Chapter 5 Problems on Options

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Problems on Options

1 A stock price is currently $100. Over of each of the next two six-month periods, it is expecte

to go up by 10% or down by 10%. The risk free rate is 8% per annum with continuous

compounding. What is the value of a one-year European call option with a strike price of

$100?

0 1 2Solution:

S0 = $100 $121

K = $100 $110

rf = 8%

u = 1.1 $100 $99

d = 0.9

h = 0.5 $90.00

p* = 0.7040539 Equation 5.3

$81.00

value of call option = exp(-2rh)(p*^2*Cuu + 2*p*(1-p*)*cud + (1-p*)^2*Cdd)

$9.61 Equation 5.19

2 Consider a six month put option on a stock with a strike price of $32. The current stock price

next six months it is expected to rise to $36 or fall to $27. The risk-free rate of interest is 6%.

i. What is the risk-neutral probability of the stock rising to $36?

ii. What position in the stock is necessary to hedge a long position in 1 put option?

iii. What is the value of the put option?

Solution: Value of op

$36 $0.00S0 = $30

Su = $36 $30

Sd = $27

$27 $5

K = $32

u = 1.20

d = 0.90

h = 0.5

rf = 6.00%

i. p* = 0.434848

ii. To find D set 36*delta = 27*delta - 5 and solve for delta.

D -0.55556

iii. $2.74

3 A stock price is currently $100.00. Over each of the next two three month periods it is expec

to increase by 10% or fall by 10%. Consider a six month European put option with a strike pri

$95. The risk-free rate is 8% per annum.



i. What is the risk-neutral probability of a 10% rise in each quarter?

ii. What is the value of the option?

iii. What is the value of the option if it is American?

iv. What is the value of the option if it is a call rather than a put?

Solution:$121.00

S0 = $100.00

K = $95 $110.00

rf = 8%

h = 0.25 $100.00 $99.00

u = 1.1

d = 0.9 $90.00

i. P* = 0.6010067 $81.00

ii. Put value = $2.14

iii. The value of the American put = $2.14

iv. The value of a call = $10.87

4 A stock price is currently $55. Over each of the next two three month periods it is expected t

down by 7%. The risk-free rate is 5.55 per annum with continuous compounding. What is the

six-month European call option with a strike price of $58? Verify that i) the risk-neutral ptob

method, ii) the Binomial method, and iii) put-call parity give the same answer.

Solution: $65.35

S0 = $55 $59.95

K = $58 $3.80

rf = 5.50% $55 $55.75

u = 1.09 $1.96

d = 0.93 $51.15

h = 0.25

p* = 0.524031 $47.57

i. Option value = $1.96

ii. Binomial option value = $1.96

iii. Put call parity value of option c + k*exp(-rt) = p + S0

where k= $58

S0 = $55

p = $3.39

Therefore, c = $1.96



5 A stock price is currently $110. It is known that at the end of six months it will be either $125

The risk-free rate of interest is 9% per annum with continuous compounding. What is the val

one year European put option with a strike price of $105? Verify that no arbitrage argument

give the same answers as risk-neutral probability method.

Solution: $142.05

S0 = $110 $125

K = $105 $107.95

rf = 9% $110

u = 1.136

d = 0.8636364 $95

h = 0.5

p* = 0.6687688 $82.05

i. Risk-neutral probability Put value = $2.30

ii. No arbitrage option value required you to compute the hedge ratio by setting $1

and solving for delta.

D -0.33333 call option value =

The value of the portfolio = $41.67 Put option value =

The present value of the portfolio = 39.83$

Hence, -.33333*S0 + f = present value of portfolio

therefore f = 3.17$

iii. To use put-call parity, first find the value of the call option.value of call = $12.79

Put-call parity --> p = c + K*exp(-rt) - S0

Therefore, c = $3.17

6 A stock price is currently $50. Over each of the next two three month periods it is expected t

The risk free rate is 10% per annum with continuous compounding.

i. What is the value of a six month European put option with a strike price of $55?

ii. What is the value of a six month American put option with a strike price of $55?

Solution: $58.32

S0 = $50 $54

K = $55 $1.84

rf = 10% $50 Early ex. $1 $49.14

u = 1.08 American $4.20

d = 0.91 European $3.77 $45.50

h = 0.25 $8.14

p* 0.6783242 Early Ex. $9.50 $41.41



i. Value of European put option = $3.77

ii. Value of American put option = $4.20

7 Consider an option on a non-dividend paying stock when the stock price is $35, the exercise

5%, the volatility is 30% per annum, and the time to maturity is 3 months.a. What is the price of the option if it is a European call?

b. What is the price of the option if it is an American call?

c. What is the price of the option if it is a European put?

d. Verify that put-call parity holds.

Solution:

S = $35.00 c = Se^(-delta*T) N(d1) - Ke^(-rT)N(d2)

K = $30.00

T = 0.25 p = Ke^(-r*T)N(-d2) -Se^(-delta*T)N(-d1)

r = 5%

0.3

0 d1 = 1.186005

N(d1) = 0.88219

d2 = 1.036005

N(d2) = 0.8499

a. c = $5.70

b. Value of an American call = $5.70

To compute put option, first compute N(-d1) and N(-d2)

N(-d1) = 0.11781

N(-d2) = 0.1501

c. p = $0.32

d. Put-call parity value = $0.32

8 For a call option on a non-dividend paying stock, the stock price is $30, the strike price is $29

is 6% per annum, the volatility is 20% per annum, and the time to maturity is three months.

Options Pricing model:

a. What is the price of the option?

b. What is the price of the option if its is a put?

c. How do the answers to (a) and (b) change if a dividend of $2 is expected in two months?

Solution:


=d

=s



K = $29.00


r = 6%

0.2

0 d1 = 0.539016

N(d1) = 0.705062

d2 = 0.439016

N(d2) = 0.669675

a. c = $2.02


N(-d1) = 0.294938

N(-d2) = 0.330325

b. p = $0.59

c. Modify the equation to account for receipt of dividends in two months.


K = $29.00


r = 6%

0.2

0 d1 = 0.702662

N(d1) = 0.758867

d2 = 0.644927

N(d2) = 0.740513

c = $1.40


N(-d1) = 0.241133

N(-d2) = 0.259487

p = $0.25

Note that the payment of dividends reduces both the call and put values.

9 The price of a European call on a non-dividend paying stock with a strike of $50 is 7%. The st

is $51, the risk-free rate (all maturities) is 6%, and the time to maturity is one year. What is t

one year European put option on the stock with a strike price of $50?

Solution:

=d

=s

=d

=s



S0 = $51

K = $50

cost of cal 7%

rf = 6%

h = 1

Price of call = $3.50

To find value of a put, use put-call parity relationship.

p = c + K/(1+r) - S0 = ($0.33)

10 Consider a European call option on a non-dividend paying stock where the stock price is $29,

strike price is $29, the risk-free rate is 3.5% per annum, the volatility is 25% per annum, and

to maturity is six months.

a. Calculate u, d, and p for a two step tree.

b. Calculate the value of the option

Solution:

S0 = $29.00 $35.22

K = $29.00

rf = 3.5% $29.00

s 0.25

t = 0.5 $23.88

a. u = 1.2144323 Equation 5.14

d = 0.82343 Equation 5.15 0.82343p* = 0.4967337

b. value of call option = $3.04



Option value

$21

$0

$0

is $30 andover the

tion

ed

ce of



Option value

0 put

$26.00 call

0 put

$4.00 call

$14.00 put

0 call

o up by 9% or

value of a

bility

Option value

$7.35 call

$2.25 0

$10.43 0



or $95.

ue of a

and put-call parity

$37.05 0

$2.95

$0.00

$22.95

*delta - $95*delta

$16.34

$2.30

o go up by 8% or down by 9%.

Option value

0

$5.86 put

$13.60 put



price is $30, the risk free rate is

, the risk-free rate

sing the Black-Scholes



ock price

he price of a



the

he time

Option value

$6.22

0

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Chapter 5 Problems on Options