Topic 10 Black-scholes pricing model PDF

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20/10/2019

Topic 10: Black-scholes pricing model

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ID: FRM.O.BSM.01A.SL

The shares of DCB Bank are currently valued at $21 and they have a volatility of 20% pa. The risk free rate is 7% pa. You many find this table showing the values of the standard normal cumulative distribution function useful. a) Calculate the price of a European call option on these shares, with strike price $24 and a term to expiration of 6 years. Give your answer in dollars and cents to the nearest cent. Price of call option = $ 6.75 DCB bank have just announced that a dividend of $0.56 will be paid in 2 years time and an additional dividend of $1.26 will be paid in 5 years time.

HINT: This question does not interpolate values from the standard normal cumulative distribution. For this particular question, when looking up z values from the distribution, first round your calculated value of z to two decimal places and then use that rounded z value to find F(z).

b) Taking this new information into account, calculate the adjusted price of the call option. Give your answer in dollars and cents to the nearest cent and assume that the time between the ex-dividend date and actual payment of the dividend has a negligible effect. Adjusted price of call option = $ 5.67

Feedback

[2 out of 2]

a) You are correct. b) You are correct.

Calculation a) The price of a call option without dividends can be calculated using the basic Black Scholes pricing formula. The Black-Scholes pricing formula for a European call is: hide variables

c

c

= price of a European call = unknown

S0

= initial share price = $21

K

= strike price of share = $24

r

= risk-free rate of interest per annum (decimal) = 0.07

T

= term to expiration (years) = 6

σ

= volatility of share per annum (decimal) = 0.2

N(x)

= cumulative probability distribution function for standardised normal distribution

= S0N(d1) - Ke-rTN(d2)

where:

d1

=

ln(S0/K) + (r + σ²/2)T σ(T)1/2

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Topic 10: Black-scholes pricing model

d2

= ln(S0/K) + (r - σ²/2)T σ(T)1/2

= d1 - σ(T)1/2

Therefore:

ln(21/24) + (0.07 + 0.22/2)×6

d1

=

d2

= 0.83 - 0.2(6)1/2

0.2(6)1/2

= 0.83

= 0.34 Substituting all the variables into the Black Scholes formula:

c

= 21 × N(0.83) - 24e-0.07 × 6 × N(0.34) = 21 × 0.7967 - 24e-0.07 × 6 × 0.6331 = 6.7472678... = $6.75

Rounded as last step

b) Dividends will affect the price of the call option because they have an effect on the future share price, which, in turn, affects the price of the call option. A share price incorporates both the risky and riskless components of the share. Once a dividend is declared and the share becomes ex-dividend, then the risky component of the share price will fall by the amount equal to the present value of the declared dividends (ignoring the effect of tax). The modified Black Scholes formula relies on the fact that if a declared dividend is payable within the life of the option, then the riskless component of the share price will not exist at the expiration of the option. This effectively means that the share price can be reduced by this riskless component at the outset of the pricing calculation. It is important to note here that this will only work with European options. The adjustment to the Black Scholes pricing formula for dividends is achieved by calculating the present value of the dividends to be paid, and subtracting this value from the current share price (S0). This new adjusted rate Sadj is then applied into the Black Scholes pricing formula. It effectively looks at the price of the underlying share if dividends were taken out. The adjusted initial share price can be calculated using the following formula: hide variables

D1

= value of first dividend = $0.56

D2

= value of second dividend = $1.26

t1

= time until payment of the first dividend (years) = 2

t2

= time until payment of the second dividend (years) = 5

r

= risk-free rate of interest

PVd

= present value of dividends = unknown

S0

= initial price of share = $21

Sadj

= adjusted price of share = unknown

PVd

= =

D1 (1 + r)t1 0.56

+

(1 + 0.07)2

D2 (1 + r)t2 +

1.26 (1 + 0.07)5

= 0.48912569... + 0.89836259... = 1.38748827... Sadj

= S0 - PVd = 21 - 1.38748827... = 19.61251173...

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Topic 10: Black-scholes pricing model

The new current option price can be calculated via the same method as above but substituting S0 with Sadj: hide variables

c

c

= price of a European call = unknown

Sadj

= adjusted share price = $19.61251173...

K

= strike price of share = $24

r

= risk-free rate of interest per annum (decimal) = 0.07

T

= term to expiration (years) = 6

σ

= volatility of share per annum (decimal) = 0.2

N(x)

= cumulative probability distribution function for standardised normal distribution

= SadjN(d1) - Ke-rTN(d2)

where:

d1

=

d2

=

ln(Sadj/K) + (r + σ²/2)T σ(T)1/2 ln(Sadj/K) + (r - σ²/2)T σ(T)1/2

= d1 - σ(T)1/2

Therefore:

ln(19.61251173.../24) + (0.07 + 0.2²/2)×6

d1

=

d2

= 0.69 - 0.2(6)1/2

0.2(6)1/2

= 0.69

= 0.20 Substituting all the variables into the Black Scholes formula:

c

= 19.61251173... × N(0.69) - 24e-0.07 × 6 × N(0.20) = 19.61251173... × 0.7549 - 24e-0.07 × 6 × 0.5793 = 5.67043176... = $5.67

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Rounded as last step

ID: FRM.OP.BSM.03.0010.SL

Calculate the prices of 3 month European call and put options using the Black-Scholes pricing HINT: This question does not formula, given that the current market price of the underlying asset is $46.42, the risk-free interpolate values from the standard continuously compounded interest rate is 8% per annum, and the volatility (standard deviation) normal cumulative distribution. For of the price of the underlying asset is 21% per annum. You may find this table useful. a) Calculate the option price for a call and a put if the strike price is $44.99. Give all answers in dollars and cents to the nearest cent. Price of a call option = $ 3.20

this particular question, when looking up z values from the distribution, first round your calculated value of z to two decimal places and then use that rounded z value to find F(z).

Price of a put option = $ 0.88 b) Calculate the option price for a call and a put if the strike price is $45.30. Give all answers in dollars and cents to the nearest cent. Price of a call option = $ 3.16 Price of a put option = $ 1.14

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Topic 10: Black-scholes pricing model

Feedback

[4 out of 4]

a) You are correct. b) You are correct.

Calculation a) To calculate the price of a European option using the Black-Scholes equation we must know the value of the five variables involved. Once these are known it is a matter of putting these values into the equation. The Black-Scholes equation for pricing a European call is: hide variables

S0

= current market price = $46.42

r

= risk-free interest rate = 0.08

T

= time to maturity = 0.25 (years)

K

= strike price = $44.99

σ

= standard deviation = 0.21

c

= European call premium = unknown

p

= European put premium = unknown

c = S0N(d1) - Ke-rTN(d2) where

d1

= ln(S0/K) + (r + σ2/2)T

d2

= d1 - σ√T

σ√T

N is the cumulative normal standard distribution function, and e is the natural exponential. Inserting the given data we find

d1

= ln(46.42/44.99) + (0.08 + 0.0441/2)0.25 0.21√0.25 = 0.5409777... = 0.54

d2

Rounded to 2 decimal places for normal distribution table use

= d1 - 0.21√0.25 = 0.4359777... = 0.44

Rounded to 2 decimal places for normal distribution table use

N(d1) = 0.7054

From normal distribution table

N(d2) = 0.6700

From normal distribution table

c

= 46.42×0.7054 - 44.99e-0.08×0.25×0.6700 = 3.19824533... = $3.20

Rounded as a last step

The Black-Scholes equation for pricing a European put is: p = c - S0 + Ke-rT Where c is the price of a European call using Black-Scholes. So once c is known we have

p

= 3.19824533... - 46.42 + 44.99e-0.08×0.25

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= 0.87738364... = $0.88

Rounded as a last step

b) This calculation is the same as part a) but with K = $45.30 instead. Inserting the values in the Black-Scholes formula, we get: hide variables

d1

S0

= current market price = $46.42

r

= risk-free interest rate = 0.08

T

= time to maturity = 0.25 (years)

K

= strike price = $45.30

σ

= standard deviation = 0.21

c

= european call price = unknown

p

= European put price = unknown

= ln(46.42/45.30) + (0.08 + 0.0441/2)0.25 0.21√0.25 = 0.4755797... = 0.48

d2

Rounded to 2 decimal places for normal distribution table use

= d1 - 0.21√0.25 = 0.3705797... = 0.37

Rounded to 2 decimal places for normal distribution table use

N(d1) = 0.6844

From normal distribution table

N(d2) = 0.6443

From normal distribution table

c

= 46.42×0.6844 - 45.30e-0.08×0.25×0.6443 = 3.16099516... = $3.16

Rounded as a last step

Then the price of a European put is

p

= 3.16099516... - 46.42 + 45.30e-0.08×0.25 = 1.14399506... = $1.14

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Rounded as a last step

ID: FRM.OP.BSM.04.0010

In order for the Black-Scholes pricing formula to be true, many assumptions need to be made. Select all the following assumptions that the Black-Scholes pricing formula requires:

there are no arbitrage opportunities trading on securities occurs continuously securities are only available in discrete units there are no dividends paid on securities there may not be any short selling of securities there are no taxes or transaction costs

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Feedback

[2 out of 2]

You are correct.

Discussion The main assumptions of the Black-Scholes pricing formula are as follows: 1) 2) 3) 4) 5) 6) 7) 8)

Changes in the underlying asset price follow a log normal distribution. There are no dividends. There are no taxes. There are no transaction costs. There are no arbitrage opportunities. The underlying asset is infinitely divisible. Short-selling is allowed. Trading occurs continuously.

4 of 4

ID: FRM.OP.BSM.06.0010.SL

The shares of DCB Bank are currently valued at $21 and they have a volatility of 19% pa. The risk free rate is 4% pa. You many find this table showing the values of the standard normal cumulative distribution function useful.

HINT: This question does not interpolate values from the standard normal cumulative distribution. For this particular question, when a) Calculate the price of a European call option on these shares, with strike price $26 and a looking up z values from the term to expiration of 6 years. Give your answer in dollars and cents to the nearest cent. distribution, first round your Price of call option = $ 4.13 calculated value of z to two decimal places and then use that rounded z DCB bank have just announced that a dividend of $0.64 will be paid in 1 years time and an value to find F(z). additional dividend of $1.31 will be paid in 5 years time.

b) Taking this new information into account, calculate the adjusted price of the call option. Give your answer in dollars and cents to the nearest cent and assume that the time between the ex-dividend date and actual payment of the dividend has a negligible effect. Adjusted price of call option = $ 3.15

Feedback

[2 out of 2]

a) You are correct. b) You are correct.

Calculation a) The price of a call option without dividends can be calculated using the basic Black Scholes pricing formula. The Black-Scholes pricing formula for a European call is: hide variables

c

c

= price of a European call = unknown

S0

= initial share price = $21

K

= strike price of share = $26

r

= risk-free rate of interest per annum (decimal) = 0.04

T

= term to expiration (years) = 6

σ

= volatility of share per annum (decimal) = 0.19

N(x)

= cumulative probability distribution function for standardised normal distribution

= S0N(d1) - Ke-rTN(d2)

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Topic 10: Black-scholes pricing model

where:

ln(S0/K) + (r + σ²/2)T σ√T ln(S0/K) + (r - σ²/2)T = σ√T

d1

=

d2

= d1 - σ√T

Therefore:

d1

ln(21/26) + (0.04 + 0.192/2)×6 0.19√6

=

= 0.29 d2

= 0.29 - 0.19√6 = -0.18

Substituting all the variables into the Black Scholes formula:

c

= 21 × N(0.29) - 26e-0.04 × 6 × N(-0.18) = 21 × 0.6141 - 26e-0.04 × 6 × 0.4286 = 4.13023377... = $4.13

Rounded as last step

b) Dividends will affect the price of the call option because they have an effect on the future share price, which, in turn, affects the price of the call option. A share price incorporates both the risky and riskless components of the share. Once a dividend is declared and the share becomes ex-dividend, then the risky component of the share price will fall by the amount equal to the present value of the declared dividends (ignoring the effect of tax). The modified Black Scholes formula relies on the fact that if a declared dividend is payable within the life of the option, then the riskless component of the share price will not exist at the expiration of the option. This effectively means that the share price can be reduced by this riskless component at the outset of the pricing calculation. It is important to note here that this will only work with European options. The adjustment to the Black Scholes pricing formula for dividends is achieved by calculating the present value of the dividends to be paid, and subtracting this value from the current share price (S0). This new adjusted rate Sadj is then applied into the Black Scholes pricing formula. It effectively looks at the price of the underlying share if dividends were taken out. The adjusted initial share price can be calculated using the following formula: hide variables

D1

= value of first dividend = $0.64

D2

= value of second dividend = $1.31

t1

= time until payment of the first dividend (years) = 1

t2

= time until payment of the second dividend (years) = 5

r

= risk-free rate of interest

PVd

= present value of dividends = unknown

S0

= initial price of share = $21

Sadj

= adjusted price of share = unknown

PVd

= D1e-rt1 + D2e-rt2 = 0.64e-0.04×1 + 1.31e-0.04×5 = 0.61490524... + 1.07253729... = 1.68744253...

Sadj

= S0 - PVd = 21 - 1.68744253... = 19.31255747...

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Topic 10: Black-scholes pricing model

The new current option price can be calculated via the same method as above but substituting S0 with Sadj: hide variables

c

c

= price of a European call = unknown

Sadj

= adjusted share price = $19.31255747...

K

= strike price of share = $26

r

= risk-free rate of interest per annum (decimal) = 0.04

T

= term to expiration (years) = 6

σ

= volatility of share per annum (decimal) = 0.19

N(x)

= cumulative probability distribution function for standardised normal distribution

= SadjN(d1) - Ke-rTN(d2)

where:

d1

=

ln(Sadj/K) + (r + σ²/2)T σ√T

d2

=

ln(Sadj/K) + (r - σ²/2)T σ√T

= d1 - σ√T

Therefore:

d1

=

ln(19.31255747.../26) + (0.04 + 0.19²/2)×6 0.19√6

= 0.11 d2

= 0.11 - 0.19√6 = -0.36

Substituting all the variables into the Black Scholes formula:

c

= 19.31255747... × N(0.11) - 26e-0.04 × 6 × N(-0.36) = 19.31255747... × 0.5438 - 26e-0.04 × 6 × 0.3594 = 3.15160337... = $3.15

Rounded as last step

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