ACCA P4 Advanced Financial Management - 2010 - Study text

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Chapter 16: Option pricing and delta hedging

Using the formulae: Step 1

. Calculate the value of d

and d

ln (0.6978/0.7000) = ln 0.99686.

Using a calculator, the value of ln 0.99686 = 0.0031478.

= 0.0031478+ (0.1247)² 0.25/2

0.1247√0.25

= 0.0050916/0.06235

= 0.08166

= 0.08166 – 0.1247√0.25

= 0.0193

Using the formulae: Step 2. Calculate the value of N(d

)

and N(d

)

From normal distribution tables, we get the following values:

 When z = 0.08166, probability = 0.0326. Therefore N(d

) = 0.0326 + 0.5000 =

0.5326.

 When z = 0.0193, probability = 0.0078. Therefore N(d

) = 0.0078 + 0.5000 = 0.5078.

(Note: d

and d

both have a positive value so add 0.5000.)

Using the formulae: Step 3. Calculate the value of an at-the-money call option

c = e

-rt

N(d

) – X N(d

)]

-rt

= 1/e

(0.055 × 0.25)

= 1/e

0.01375

= 0.9863.

c = 0.9863 × [(0.6978 × 0.5326) – (0.7000 × 0.5078)]

c = 0.9863 × 0.01619 = 0.016

The value of an at-the-money call option is £0.016 or 1.6 pence per euro.

The cost of an at-the-money call option on 100,000 euros is therefore £1,600.

Using the formulae: Step 4. Calculate the value of an at-the-money put option

p = e

-rt

[X N(- d

) – F

N(- d

)]

= 0.0193, so – d

= – 0.0193 and N(– d

) = 0.5000 – 0.0078 = 0.4922.

= 0.08166, so – d

= – 0.08166 and N(– d

) = 0.5000 – 0.0326 = 0.4674.

p = 0.9863 ×[(0.7000 × 0.4922) – (0.6978 × 0.4676)]

p = 0.9863 × 0.01825 = 0.018

The value of an at-the-money call option is £0.018 or 1.8 pence per euro.

The cost of an at-the-money call option on 100,000 euros is therefore £1,800.

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Delta hedging

 Delta defined

 Constructing a delta hedge for a call options exposure

 Constructing a delta hedge for a put options exposure

 The Greeks

5 Delta hedging

Buying call options or put options is a method of hedging financial risk. Options are

a zero-sum game, and to the extent that an option holder is protected against risk,

the option writer is exposed to risk. For banks and other institutions that write

options, their own exposure to risk from writing options can be hedged with a delta

hedge.

5.1 Delta defined

The delta of an option is defined as the change in the price of an option in

proportion to the change in the value of the underlying item.

Delta =

Change in value of the option

Change in market value of underlying item

For example, in the case of an option on a company’s shares, if the share price goes

up by $1, the option price will go up if it is a call option or down if it is a put option.

However, the change in the price of the option will not exceed $1. The value of an

option delta is therefore in the range – 1 to + 1.

 A call option that is deeply in-the-money has a delta close to + 1

 A put option that is deeply in-the-money has a delta close to – 1.

 An option that is deeply out-of-the-money has a delta close to 0.

 An option that is at-the-money has a delta close to + 0.5 (call option) or – 0.50

(put option).

The value of an option’s delta can be calculated. It is related the value of N(– d

) or

N(d

) in the Black-Scholes option pricing model.

5.2 Constructing a delta hedge for a call options exposure

A bank that writes a large number of options has an option portfolio. It might want

to create a hedge for its exposure to adverse price movements.

 An option portfolio is said to be delta neutral when its delta is 0.

 A bank that writes call options can create an option position that is delta neutral

by purchasing a quantity of the underlying item. For example, a bank that has

written call options on the shares of XYZ Company can hedge the position by

holding some shares in XYZ.

Chapter 16: Option pricing and delta hedging

 If the value of the underlying shares goes up, the value of the call options will

also go up. The bank will incur a loss on its options position, because it has

written options. However, it makes a gain on the rise in the value of the

underlying shares.

 A delta neutral position will exist when the rise in the value of the options (=

benefit to the option holders and loss for the option writer) is matched by an

equal rise in the value of the shares held by the option writer (bank). This will

leave the bank with neither a loss nor a gain.

The number of shares that a call option writer should hold to create a delta hedge =

[Quantity of the underlying item on which there are options] × [Option delta.]

For example if the delta value for call options on 1,000,000 shares of XYZ Company

at an exercise price of $15 is 0.45, a delta hedge will be created by holding 450,000 of

the shares (1,000,000 × 0.45).

Calculating delta for call options

The value of delta for a position in call options is the value of N(d

) in the Black-

Scholes model formulae:

Amount of underlying item to be hedged × N(d

) = Amount of underlying to hold

as a hedge

Example

A UK bank is writing currency call options on €200 million in exchange for British

pounds. The value of delta of the options is 0.5326, measured as N(d

In order for the bank to be certain that it will not make a loss, it could hedge its

options position by holding a quantity of euros.

The quantity of euros that it would need to hold to create a hedge is €200 million ×

0.5326 = €106.52 million.

Example

In principle, the same formula can be used to calculate how many at-the-money call

options it would be necessary to hold to eliminate exchange rate risk entirely.

For example, suppose that a UK company needs to buy €100 million in three months

time and wants to use currency call options to create a hedge that would eliminate

the exchange rate risk entirely. At-the-money call options on euros have a delta

value of 0.5326. Each option is for €100,000.

Re-arranging the previous formula, we can calculate the amount of euros for which

at-the-money call options should be purchased is:

€100 million/ N(d

)

= €100 million/0.5326

= €187.758 million.

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Since each option contract is for €100,000, this would mean buying 1,878 contracts.

Any increase in the value of the euro would mean that the cost of buying the €100

million at the option exercise date would increase, but this ‘loss’ would be offset

entirely by a gain on the call options position.

In practice however, creating a hedge with options that eliminates the risk entirely is

very expensive, because of the cost of the options. In this example, if the company

did decide to use options to create a hedge, the number of options that it would buy

is likely to be €100 million/€100,000 per contract = 1,000 contracts.

5.3 Constructing a delta hedge for a put options exposure

A similar technique can be applied to calculate a hedge position for a writer of put

options. The difference is that if a bank writes put options, it will incur a loss on the

options position if the price of the underlying item falls.

To create a hedge, it therefore needs to be ‘short’ in the underlying item. It could do

this by borrowing the underlying item in the case of shares, or selling the item

forward if it is currency.

For example if the delta value for put options on 1,000,000 shares of XYZ Company

at an exercise price of $15 is – 0.27, a delta hedge will be created by borrowing

270,000 of the shares (1,000,000 × 0.27), and the settlement date for returning the

borrowed shares should be the same as the expiry date for the option..

Calculating delta for put options

The value of delta for a position in put options is the value of N(– d

) in the Black-

Scholes model formulae:

Amount of underlying item to be hedged × N(– d

) = Amount of underlying to hold

as a hedge

Example

A UK bank is writing currency put options on €50 million in exchange for British

pounds. The value of delta of the options is 0.4674, measured as N(– d

In order for the bank to be certain that it will not make a loss, it could hedge its

options position by selling forward a quantity of euros, for settlement on the same

date as the expiry date for the options.

The quantity of euros that it would need to sell forward to create a hedge is €50

million × 0.4674 = €23.37 million.

Chapter 16: Option pricing and delta hedging

Example

In principle, the same formula can be used to calculate how many at-the-money put

options it would be necessary to hold to eliminate exchange rate risk entirely.

For example, suppose that a UK company will receive €20 million in three months

time and wants to use currency put options to create a hedge that would eliminate

the exchange rate risk entirely. At-the-money put options on euros have a delta

value of 0.4764 (N(– d

). Each option is for €100,000.

Re-arranging the previous formula, we can calculate the amount of euros for which

at-the-money call options should be purchased is:

€20 million/ N(– d

)

= €20 million/0.4764

= €41,981,528.

Since each option is for €100,000 this would mean having to buy 420 put option

contracts.

As stated earlier, creating a hedge with options that eliminates the risk entirely is

very expensive, because of the cost of the options. In this example, if the company

did decide to use options to create a hedge, the number of options that it would buy

is likely to be €20 million/€100,000 per contract = 200 contracts.

5.4 The Greeks

You might come across the term ‘the Greeks’. Organisations that deal in options

calculate a variety of option ratios for establishing and monitoring their option

positions. These ratios are called ‘the Greeks’, because they are all designated by a

Greek letter.

Delta

Delta has already been explained. It is the change in the price of an option (in

money value) for a given change in the price of the underlying item.

The value of delta changes as the price of the underlying item changes. For example,

if the market price of a share rises, the delta value of a call option on the shares will

increase.

The delta value of an option may also be seen as a probability estimate that the

option will expire in-the-money. For example, a deeply in-the-money option has a

delta close to + 1 (call option) or – 1 (put option) and the probability estimate that it

will expire in the money is therefore close to 100%.

A deeply out-of the-money option has a delta close to 0, and the probability that it

will expire in-the-money is therefore close to 0%.

For an at-the-money option, the probability that it will expire in the money is about

50%.

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Gamma

The gamma for an option is the ratio of the change in the delta of an option for a

given change in the value of the underlying item. For example, it is the change in

the delta of a share option for an increase or decrease of $0.01 in the share price.

The value of gamma for an option is very low as the certainty increases that the

option will expire in-the-money or out-of-the-money. The value of gamma increases

with uncertainty as to whether the option will expire in-the-money or out of-the-

money.

Vega

Vega measures the ratio of the change in the option price for a 1% change in the

volatility of the underlying item. If volatility in the underlying item increases, vega

increases.

(It might help to remember that

Vega measures sensitivity of the option price to

changes in

Volatility – two Vs.)

Theta

Theta is a measure of the sensitivity of the option price to the remaining time to

expiry of the option. Theta decreases as the remaining time to expiry falls, and is

higher for options with a long time remaining to expiry.

(It might help to remember that

Theta measures sensitivity of the option price to

changes in

Time to expiry – two Ts.)

The most important of the Greeks is the delta, and (as described earlier) you might

be expected in the examination to show your understanding of what it represents

and how a delta hedge might be constructed.

Paper P4

Advanced Financial Management

CHAPTER

Financial management

and multinationals

Contents

1 Problems for multinational companies

2 Transfer pricing and taxation

3 Free trade and protectionism

4 International and financial markets

Paper P4: Advanced Financial Management

Problems for multinational companies

 Global competition

 MNCs and national governments: political risk and regulatory risk

 MNCs and management co-ordination

1 Problems for multinational companies

1.1 Global competition

In many industries, the market place and the competition is global. Global

competition exists largely for products, but there is also global competition in many

service industries, such as banking and financial services and telecommunications.

When an industry becomes global, there is usually a trend towards the creation of a

number of large multinational companies or corporations (MNCs). Powerful MNCs

make important economic decisions about:

 where to locate production facilities and who to employ

 pricing goods (and services).

Some large MNCs control very large amounts of resources, and have more

economic power than many of the countries in which they invest.

1.2 MNCs and national governments: political risk and regulatory risk

The main objective of a MNC is presumably to maximise the wealth of its

shareholders. The main objective of many national governments is to provide for

the well being of their population. The objectives of MNCs and national

governments therefore differ, particularly when the MNC is a ‘foreign company’

with its head office in another country.

National governments might expect MNCs to:

 invest in their country, and provide employment

 through investment, add to the economic wealth of the country and help to

develop the country’s economy

 pay taxes on their profits, that the government can spend on the well-being of its

population.

MNCs might therefore be exposed to large political and regulatory risks when they

invest in another country, particularly when the local government and population

suspect that the multinational is more concerned about the interests of its

shareholders than in the welfare of the country. Potential risks include:

 the risk of confiscation of their assets by the government

 the risk of new regulations of the industry, such as regulations requiring

minimum ownership in all businesses by ‘nationals’ of the country

 the risks of political unrest and violence.

Chapter 17: Financial management and multinationals

Some MNCs are aware of their corporate social responsibility to the countries in

which they invest, and are aware of the need to educate the local population,

preserve the local environment and create a sustainable industry within that

country. On the other hand, some MNCs might show little awareness of corporate

social responsibility, and seek to avoid paying taxes if possible. As a consequence, in

many countries, MNCs are treated with dislike and suspicion.

Investment decisions by multinationals might be affected by the perceived political

and regulatory risks of investing in particular countries.

1.3 MNCs and management co-ordination

Corporate governance issues were described in an earlier chapter. Corporate

governance is a particular problem for MNCs. A part of the problem is that MNCs

need to recognise the interests of local governments and populations as

stakeholders in their foreign subsidiaries.

Another issue is the agency problem, and the requirement for management to act in

the bests interests of the company’s shareholders. In large multinational companies,

a considerable amount of authority is delegated to ‘local’ management in divisions

and subsidiaries around the world. Delegation of authority is essential for the

effective management of large global businesses.

On the other hand, when authority is delegated to local managers, head office

management need to co-ordinate the worldwide activities of the MNC, and try to

ensure that the strategic objectives of the MNC are met and the interests of its

shareholders are satisfied.

Decisions by head office management might be resisted by local management in

some countries, where head office managers might be trying to act in the interests of

the shareholders, whereas local managers have more concern for their own position

or for the interests of employees, customers and economic conditions in their own

country.

Paper P4: Advanced Financial Management

Transfer pricing and taxation

 Corporate objectives and the relevance of tax and transfer pricing

 MNCs: transfer pricing and tax rules

 Multinationals and tax planning

2 Transfer pricing and taxation

2.1 Corporate objectives and the relevance of tax and transfer pricing

The corporate objective of a multinational company, as for any other company, is

linked to profitability and returns for shareholders. Multinational companies

operate in different countries, and are therefore subject to differing tax regulations

and systems. MNCs will try to improve earnings for shareholders by minimising

their worldwide tax liabilities.

 Multinational companies will seek to invest in countries where the taxation

system is more favourable. For example, companies might seek relief from

taxation for making capital investments in a country, and will choose one

country in preference to another on the basis of the tax advantages offered

 Some MNCs have been accused of using transfer pricing to minimise their tax

liabilities. Transfer prices are used to charge for goods and services transferred

between companies in the group. Transfer prices can be charged in such as way

that:

− transfers from a subsidiary in a high-tax country to a subsidiary in a low-tax

country are fixed at high levels, and

− low transfer prices are set for transfers from a low-tax to a high-tax country.

In this way, profits in the high-tax subsidiaries are reduced, and more profits are

earned in the low-tax countries.

2.2 MNCs: transfer pricing and tax rules

Transfer prices are prices charged internally within a group, and have no effect on

the total pre-tax profit of the group. Transfer prices are simply a way of sharing the

profit between the two companies concerned.

Transfer prices between a company and a foreign subsidiary have implications for

taxation, and the aim of a group of companies should be to minimise total tax

payable on the group profits. Issues to consider are:

 the rate of taxation on profits in the two countries

 the existence of a double taxation agreement

 the existence of any withholding tax in the country of the foreign subsidiary

 tax rules on transfer pricing.

ACCA P4 Advanced Financial Management - 2010 - Study text - Emile Woolf Publishing

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