Fin 674 Problems Ch01

8/8/2019 Fin 674 Problems Ch01

http://slidepdf.com/reader/full/fin-674-problems-ch01 1/9

Chapter 1

Introduction

SOLUTIONS TO QUESTIONS AND PROBLEMS

Problem 1 .1.

When a trader en ters into a long fo rward con tract, she is ag reeing to buy the underlying asset

for a certain price at a certain time in the future. When a trader enters into a short forward

contrac t, she is agree ing to sell the underlying asset fo r a certain p rice at a certain time in the

future.

,P roblem 1.2 .

A trader is hedging when she has an exposure to the price of an asset and takes a position in a

derivative to offset the exposure. In a speculation the trader has no exposure to offset. She is

betting on the future movemen ts in the price o f the asset. Arbitrage involves taking a position

in two or more differen t markets to lock in a pro fit.

Problem 1.3.

In the first case the trader is obligated to buy the asset for $50. (The trader does not have a

choice.) In the second case the trader has an option to buy the asset for $50. (The trader do~s

not have to exercise the option.)

Prob lem 1.4.

Selling a call option invo lves giv ing someone else the right to buy an asset from you . It g ives

you a payoff of

- max(ST - K, 0) = min(K - ST , 0)

Buying a put op tion involves buying an option from someone else. It gives a payoff of

max(K - ST , 0)

In both cases the potential payoff is K - ST. When you write a call option, the payoff is

negative o r zero . (This is because the coun terparty chooses whether to exercise.) When you

buy a put option, the payoff is zero or positive. (This is because you choose whether to

exercise. )



2Chapter 1 . In troduction

Problem 1.5.

(a) The investor is obligated to sell pounds for 1.5000 when they are worth 1.4900. The gain

is (1.5000 - 1.4900) x 100,000 = $1,000.

(b) The investor is obligated to sell pounds for 1.5000 when they are worth 1.5200. The loss

is (1.5200-1.5000) x 100,000= $2 ,000.

Problem 1.6.

(a) The trader sells for 50 cents per pound something that is worth 48.20 cents per pound.

Gain = ($0.5000 - $0.4820) x 50,000 = $900.

(b) The trader sells for 50 cents per pound something that is worth 51.30 cents per pound.

Loss = ($0.5130 - $0.5000) x 50,000 = $650.

Problem 1.7.

You have sold a put option. You have agreed to buy 100 shares for $40 per share if the party

on the other side o f the contract chooses to exercise the right to sell fo r th is p rice. The option

w ill be exercised only when the price of stock is below $40. Suppose, for example, that the

option is exercised when the price is $30. You have to buy at $40 shares that are worth $30;

you lose $10 per share, or $1,000 in total. If the op tion is exercised when the price is $20 , you

lose $20 per share, or $2,000 in total. The worst that can happen is that the price of the stock

dec lines to almost zero during the three -month period. This h ighly unlikely event would cost

you $4,000. In return for the possible future losses, you receive the price of the option from

the purchaser.

Problem 1.8.

The over-the-counte r market is a te lephone- and compute r-linked network of financia l insti-

tu tions, fund managers, and co rpo rate treasurers where two participants can en ter in to any

mutually acceptable contract. An exchange-traded market is a market organized by an ex-

change where traders e ither mee t physically or communica te electronically and the contracts

that can be traded have been defined by the exchange. When a market maker quotes a bid and

an offer, the bid is the price at which the market maker is prepared to buy and the offer is the

price at which the market maker is prepared to sell.

Problem 1.9.

One strategy wou ld be to buy 200 shares. Ano ther wou ld be to buy 2,000 op tions. If the share

price does well the second strategy w ill give rise to greater gains. For example, if the share

price goes up to $40 you gain [2,000 x ($40 - $30)] - $5,800 = $14,200 from the seconds tra tegy and only 200 x ($40 - $29) = $2,200 from the first strategy. However, if the share

price does badly , the second stra tegy gives greater losses. For example , if the share price goes

down to $25 , the first strategy leads to a loss o f 200 x ($29 - $25) = $800, whereas the second



3

strategy leads to a loss of the whole $5,800 investment. This example shows that options

contain built in leverage.

Problem 1.10.

You could buy 5,000 put options (or 50 contracts) with a strike price of $25 and an expirationdate in 4 months. This provides a type of insurance. If at the end of 4 months the stock price

proves to be less than $25 you can exercise the options and sell the shares for $25 each. The

cost of this strategy is the price you pay for the put options.

Problem 1.11.

A stock option provides no funds for the company. It is a security sold by one trader to another.

The company is not involved. By contrast, a stock when it is first issued is a claim sold by the

company to investors and does provide funds for the company.

Problem 1.12.

If a trader has an exposure to the price of an asset, she can hedge w ith a forward contract.

If the exposure is such that the trader will gain when the price decreases and lose when the

price increases, a long forward position w ill hedge the risk. If the exposure is such that the

trader will lose when the price decreases and gain when the price increases, a sho rt forward

position will hedge the risk. Thus either a long o r a sho rt forward position can be en tered in to

fo r hedging pu rposes. If the trader has no other exposu re to the price of the underlying asset,

enter ing into a forward contract is speculation.

Prob lem 1.13.

Ignoring the time value of money , the holder o f the op tion will make a profit if the stock p rice

in March is greater than $52.50. This is because the payoff to the holder of the option is, in

these circumstances, greater than the $2 .50 paid for the option . The op tion will be exercised

if the stock price at maturity is greater than $50.00. Note that if the stock price is between

$50.00 and $52.50 the option is exercised, but the holder of the option takes a loss overall.

The profit from a long position is as shown in Figu re S1 .1.

Problem 1 .14 .

Igno ring the time value o f money , the seller of the option will make a pro fit if the stock price

in June is greater than $56.00. This is because the cost to the seller of the option is in thesecircumstances less than the price received fo r the option. The op tion will be exercised if the

stock price at maturity is less than $60.00. Note that if the stock price is between $56.00 and

$60.00 the seller of the option makes a p rofit even though the op tion is exercised. The p rofit

from the short position is as shown in Figure S 1.2 .



4

C ha pter 1. Introd uctio n

Figure S1.1 : P rofit from long position in P roble m 1.13

8

6

-4

Problem 1.15.

The trader receives an inflow of $2 in May. Since the option is exercised, the trader also has

an outflow of $5 in September. The $2 is the cash received from the sale of the option. The

$5 is the result of buying the stock for $25 in September and selling it to the purchaser of the

option for $20.

P ro blem 1.16.

The trader m akes a gain if the price of the stock is above $26 in D ecem ber. (This ignores the

time value of money .) .

P ro ble m 1 .1 7.

A long position in a fou r-mon th put option can provide in surance again st the exchange rate

falling below the strike price. It ensures that the foreign currency can be sold for at least the

str ike price .



4;..::96.:

2

Stock Price

560 65 70

-2

5

Figure S1.2: Profit from short position in Problem 1.14

6

-4

-6

-8

Problem 1 .18.

The company could enter into a long forward contract to buy I m illion Canadian dollars in

six months. This would have the effect of locking in an exchange rate equal to the current

forward exchange rate. A lternatively the company cou ld buy a calI op tion giving it the righ t

(but not the ob ligation) to purchase I m illion Canad ian dollar at a certain exchange rate in six

months. This would p rovide insu rance against a strong Canad ian do llar in six mon ths while

still allowing the company to benefit from a weak Canadian dolIa r a t that time.

Problem 1 .19 .

(a) The trader sells 100 m illion yen for $0.0080 per yen when the exchange rate is $0.0074

per yen. The gain is 100 x 0.0006 millions of dollars or $60,000.

(b) The trader sells 100 m illion yen for $0.0080 per yen when the exchange rate is $0.0091

per yen . The loss is 100 x 0.00 11millions of dollars or $110,000 .



6Chapter 1. Introduction

Prob lem 1 .20.

Most traders who use the contract will wish to do one of the following:

(a) Hedge their exposure to long-term interest rates

(b) Speculate on the future direction of long-term interest rates

(c) Arbitrage betw een cash and futures m arkets

This contract is discussed in Chapter 6.

Problem 1.21.

The statement m eans that the gain (loss) to the party with a short position in an option is always

equal to the loss (gain) to the party with the long position. The sum of the gains is zero.

Problem 1.22 .

The te rminal value of the long forward contract is:

ST - Fo

where ST is th e price o f the asset at maturity and Fa i s the forward p rice of the asset at th e time

the portfo lio is set up . (The deliv ery price in the forward con tract is Fa.)

The terminal value of the put op tion is:

max (Fa - ST, 0)

The te rminal value of the portfolio is therefore

ST - Fa + max(Fo -ST, 0)

= max (0, ST - Fa]

This is the same as the term inal value of a European call option w ith the same maturity as the

forward contrac t and an exerc ise price equal to Fa. This resu lt is illustra ted in the Figure S 1 .3 .

The profit equa ls the te rminal value le ss the amount paid fo r the option.

P roblem 1.23.

Suppose that th e yen exchange rate (yen per dollar) at maturity of the ICON is ST . The payoff

from the ICON is1

1,000 if ST > 169

1 000 - 1 OOO( 169- 1),

STif 84.5 -s. ST - s. 169

0 if ST < 84.5

1There was a typo in the first printing of the book. The exp re ss io ng iv en fo r th e payon should be 1,000-

max[O, I,OOO( ~~9-1)1.



Terminal Terminal Terminal Terminalalue of Value ofValue of Value ofegular Bond Short Calls Long Calls Whole Positionr> 169 1,000 0 0 10004.5 :S S r :S 169 1000 -169,000

0 2000 - ~SrT < 84.5 1000 -169,0000

7

Figure S1.3: Payoff from portfolio in Problem 1.22

'-

Asse t Pric e

::>0::

Forward

- Put- Total

.

When 84.5 :::; Sr :::; 169 the payoff can be written

169,0002,000 - Sr

The payoff from an ICON is the payoff from:

(a) A regular bond

(b) A short position in caJI options to buy 169,000 yen with an exercise price of 1/169

(c) A long position in call options to buy 169,000 yen with an exercise price of 1/84.5

This is demonstrated by the following table:



8

Chapter 1. Introduction

Problem 1.24.

Suppose that the forward price for the contract entered into on July I, 2005 is F] and that the

forward price for the contract entered into on September 1, 2005 is F2 with both FJ and F2

being measured as dollars per yen. If the value of one Japanese yen (measured in U.S. dollars)

is ST on January 1, 2006, then the value of the first contract (in millions of dollars) at that

time is

lO(ST - F1)

while the value of the second contract (per yen sold) at that time is:

1O(F2 - ST)

The total payoff from the two contracts is therefore

lO(ST - F)) + 10(F2 - ST) = 1O(F2- FJ)

Thus if the forward price for delivery on January 1, 2006 increases between July 1, 2005 and

September 1,2005 the company will make a profit.

Problem 1.25.

(a) The trader buys a 180-day call option and takes a short position in a 180-day forward

contrac t. I f ST is the term inal spot rate, the profit from the call option is

max(ST -1.57, 0) -0.02

The profit from the short forward contract is

1.6018-ST

The profit from the strategy is therefo re

"

max (ST - 1.57,0) - 0.02 + 1.6018 - ST

or

max (ST - 1.57,0) + 1.5818 - ST

This is

1.5818 - ST

0.0118

when ST < 1.57

when ST > 1.57This shows that the profit is always positive. The time value of money has been ignored

in these calcula tions. However, when it is taken into account the strategy is still likely to

be profitable in all circum stances. (W e would require an extremely high interest rate for

$0.0118 interest to be required on an outlay of $0.02 over a 180-day period .)



9

(b ) The trader buys 90 -day put options and takes a long position in a 90 day forward con tract.

If Sr is the term inal spot rate, the profit from the put option is

max (1 .64 - ST, 0) - 0.020

The pro fit from the long forward contract is

S1' - 1.6056

The profit from this strategy is there fore

max(1.64-ST, 0) -0.020+S1' -1.6056

or

m ax (1.64 - S1', 0) + Sr - 1.6256

This is

S1 ' - 1.6256 whenS1' > 1.64

0.0144 when ST < 1.64

The profit is therefore always positive. Again, the time value of money has been igno red

but is unlikely to affect the overall profi tabi li ty of the s tra tegy. (We would require interest

rates to be extrem ely high for $0.0144 interest to be required on an outlay of $0.02 over

a 90-day period.)

Documents

Fin 674 Problems Ch01