consider a finite market model that is viable and complete. Formulate a notion of an American contingent claim in this setting. Is there a (minimal) superhedging strategy for any American contingent claim? (Hint: you may need to use the martingale representation property.) Is there a unique arbitrage free initial price for any American contingent claim? Prove any claims that you make.