ACCA P4 Advanced Financial Management - 2010 - Study text

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Chapter 16: Option pricing and delta hedging

The value of e

-rt

is the inverse of the constant e to the power of rt. In other words, e

-rt

is 1/

. Use your calculator to calculate the value of e

, then calculate the inverse of

your answer.

2.3 Using the Black-Scholes model

The formulae might seem very complex. The following example shows how they

should be applied:

Example

The current share price of BS Company is $16. What should be the price of a

European-style call option on the company’s shares at an exercise price of $16.50, if

the expiry date is in six months, the standard deviation of annual returns on the

share is 12% and the risk-free rate of return is 7% per year?

Answer

Step 1. Calculate d

= ln(16.00/16.50) + [0.07 + (0.5 × 0.12²) 0.5

0.50.12

16.50

⎛

⎝

⎜

⎞

⎠

⎟

= ln 0.9697. Using a calculator, the natural logarithm of 0.9697 can be

obtained. This is – 0.03077.

0.50.12

0.03860.03077

+−

= 0.00783/0.08485

= + 0.092

Step 2. Calculate d

()

0.50.120.092tdd

−=−= s

= + 0.0071.

Step 3. Obtain values for N(d

) and N(d

)

(a) N(d

). When d

= 0.092, the corresponding value in the normal distribution table

is 0.3212. The value of d

is positive, so add 0.50. N(d

) = 0.3212 + 0.50 = 0.8212.

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(b) N(d

). When d

= 0.007, the corresponding value in the normal distribution table

is 0.0279. The value of d

is positive, so add 0.50. N(d

) = 0.0279 + 0.50 = 0.5279.

Step 4. Calculate the option price (for a call option)

It might help to calculate the value of e

-rt

first. The value of rt is (0.07)(0.5) = 0.035.

The value of e

-0.035

is the same as 1/e

0.035

. Using a calculator, the value of e

0.035

can be

obtained: this is 1.0356. Therefore e

-0.035

= 1/1.0356 = 0.9656.

The exercise price (P

) is 16.50. The option price can now be calculated as follows:

() ()

2e1s

e dNPdNPpriceOption

−

−=

= 16(0.8212) – 16.50 (0.5279) (0.9656)

= 13.14 – 8.16

= 4.98

The option price is $4.98. This is high relative to the current market price, but the

high option value is due to the fact that the strike price is close to the current market

price, and the returns on the share are fairly volatile.

Example

In this similar example, the values of d

and d

are negative.

The current share price of CB Company is 400. What should be the price of a

European-style call option on the company’s shares at an exercise price of 440, if the

expiry date is in six months, the standard deviation of annual returns on the share is

30% and the risk-free rate of return is 6% per year?

Answer

Step 1. Calculate d

= ln(400/440) + [0.06 + (0.5 × 0.30²)] 0.5

0.50.30

ln (400/440) = ln 0.9091. Using a calculator, the natural logarithm of 0.95238 can be

obtained. This is – 0.0953.

0.50.30

0.05250.0953

−

= – 0.0428/0.2121

= – 0.202

Chapter 16: Option pricing and delta hedging

Step 2. Calculate d

()

0.50.30220.0tdd

−−=−= s

= – 0.414

Step 3. Obtain values for N(d

) and N(d

)

N(d

). When d

= – 0.202, the corresponding value in the normal distribution table is

about 0.0801. The value of d

is negative, so subtract 0.0801 from 0.50. N(d

) = 0.5000 –

0.0801 = 0.4199

N(d

). When d

= – 0.414, the corresponding value in the normal distribution table is

about 0.1613. The value of d

is negative, so subtract 0.1613 from 0.50. N(d

) = 0.5000 –

0.1613 = 0.3387.

Step 4. Calculate the option price (for a call option)

It might help to calculate the value of e

-rt

first. The value of rt is (0.06)(0.5) = 0.030.

The value of e

-0.030

is the same as 1/e

0.03

. Using a calculator, the value of e

0.03

can be

obtained: this is 1.03045. Therefore e

-0.03

= 1/1.03045 = 0.9704.

The exercise price (P

) is 440. The option price can now be calculated as follows:

()

(

)

2e1

e dNPdPaNpriceOption

−

−=

= 400(0.4199) – 440 (0.3387) (0.9704)

= 167.96 – 144.62

= 23.34.

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Value of a put option: put-call parity

 Value of a put option: put-call parity pricing model

 Value of a put option: alternative put-call parity pricing model

3 Value of a put option: put-call parity

3.1 Value of a put option: put-call parity pricing model

The Black-Scholes model calculates the value of a European-style call option. It can

also be used to calculate the value of a European-style put option with the same

expiry date and exercise price, using the put-call parity pricing model.

The value of a European-style put option is calculated as:

Value of put + Current market price of underlying item = Value of call + Present

value of the strike price.

PUT

+ MV = V

CALL

+ PV

STRIKE

Example

Shares in BS Company have a current market value of $16. The value of a call option

with a strike price of $16.50 and an expiry date in six months is $4.98. The risk-free

rate of return is 7% per year, so the present value of the exercise price is $16.50 ×

1/(1.07)

0.5

= $16.50 × 0.9667 = $15.95.

Applying the put-call parity pricing model:

Value of put + $16 = $4.98 + $15.95

Value of put = $4.93.

3.2 Value of a put option: alternative put-call parity pricing model

An alternative formula for the put-call parity is given in the formula sheet in the

examination. Allowing for rounding differences, it gives the same valuation for a

put option as the formula described above.

The alternative formula is:

p = c – P

+ P

-rt

where

p = the price of a put option

Chapter 16: Option pricing and delta hedging

c = the price of a call option with the same exercise price and expiry date

is the current market price of the underlying item

is the exercise price of the option

r = the risk free rate of interest

t = the time to expiry in years

Example

The same example used above to demonstrate put-cal parity will be used to show

how the alternative formula is used.

Shares in BS Company have a current market value of $16. The value of a call option

with a strike price of $16.50 and an expiry date in six months is $4.98.

The risk-free rate of return is 7% per year, and the option has six months to expiry,

so rt = (0.07)(0.5) = 0.035. The value of e

0.035

is 1.03562 so e

- 0.035

is 0.9656.

p = 4.98 – 16 + (16.5)(0.9656)

p = 4.91.

This differs from the valuation above using the alternative formula by 0.02, due to

mathematical rounding differences.

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The forex variant for pricing FX options

 The Grabbe variant of the Black-Scholes model

 Initial problems with applying the formula

 Using the forex variant for pricing FX options

4 The forex variant for pricing FX options

4.1 The Grabbe variant of the Black-Scholes model

A variant of the Black-Scholes model can be used to calculate a price for European-

style

at-the-money currency options. It is sometimes called the Grabbe variant of

the Black-Scholes model. The formulae are provided in the formula sheet for your

examination, and you need to be able to use them.

The meaning of ‘at-the-money’

An at-the-money currency option is an option whose exercise rate or strike rate is

equal to the current forward exchange rate for a forward contract whose settlement

date is the expiry date for the option. For example, a six-month at-the-money

currency option is a currency option whose stroke rate is the current six-month

forward rate.

The formulae for the pricing model

There is a formula for calculating the price of a call option (c) and another formula

for the price of a put option (p). The formulae are as follows.

c = e

-rt

N(d

) – X N(d

)]

p = e

-rt

[X N(- d

) – F

N(- d

)]

where

= ln(F

/X) + s²t/2

s√t

= d

– s√t

Some of these symbols have the same meaning as in the basic Black-Scholes

formula.

X is the current spot exchange rate

is the forward exchange rate for a settlement date at the expiry of the currency

option. This is also the exercise rate or strike rate for an at-the-money option.

Chapter 16: Option pricing and delta hedging

ln(F

/X) is the log normal value of F

s is the

annualised standard deviation of the exchange rate (and s² is the annual

variance of the exchange rate)

t is the time period to expiry of the option in years

r is the risk-free interest rate. The risk-free rate is normally assumed to be an inter-

bank money market rate such as LIBOR.

The risk-free interest rate

With currency options, there are two currencies and each currency has a different

risk-free interest rate. It is therefore important to know which risk-free interest

currency to use in the option pricing formula. The interest rate for the base currency

should be used. This is the currency in which the option is priced.

Example

To calculate a price for traded options in sterling (British pounds), you should use

the three-month sterling LIBOR rate for pricing three-month options, the six-month

sterling LIBOR rate for pricing six-month options and the nine-month sterling

LIBOR rate for pricing nine-month options. These money market rates are all

expressed as annual percentage rates, and should be used as the rate for r in the

option pricing model.

The annual standard deviation (volatility) of the exchange rate

The standard deviation of the exchange rate, which is the measure of the volatility

of the exchange rate, is an annual standard deviation. If an examination question

gives you a standard deviation (or ‘volatility’) for any time period other than one

year, you will need to convert the figure you are given to an annual volatility.

 If you are given a monthly volatility (standard deviation) the annual standard

deviation is the monthly standard deviation multiplied by a factor × √12

(× 3.4641).

 If you are given a three-monthly volatility (standard deviation) the annual

standard deviation is the monthly standard deviation multiplied by a factor × √4

(× 2.000).

 If you are given a six-monthly volatility (standard deviation) the annual

standard deviation is the monthly standard deviation multiplied by a factor × √2

(× 1.4142).

4.2 Initial problems with applying the formula

Two problems have to be dealt with before the formulae can be used. These are:

 Sorting out which is the fixed currency and which is the variable currency, and

the currency in which the option is priced.

 Calculating the forward rate when this is not given in the examination question.

Paper P4: Advanced Financial Management

Sorting out the base currency and variable currency

It is important to identify which currency is the base currency and which is the

variable currency. Suppose there are currency options for euros-US dollars.

 When options are options on euros, each option is for a fixed quantity of euros in

exchange for US dollars at an exchange rate equal to the strike rate or exercise

rate for the option. The exercise rate is a rate for the number of dollars it will cost

to buy (call option) or sell (put option) one euro.

 The option pricing model should therefore use a spot rate and a forward rate

where euros are the base currency and US dollars are the variable currency. In

other words, we need an exchange rate of €1 = US$X for both the spot rate and

the forward rate, to use in the option pricing model.

 The option pricing model will give a value or price for the option in US currency

(cents per €1).

 The risk-free interest rate to use in the pricing formula is the risk-free rate for the

currency in which the option is priced.

Example

Traded currency options for euros-US dollars are 100,000 euros per contract. The

spot exchange rate is €1 = US$1.3250.

Since options are for a fixed quantity of euros, the exchange rate for €1 = $X should

be used in the option pricing formula. This is €1 = US$1.3250.

Using the Black-Scholes formula to calculate the value of an option will give a price

in cents per €1.

For an option expiring in six months, the option pricing model should use a risk-

free six-month interest rate for the US dollar, such as the six-month US dollar LIBOR

rate.

Calculating the forward rate

An examination question might give you the spot rate for a currency and the risk-

free interest rate in each currency, but might not give you the forward exchange

rate.

When this happens, you need to calculate the forward rate from the spot rate and

the two risk-free interest rates. A technique for doing this is as follows:

 Start with the spot rate. This will be given as 1 unit of base currency = X units of

variable currency.

 The option expires in n months’ time.

 Calculate how much 1 unit of base currency will be worth in n months if it is

invested now for n months at the risk-free rate of interest.

 Similarly, calculate how much X units of variable currency will be worth in n

months if they are invested now for n months at the risk-free rate of interest.

Chapter 16: Option pricing and delta hedging

 The forward rate for settlement in n months’ time is the value of the variable

currency after n months divided by the value of the base currency after n

months.

Example

The spot exchange rate for British pounds/US dollars is £1 = US$1.9500.

The three-month sterling LIBOR interest rate is 5.80%.

The three-month US dollar LIBOR rate is 4.40%.

Required

Calculate the three-month forward exchange rate.

Answer

The three-month interest rates are annual percentages. If US dollars are invested for

three months, the amount of interest earned will be approximately 4.40% × 3/12 =

1.10%.

 Similarly, if sterling is invested for three months, the amount of interest earned

will be approximately 5.80% × 3/12 = 1.45%.

 The spot exchange rate is $1.9500.

 If £1 is invested for three months, its value after three months will be £1 × 1.0145

= £1.0145.

 If $1.9500 is invested for three months, its value after three months will be $1.95

× 1.011 = 1.97145.

The three month forward rate is $1.97145/£1.0145 = 1.9433.

Alternative method of calculating a forward rate: interest rate parity model

The same forward rate can be calculated using the interest rate parity model,

although this is effectively the same as the method explained above.

The interest rate parity model is:

= S

× (1 + i

)

(1 + i

)

where

is the forward rate for settlement after n months

is the current spot rate

is the interest that would be earned by investing 1 unit of the variable currency for

n months (so if the three-month rate of interest is 4% at an annualised rate, i

= 4% ×

3/12 = 1% or 0.01

is the interest that would be earned by investing 1 unit of the base currency for n

months.

Paper P4: Advanced Financial Management

Example

In the previous example, the forward rate could be calculated as:

= 1.9500 × 1.011

1.0145

= 1.9433.

4.3 Using the forex variant for pricing FX options

The forex variant of the Black-Scholes model is similar to the basic Black-Scholes

model, and the following example illustrates how it should be used:

Example

Traded currency options for sterling-euros are 100,000 euros per contract. The spot

exchange rate is €0.7000 = £1.

The three-month euro Libor rate is 4.25% and the three-month sterling Libor rate is

5.5%.

Monthly volatility in the sterling-euro exchange rate is 3.6% (measured as the

monthly standard deviation of the exchange rate).

Required

(a) Calculate the current price of a three-month at-the-money currency call option

on €100,000.

(b)

Calculate the current price of a three-month at-the-money currency put option

on €100,000.

Answer

We are given the one-month volatility of the exchange rate. The annual volatility is

3.6% × √12

= 12.47%.

The options are for euros, which means that euros are the variable currency and

sterling is the base currency. We want the exchange rate expressed as £1 = €X. The

spot exchange rate is given as £1 = €0.7000.

The three-month forward rate (at-the-money strike rate) is calculated as follows:

 If euros are invested for three months, the amount of interest earned will be

approximately 4.25% × 3/12 = 1.0625%.

 If sterling is invested for three months, the amount of interest earned will be

approximately 5.5% × 3/12 = 1.375%.

 The three-month forward rate is:

= 0.7000 × 1.010625

1.01375

= 0.6978.

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

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