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ASSIGNMENT 1

Math 174E – Winter 2024

Release date: Wednesday, January 10, 2024

Exercises for Discussion Sections in Week 1

Exercise 1.1 (Forward contracts)

A relatively simple financial derivative is a forward contract. It is an agreement (binding commitment) to buy or sell an asset on a certain specified future date T > 0 for a certain specified price F.

The predetermined price F is called forward price and the future date T is called maturity date or delivery date.

One of the parties of a forward contract assumes a long position and agrees to buy the underlying asset. The other party assumes a short position and agrees to sell the asset.

By convention, it does not cost anything to enter into a forward contract (more precisely, the forward price is determined in such a way that the value of the forward contract is zero when it is initiated; see Chapter 5).

Forward contracts on foreign exchange are very popular and can be used to hedge foreign currency risk. Suppose that, on May 21, 2020, the treasurer of a U.S. corporation knows that the corporation will pay £1 million in 6 months (i.e., on November 21, 2020). In order to hedge against exchange rate moves the treasurer of the corporation decides to enter into a forward contract with an investment bank. The forward price is F “ $1.2230 per British pound and the maturity date is November 21, 2020.

(a) What position in the forward contract should the treasurer take? What is the correspond-ing position of the investment bank in the forward contract? What are their commitments?

(b) What is the market value (also called payoff ) of the forward contract for the corporation on November 21, 2020, if the GBP/USD exchange rate on that day turns out to be (i) $1.3000 per British pound or (ii) $1.2000 per British pound? And what is the market value of the forward contract for the investment bank in these two cases?

Exercise 1.2 (Stock options)

Stock options are typical examples of financial derivatives (i.e., a financial derivative written on a stock). Following are two basic types of stock options.

1. A European call option on a stock gives the holder (buyer of the call option) the right, but not the obligation, to buy the stock for a certain predetermined price K > 0 on a certain specified future date T > 0 from the seller (or writer) of the call option.

2. A European put option on a stock gives the holder (buyer of the put option) the right, but not the obligation, to sell the stock for a certain predetermined price K > 0 on a certain specified future date T > 0 to the seller (or writer) of the put option.

The predetermined price K in the option contract is called strike price or exercise price. The date T in the contract is called maturity or expiration date.

An investor who buys a stock option is said to have a long position in the stock option. An investor who sells (or writes) a stock option is said to have a short position in the stock option.

(a) What is the crucial difference between a forward contract written on a stock and an option written on a stock?

(b) Would the holder of a European call option (put option) always exercise her right to buy the stock (sell the stock) at maturity T for the predetermined strike price K?

(c) Discuss the purpose of a European call and a European put option on a stock. Describe a situation where it would make sense for an investor to buy a call option or to buy a put option.

(d) Describe the market value (also called payoff ) at maturity T of following option positions as a function of the price of the underlying stock ST at time T:

(i) long position in a call,

(ii) short position in a call,

(iii) long position in a put,

(iv) short position in a put.

(e) The buyer of an option (long position) has to pay a premium (the price of the option) to the seller/writer of the option (short position) at time t “ 0 when both parties agree on the deal. Denote by C0pK, Tq the price of a European call option at time t “ 0 with strike price K and maturity T, and by P0pK, Tq the price of a European put option at time t “ 0 with strike price K and maturity T.

Describe the profit and loss at maturity T of all option positions (i)-(iv) in (d) as a function of the price of the underlying stock ST at time T.

Exercise 1.3 (Hull, Question 1.37)

Trader A enters into a forward contract to buy an asset for $1,000 in one year. Trader B buys a call option to buy the asset for $1,000 in one year. The cost of the option is $100. What is the difference between the positions of the traders? Show the profit as a function of the price of the asset in one year for the two traders.

Exercise 1.4 (Hull, Question 1.44)

Suppose today a corporate treasurer from a U.S. company is saying the following: “I will have £1 million to sell in 6 months. If the exchange rate is less than 1.19, I want you to give me 1.19. If it is greater than 1.25, I will accept 1.25. If the exchange rate is between 1.19 and 1.25, I will sell the sterling for the exchange rate”. How could you use options written on the GBP/USD exchange rate to satisfy the treasurer?