Solutions to Homework 1

FM 5021 Mathematical Theory Applied to Finance

1.1 What is the difference between a long forward position and a short for-ward position?When entering into a long forward contract, a trader agrees to buy the underlying assetfor a certain price at a certain time in the future. When entering into a short forwardcontract, the trader agrees to sell the underlying asset for a certain price at a certain timein the future.

1.4 Explain carefully the difference between selling a call option and buyinga put option.Selling a call option involves giving someone else the right to buy an asset from you for aspecified strike price K on (assuming the option is European) a specified maturity date T .The payoff of selling a call option is

−max(ST −K, 0) = min(K − ST , 0).

Buying a put option gives you the right to sell an asset to someone else for a specifiedstrike price K on (again, assuming the option is European) a specified maturity date T .The payoff of buying a put option is

max(K − ST , 0)

When you write a call option, the payoff is negative or zero, because the counterpartychooses whether to exercise. The maximum profit from writing a call option is then limitedto the price of the call option. When you buy a put option, the payoff is zero or positive,because you choose whether to exercise your right or not. The maximum loss of buying aput option is thus restricted to the price paid for the put option.

1.5 An investor enters into a short forward contract to sell 100, 000 Britishpounds for US dollars at an exchange rate of 1.5000 US dollars per pound. Howmuch does the investor gain or lose if the exchange rate at the end of thecontract is (a) 1.4900 and (b) 1.5200?(a) The investor is obliged to sell pounds for $1.5000 when their market price is only$1.4900. She, therefore, makes a gain of

($1.5000− $1.4900) · 100, 000 = $1, 000.

(b) In this situation, the investor is obliged to sell pounds for $1.5000 when their marketprice is $1.5200. She, therefore, loses

($1.5200− $1.5000) · 100, 000 = $2, 000.

1

1.7 Suppose that you write a put contract with a strike price of $40 and anexpiration date in 3 months. The current stock price is $41 and the contractis on 100 shares. What have you committed yourself to? How much could yougain or lose?You have agreed to buy 100 shares for $40 per share if the party on the other side of thecontract chooses to exercise the right to sell you the shares for this price. The option willbe exercised only when the market price of the stock is below $40, otherwise the otherparty would sell the shares in the market.Suppose the option is exercised when the price is $30, so that you have to buy shares for$40 each when they are only worth $30 each. You lose $10 per share, or $1, 000 in total. Ifthe option is exercised when the market price of the stock is $20, you lose $20 per share,or $2, 000 in total. The more the stock price declines during the 3-month period, the moreyou lose, with the worst case being when the price declines to almost zero during the 3-month period when you can lose up to $4, 000. You profit only when the other party doesnot exercise the option, so that your profit is limited to the price you received for the option.

1.9 You would like to speculate on a rise in the price of a certain stock. Thecurrent stock price is $29, and a 3-month call with a strike price of $30 costs$2.90. You have $5, 800 to invest. Identify two alternative investment strategies,one in the stock and the other in an option on the stock. What are the potentialgains and losses from each?One strategy is to buy 200 shares. Another is to buy 2, 000 call options. Investing inoptions magnifies the potential gains and losses. If the share price does well, investing inoptions will give rise to greater gains. For example, if the share price increases to $40, thesecond strategy ends up in a gain of (2, 000 ·($40−$30))−$5, 800 = $14, 200, while the firststrategy brings a gain of 200 · ($40− $29) = $2, 200. If the share price does badly, however,investing in options gives also a greater loss. For example, if the share price goes down to$25, investing in shares leads to a loss of 200 · ($29 − $25) = $800, whereas investing inoptions leads to a loss of the whole investment of $5, 800, since then the option would notbe exercised.

1.16 A trader writes a December put option with a strike price of $30. Theprice of the option is $4. Under what circumstances does the trader make again?The trader makes a profit if the price of the underlying asset at expiry is above $26 (igoringthe time value of money). Suppose the underlying asset’s price at expiry is $24. Then theparty on the other side of the contract will exercise their right to sell the asset at $30. Thetrader loses $6 from the transaction, but because he received $4 for the option, his totalloss is $2. On the other hand, suppose the asset’s price at expiry is $28. The other partystill exercises the right to sell the asset at $30. The trader loses $2 from the transaction,but because he received $4 dollars for the option, his total gain is $2. The writer of theput option breaks even when the price of the underlying asset is $26 at expiry.

2

2.1 Distinguish between the terms open interest and trading volume.The open interest of a futures contract at a particular time is the total number of longpositions outstanding (or, equivalently, the total number of short positions outstanding).The trading volume during a certain period of time is the number of contracts traded duringthat period.

2.4 Suppose that in September 2006 you take a long position in a contract onMay 2007 crude oil futures. You close out your position in March 2007. Thefutures price (per barrel) is $18.30 when you enter into your contract, $20.50when you close out your position, and $19.10 at the end of December 2006. Onecontract is for the delivery of 1, 000 barrels. What is your total profit? Whenis it realized? How is it taxed if you are (a) a hedger and (b) a speculator?Assume that you have a December 31 year-end.The total profit is 1, 000·($20.50−$18.30) = $2, 200. Of this 1, 000·($19.10−$18.30) = $800is realized on a day-to-day basis between September 2006 and December 31, 2006. Therest $1, 400 = 1, 000 · ($20.50 − $19.10) is realized on a day-to-day basis between January1, 2007, and March 2007.A hedger would be taxed on the whole profit in 2007, while a speculator would be taxedon $800 in 2006 and $1, 400 in 2007.

2.11 An investor enters into two long July futures contracts on orange juice.Each contract is for the delivery of 15, 000 pounds. The current futures price is160 cents per pound, the initial margin is $6, 000 per contract, and the mainte-nance margin is $4, 500 per contract. What price change would lead to a margincall? Under what circumstances could $2, 000 be withdrawn from the marginaccount?There would be a margin call if $1, 500 is lost on one contract. This happens if the futuresprice on orange juice falls by 10 cents to 150 cents per pound.$2, 000 can be withdrawn from the margin account if there is a gain on each contract of$1, 000. This will happen if the futures price rises by $1,000

15,000= $0.0667 (6.67 cents) to 166.7

cents per pound.

2.28 What position is equivalent to a long forward contract to buy an assetat K on a certain date and a put option to sell it for K on that date?The payoff of a long forward contract to buy an asset at K on a certain date is ST − K(could be negative), where ST is the asset’s price at expiry.The payoff of a put option to sell the asset for K on that same date is

max (K − ST , 0).

If ST > K, the total payoff of the long forward contract and the put option is ST −K. IfST ≤ K, the total payoff is 0. Therefore, we can write the total payoff of the long forwardand the put option as

max (ST −K, 0).

3

Note that this is exactly the payoff of a European call option that gives an investor theright to purchase the asset on the specified date at the strike price K. Since the Europeancall option will be exercised only when ST > K for a payoff of ST −K, and expire worthlessif ST ≤ K, the payoff of owning a European call option is

max (ST −K, 0),

exactly the same as a portfolio of one put option and one long forward contract.

4