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Exam MFE/3F Solutions Chapter 10 Binomial Option Pricing: I

ActuarialBrew.com 2012 Page S10-4

Solution 7

C Chapter 10, Graphical Interpretation of the Binomial Formula

Delta is the slope of the line that passes through , dSd C and , uSu C :

( )h u d u dV V C C

e S u d Su Sd

Although we are not given the value of Su , we can still find the slope of the line becausewe have two points on the line:

(0, 26.58) and (40.53, 0)

The rise over run for the two points is:

0 ( 26.58) 26.580.656

40.53 0.00 40.53- -D = = =-

Solution 8D Chapter 10, Graphical Interpretation of the Binomial Formula

Delta is the slope of the line that passes through , dSd C and , uSu C :

( )h u d u dV V C C e

S u d Su Sd

Although we are not given the value of Sd , we can still find the slope of the line becausewe have two points on the line:

(0, 32.38) and (88.62, 26.62)

The rise-over-run for the two points is:26.62 ( 32.38) 59.00

0.665888.62 0.00 88.62

The intercept at the vertical axis is the value of the replicating portfolio if the stock priceis zero at the end of 1 year:

0.090 32.3829.5931

D + = -= -

Be B

The option can be replicated by purchasing 0.6658 shares of stock and borrowing$29.5931:

0 60(0.6658) 29.5931 10.35= D + = - =C S B

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Solution 9

E Chapter 10, Risk-Neutral Probability

To find the price using the risk-free rate of return, we must use risk-neutral probabilities,not realistic probabilities. The 45% probability of an upward movement is a red herring inthis question.

The values of u and d are:

( ) (0.05 0.02)(1) 0.30 1

( ) (0.05 0.02)(1) 0.30 1

1.39097

0.76338

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e

If the stock price moves up, then the option pays $0. If the stock price moves down, thenthe option pays $14.5634:

97.3678 0.000070 V

53.4366 14.5634

The risk-neutral probability of an upward movement is:

( ) (0.05 0.02)(1) 0.76338* 0.42556

1.39097 0.76338

d - -- -= = =- -r he d e

pu d

The value of the put option is:

[ ]0.05(1)( *) (1 *) 0.42556(0.00) (1 0.42556)(14.5634)7.96

- -= + - = + - =

rhu dV e p V p V e

Solution 10E Chapter 10, Arbitrage

Unless the following inequality is satisfied, arbitrage is available:

( )d -<

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The final row violates the inequality because the risk-free investment outperforms therisky asset in both scenarios. Therefore, the model described in Choice E permitsarbitrage.

Solution 11

B Chapter 10, One-Period Binomial Tree

This question can be answered using put-call parity:

00.08(0.5) 0.02(0.5)

( , ) ( , )

2.58 55 50 (55,0.5)

(55,0.5) 5.92

d - -

- -+ = +

+ = +=

rT T Eur Eur

Eur

Eur

C K T Ke S e P K T

e e P

P

Solution 12

B Chapter 10, Two-Period Binomial Tree


( ) (0.08 0.03)(1) 0.23 1

( ) (0.08 0.03)(1) 0.23 1

1.32313

0.83527

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.08 0.03)(1) 0.83527* 0.44275

1.32313 0.83527

d - -- -= = =- -r he d e

pu d

The stock price tree and its corresponding tree of option prices are:Stock American Call

126.0484 52.048495.2653 24.1392

72.0000 79.5723 11.0375 5.572360.1395 2.2775

50.2327 0.0000

The tree of prices for the call option is found by working from right to left. The rightmostcolumn is found as follows:

[ ][ ][ ]

0, 126.0484 74 52.04840, 79.5723 74 5.5723

0, 50.232 74 0.0000

- =- =- =

Max Max

Max

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The prices after 1 year are found using the risk-neutral probabilities:

[ ][ ]

0.08(1)

0.08(1)

(0.44275)(52.0484) (1 0.44275)(5.5723) 24.1392

(0.44275)(5.5723) (1 0.44275)(0.0000) 2.2775

-

-+ - =

+ - =

e

e

There is only one opportunity to call the option early, and this occurs after 1 year if thestock price has risen. Early exercise is not optimal since:

95.2653 74 21.2653 and 21.2653 24.1392- = < The current price of the option is:

[ ]0.08(1) (0.44275)(24.1392) (1 0.44275)(2.2775) 11.0375- + - =e

Solution 13

D Chapter 10, Two-Period Binomial Tree


( ) (0.08 0.03)(1) 0.23 1

( ) (0.08 0.03)(1) 0.23 1

1.32313

0.83527

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.08 0.03)(1) 0.83527* 0.44275

1.32313 0.83527

d - -- -= = =- -r he d e

pu d

The stock price tree and its corresponding tree of option prices are:

Stock American Call126.0484 52.0484

95.2653 24.139272.0000 79.5723 11.0375 5.5723

60.1395 2.277550.2327 0.0000

We are told that after 1 year, the stock price has increased. Therefore, the new stockprice is $95.2653. At this point the value of delta is:

0.03(1) 52.0484 5.5723 0.9704( ) 126.0484 79.5723

d - -- -D = = =

- -

h u dV V e eS u d

Therefore, the investor must hold 0.9704 shares at the end of 1 year in order to replicatethe call option.

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Stock American Put126.0484 0.0000

95.2653 0.000072.0000 79.5723 7.1299 0.0000

60.1395 13.860550.2327 23.7673

If the stock price initially moves down, then the resulting put price is $13.8605. Thisprice is in bold type above to indicate that it is optimal to exercise early at this node:

74 60.1395 13.8605- = The exercise value of 13.8605 is greater than the value of holding the option, which is:

[ ]0.08(1) (0.44275)0.0000 (1 0.44275)(23.7673) 12.2260- + - =e

The current value of the American option is:

[ ]0.08(1) (0.44275)0.0000 (1 0.44275)(13.8605) 7.1299- + - =e

Solution 16

B Chapter 10, Three-Period Binomial Tree

The values of h , u and d are:

( ) (0.10 0.08)(0.25) 0.32 0.25

( ) (0.10 0.08)(0.25) 0.32 0.25

9 /12 1 / 3 1 / 4

1.17939

0.85642

r h h

r h h

h

u e e

d e e


( ) (0.10 0.08)(0.25) 0.85642* 0.46009

1.17939 0.85642

d - -- -= = =- -r he d e

pu d


Stock 180.4548 European Call 80.4548153.0065 52.4458

129.7332 131.0371 30.8676 31.0371

110.0000 111.1055 17.1419 13.927194.2057 95.1525 6.2495 0.000080.6792 0.0000

69.0949 0.0000

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Although we have shown all of the prices in the tree above, we can also solve moredirectly for the price of the call option ( n is 3) as follows:

( )0 0

0

0.10(0.75) 2 3

( , ,0) ( *) (1 *) ( , , )

0 0 3(0.46009) (1 0.46009)(31.0371) (0.46009) (80.4548)

17.14

nr hn j n j j n j

j

nV S K e p p V S u d K hn

j

e

Solution 17

A Chapter 10, Three-Period Binomial Tree


( ) (0.10 0.08)(0.25) 0.32 0.25

( ) (0.10 0.08)(0.25) 0.32 0.25

1.17939

0.85642

d s

d s

- + - +

- - - -

= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.10 0.08)(0.25) 0.85642* 0.46009

1.17939 0.85642

d - -- -= = =- -r he d e

pu d


Stock 180.4548 American Call 80.4548153.0065 53.0065

129.7332 131.0371 31.1192 31.0371110.0000 111.1055 17.2548 13.9271

94.2057 95.1525 6.2495 0.000080.6792 0.0000

69.0949 0.0000

At each node, the value of holding the option must be compared with the value obtainedby exercising it. If the stock price reaches $153.0065, then it is optimal to exercise theoption early because:

0.10(0.25)153.0065 100 (0.46009) 80.4548 (1 0.46009) 31.037153.0065 52.4458

e

Working from right to left, we calculate the current value of the option to be $17.25.

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Solution 18



( ) (0.10 0.08)(0.25) 0.32 0.25

( ) (0.10 0.08)(0.25) 0.32 0.25

1.17939

0.85642

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.10 0.08)(0.25) 0.85642* 0.46009

1.17939 0.85642

d - -- -= = =- -r he d e

pu d


Stock 180.4548 European Put 0.0000153.0065 0.0000

129.7332 131.0371 1.3442 0.0000

110.0000 111.1055 6.3222 2.552694.2057 95.1525 10.8606 4.8475

80.6792 18.449469.0949 30.9051

Although we have shown all of the prices in the trees above, we can also solve moredirectly for the price of the European put option as follows:

( )0 0

0

0.10(0.75) 3 2

( , ,0) ( *) (1 *) ( , , )

(1 0.46009) (30.9051) 3(0.46009)(1 0.46009) (4.8475) 0 0 6.3222

nr hn j n j j n j

j


j

e

Solution 19

C Chapter 10, Three-Period Binomial Tree


( ) (0.10 0.08)(0.25) 0.32 0.25

( ) (0.10 0.08)(0.25) 0.32 0.25

1.17939

0.85642

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.10 0.08)(0.25) 0.85642* 0.46009

1.17939 0.85642

d - -- -= = =- -r he d e

pu d

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Solution 21

C Chapter 10, Three-Period Binomial Tree


( ) (0.04 0.03)(1 / 3) 0.30 1 / 3

( ) (0.04 0.03)(1 / 3) 0.30 1 / 3

1.19308

0.84377

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.04 0.03)(1 / 3) 0.84377* 0.45681

1.19308 0.84377

d - -- -= = =- -

r he d e p

u d

The stock price tree and its corresponding tree of option prices are below:

Stock 152.8451 American Put 0.0000128.1096 0.0000

107.3772 108.0955 2.7444 0.0000

90.0000 90.6020 8.7728 5.120175.9396 76.4475 14.0593 9.5525

64.0758 21.924254.0654 31.9346

At each node, the value of holding the option must be compared with the value obtainedby exercising it. If the stock price reaches $64.0758, then it is optimal to exercise theoption early because:

0.04(1 / 3)86 64.0758 (0.45681)9.5525 (1 0.45681)31.934621.9242 21.4227

e

When the stock price is $75.9396, the value of the put option is:

0.04(1 / 3) (0.45681)5.1201 (1 0.45681)21.9242 14.0593e

Solution 22

B Chapter 10, Three-Period Binomial Tree


( ) (0.04 0.03)(1 / 3) 0.30 1 / 3

( ) (0.04 0.03)(1 / 3) 0.30 1 / 3

1.19308

0.84377

d s

d s

- + - +

- - - -

= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.04 0.03)(1 / 3) 0.84377* 0.45681

1.19308 0.84377

d - -- -= = =- -

r he d e p

u d

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The stock price tree and its corresponding tree of option prices are below:

Stock 152.8451 American Put 0.0000128.1096 0.0000

107.3772 108.0955 2.7444 0.000090.0000 90.6020 8.7728 5.1201

75.9396 76.4475 14.0593 9.552564.0758 21.9242

54.0654 31.9346

We are told that during the first period, the stock price decreases. Therefore, the newstock price is $75.9396. At this point the value of delta is:

0.03(1 / 3) 5.1201 21.9242 0.627( ) 90.6020 64.0758

d - -- -D = = = -- -h u dV V e e

S u d

Solution 23

B Chapter 10, Four-Period Binomial Tree


( ) (0.07)(0.25) 0.30 0.25

( ) (0.07)(0.25) 0.30 0.25

1.18235

0.87590

d s

d s

- + +

- - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.07)(0.25) 0.87590* 0.46257

1.18235 0.87590

d - - -= = =- -r he d e

pu d

Since the stock does not pay dividends, the price of the American call option is equal tothe price of an otherwise equivalent European call option. Therefore, we do not need tocalculate the possible values of the stock at times 0.25, 0.50, and 0.75. Furthermore, sincethe option expires out of the money if the final price is less than $131, it makes sense tobegin by calculating the highest possible final prices:

4 4

3 3

90 90(1.18235) 175.8814

90 90(1.18235) (0.87590) 130.2961

= = == = =

uuuu

uuud

S u

S u d

The portions of the stock price tree that we need are below:

Stock 175.8814 130.2961

90.0000

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We did not complete the stock price tree above because the option is in-the-money at theend of the year only if the final stock price is $175.8814. Since it is not in-the-money ifthe stock price is $130.2961, it clearly will not be in-the-money at any lower stock price.Since the end of the year payoff is zero for all of the nodes except the uppermost one, it is

relatively quick to find the value of the option:

( )0 0

0

0.07(1) 4

( , ,0) ( *) (1 *) ( , , )

(0.46257) (175.8814 131) 1.9159

nr hn j n j j n j

j


j

e

Solution 24

E Chapter 10, Three-Period Binomial Tree

The values of h , u and d are provided in the question:

1 / 4 1 / 3 1 /121.200.90

hud


( ) (0.03 0.05)(1 /12) 0.90* 0.32778

1.20 0.90

d - -- -= = =- -r he d e

pu d

The tree of stock prices and the tree of option prices are below:

Stock 69.1200 European Call 36.120057.6000 24.4429

48.0000 51.8400 14.7663 18.840040.0000 43.2000 7.9074 10.1028

36.0000 38.8800 4.5924 5.880032.4000 1.9225

29.1600 0.0000

Although we have shown all of the prices in the trees above, we can also solve moredirectly for the price of the European option as follows:

( )0 0

0

0.03(3 / 12) 2

2 3

( , ,0) ( *) (1 *) ( , , )

[0 3(0.32778)(1 0.32778) (5.88)

3(0.32778) (1 0.32778)(18.84) (0.32778) (36.12)] 7.9074

nr hn j n j j n j

j


j

e

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Solution 25

D Chapter 10, Three-Period Binomial Tree

The values of h , u and d are provided in the question:

1 / 4 1 / 3 1 /12

1.200.90

h

ud


( ) (0.03 0.05)(1 /12) 0.90* 0.32778

1.20 0.90

d - -- -= = =- -r he d e

pu d

The tree of stock prices and the tree of option prices are below:

Stock 69.1200 American Call 36.120057.6000 24.6000

48.0000 51.8400 15.0000 18.840040.0000 43.2000 8.0052 10.2000

36.0000 38.8800 4.6242 5.880032.4000 1.9225

29.1600 0.0000

Early exercise is optimal at the three nodes shown in bold type. After two up jumps, theoption exercise value of 24.60 exceeds the option expected present value of 24.4429, whichwe calculated in the prior solution. After one up jump and one down jump, the optionexercise value of 10.20 exceeds the option expected present value of 10.1028, which wealso calculated in the prior solution. After one up jump, the option exercise value of 15.00

exceeds the option expected present value of: 0.03(1 /12) (0.32778)24.6000 (1 0.32778)10.2000 14.8828e

The value of the American call option is:

0.03(1 / 12) (0.32778)15.0000 (1 0.32778)4.6242 8.0052e

Solution 26

D Chapter 10, Multiple-Period Binomial Tree

It isnt very reasonable to expect us to work through an 8-period binomial tree during theexam, so there must be a shortcut to the answer. As it turns out, the option is in-the-money at only one final node. That makes it fairly easy to find the correct answer.

The value of h is 1/12 since the intervals are monthly periods. The values of u and d are:

( ) (0.07 0.02)(1 / 12) 0.35 1 /12

( ) (0.07 0.02)(1 /12) 0.35 1 /12

1.11094

0.90767

r h h

r h h

u e e

d e e

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( ) (0.07 0.02)(1 /12) 0.90767* 0.47476

1.11094 0.90767

d - -- -= = =- -r he d e

pu d

If the stock price increases each month, then at the end of 8 months, the price is:

8 8130 130(1.11094) 301.6170= =u

If the stock price increases for 7 of the months and decreases for 1 of the months, then theprice is:

7 7130 130(1.11094) (0.90767) 246.4319= =u d

Since the price of $246.4319 is out-of-the-money, all other lower prices are also out-of-the-money. This means that the only price at which the option expires in-the-money is thehighest possible price, which is $301.6170. Consequently, the value of the option is:

( )0 00

0.07(8 /12) 8

( , ,0) ( *) (1 *) ( , , )

[0 0 0 0 0 0 0 (0.47476) (301.6170 247)] 0.1345

- - -=

-

= -

= + + + + + + + -=

n

r hn j n j j n j j

nV S K e p p V S u d K hn j

e

Solution 27

C Chapter 10, Multiple-Period Binomial Tree

If we work through the entire binomial tree, this is a very time-consuming problem. Evenusing the direct method takes a lot of time since the put option provides a positive payoff

at multiple final nodes. But once we notice that the value of the corresponding call optioncan be calculated fairly quickly, we can use put-call parity to find the value of the putoption.

The value of h is 1/12 since the intervals are monthly periods. The values of u and d are:

( ) (0.07 0.02)(1 / 12) 0.35 1 /12

( ) (0.07 0.02)(1 /12) 0.35 1 /12

1.11094

0.90767

r h h

r h h

u e e

d e e


( ) (0.07 0.02)(1 /12) 0.90767* 0.474761.11094 0.90767

d - -- -= = =- -

r he d e p u d

If the stock price increases each month, then at the end of 8 months, the price is:

8 8130 130(1.11094) 301.6170= =u

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We can use put-call parity to find the value of the corresponding put option:

00.07(4 /12) 0.02(4 /12)

( , ) ( , )

0 200 130 ( , )

( , ) 66.25

d - -

- -+ = +

+ = +=

rT T Eur Eur

Eur

Eur

C K T Ke S e P K T

e e P K T

P K T

Solution 29

C Chapter 10, Replication


( ) (0.07 0.02)(1 / 12) 0.35 1 /12

( ) (0.07 0.02)(1 /12) 0.35 1 /12

1.11094

0.90767

r h h

r h h

u e e

d e e


( ) (0.07 0.02)(1 /12) 0.90767* 0.47476

1.11094 0.90767

d - -- -= = =- -r he d e

pu d

If the stock price increases during each of the first 3 months, then the price is:

3 3130 130(1.11094) 178.2422= =u

The relevant portions of the stock price tree and the option price tree are shown below:

Stock European Put198.0157 1.9843

178.2422

161.786 38.2141

The amount that must be lent at the risk-free rate of return is:

0.07(1/12) 1.11094(38.2141) 0.90767(1.9843) 198.841.11094 0.90767

- -- -= = =- -rh d uuV dV B e e

u d

Solution 30

C Chapter 10, Option on a Stock Index

The value of h is 1 since the intervals are annual periods. The values of u and d are:

( ) (0.05 0.04)(1) 0.34 1

( ) (0.05 0.04)(1) 0.34 1

1.41907

0.71892

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.05 0.04)(1) 0.71892* 0.41581

1.41907 0.71892

d - -- -= = =- -r he d e

pu d

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The tree of stock index prices and the tree of option prices are below:

Stock Index 285.7651 European Call 190.7651201.3753 103.1124

141.9068 144.7735 51.7241 49.7735100.0000 102.0201 24.7855 19.6869

71.8924 73.3447 7.7868 0.000051.6851 0.0000

37.1577 0.0000


( )0 0

0

0.05(3) 2 3

( , ,0) ( *) (1 *) ( , , )

[0 0 3(0.41581) (1 0.41581)(49.7735) (0.41581) (190.7651)]

24.7855

nr hn j n j j n j

j


j

e

Solution 31

A Chapter 10, Option on a Stock Index


( ) (0.05 0.04)(1) 0.34 1

( ) (0.05 0.04)(1) 0.34 1

1.41907

0.71892

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.05 0.04)(1) 0.71892* 0.41581

1.41907 0.71892

d - -- -= = =- -r he d e

pu d


Stock Index 285.7651 European Put 0.0000201.3753 0.0000

141.9068 144.7735 6.6872 0.0000100.0000 102.0201 17.8608 12.0338

71.8924 73.3447 27.3813 21.655351.6851 40.7083

37.1577 57.8423

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( )0 0

0

0.05(3) 3 2

( , ,0) ( *) (1 *) ( , , )

[(1 0.41581) (57.8423) 3(0.41581)(1 0.41581) (21.6553) 0 0] 17.8608

- - -

=

-

= -

= - + - + +=

n

r hn j n j j n j

j


j

e

Solution 32

A Chapter 10, Option on a Stock Index


( ) (0.05 0.04)(1) 0.34 1

( ) (0.05 0.04)(1) 0.34 1

1.41907

0.71892

d s

d s

- + - +

- - - -= = =

= = =

r h h

r h h

u e e

d e e


( ) (0.05 0.04)(1) 0.71892* 0.41581

1.41907 0.71892

d - -- -= = =- -r he d e

pu d


Stock Index 285.7651 American Put 0.0000201.3753 0.0000

141.9068 144.7735 6.6872 0.0000100.0000 102.0201 18.6657 12.0338

71.8924 73.3447 28.8298 21.655351.6851 43.314937.1577 57.8423

If the stock price reaches $51.6851 after 2 years, then early exercise is optimal since theexercise value of the option exceeds the expected present value of the option at that node.

We have:

0.05 (0.41581)(12.0338) (1 0.41581)(43.3149) 28.8298e

The value of the American put option is:

0.05 (0.41581)(6.6872) (1 0.41581)(28.8298) 18.6657e

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Solution 33

E Chapter 10, Options on Currencies

The value of h is 1/4 since the intervals are quarterly periods. The values of u and d are:

( ) (0.05 0.09)0.25 0.15 0.25

( ) (0.05 0.09)0.25 0.15 0.25

1.06716

0.91851

s

s

- + - +

- - - -

= = =

= = =

f

f

r r h h

r r h h

u e e

d e e


( ) (0.05 0.09)(0.25) 0.91851* 0.48126

1.06716 0.91851

d - -- -= = =- -r he d e

pu d

The tree of euro prices and the tree of option prices are below:

Euros 1.4584 European Call 0.35841.3666 0.2499

1.2806 1.2552 0.1565 0.15521.2000 1.1762 0.0924 0.0738

1.1022 1.0804 0.0351 0.00001.0124 0.0000

0.9299 0.0000


( )0 0

0

0.05(0.75) 2 3

( , ,0) ( *) (1 *) ( , , )

[0 0 3(0.48126) (1 0.48126)(0.1552) (0.48126) (0.3584)] 0.0924

- - -

=

-

= -

= + + - +=

n

r hn j n j j n j

j


j

e

Solution 34


The value of h is 1/4 since the intervals are quarterly periods. The values of u and d are:

( ) (0.05 0.09)0.25 0.15 0.25

( ) (0.05 0.09)0.25 0.15 0.25

1.06716

0.91851

s

s

- + - +

- - - -

= = =

= = =

f

f

r r h h

r r h h

u e e

d e e


( ) (0.05 0.09)(0.25) 0.91851* 0.48126

1.06716 0.91851

d - -- -= = =- -r he d e

pu d

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The tree of euro prices and the tree of option prices are below:

Euros 1.4584 American Call 0.35841.3666 0.2666

1.2806 1.2552 0.1806 0.15521.2000 1.1762 0.1044 0.0762

1.1022 1.0804 0.0362 0.00001.0124 0.0000

0.9299 0.0000

The three nodes that are in bold type in the American call option tree above indicate thatearly exercise is optimal at those nodes since the option exercise value exceeds the optionexpected present value at those nodes.


[ ]0.05(0.25) (0.48126)(0.1806) (1 0.48126)(0.0362) 0.1044- + - =e

Solution 35

D Chapter 10, Options on Currencies

The dollar call option is an option to buy a dollar. Since it is yen-denominated, the strike price is denominated in yen. Since it is at-the-money, the strike price is 118 yen.

Since the call is yen-denominated, we treat the yen as the base currency.

Unless told otherwise, we assume that the price of a yen-denominated option isdenominated in yen. Therefore, the answer choices are in yen, not dollars.

The value of h is 1/3 since the intervals are 1/3 of a year. The values of u and d are:

$

$

( ) (0.01 0.06)(1 / 3 ) 0.11 1 / 3

( ) (0.01 0.06)(1 / 3 ) 0.11 1 / 3

1.04796

0.92295

s

s

- + - +

- - - -= = =

= = =

r r h h

r r h h

u e e

d e e


$( ) (0.01 0.06)(1/ 3) 0.92295* 0.48413

1.04796 0.92295

- -- -= = =- -

r r he d e p

u d

The tree of prices for a dollar and the tree of option prices are below. Both trees areexpressed in yen:

Dollar 135.8037 American Call 17.8037129.5891 11.5891

123.6588 119.6048 5.9901 1.6048118.0000 114.1315 3.0824 0.7744

108.9086 105.3382 0.3736 0.0000100.5177 0.0000

92.7733 0.0000

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The node that is in bold type in the American call option tree above indicates that earlyexercise is optimal since the option exercise value exceeds the option expected presentvalue at that node.


[ ]0.01(1 / 3) (0.48413)(5.9901) (1 0.48413)(0.3736) 3.0824- + - =e

Solution 36


The dollar put option is an option to sell a dollar. Since it is yen-denominated, the strike price is denominated in yen. Since it is at-the-money, the strike price is 118 yen.

Since the put is yen-denominated, we treat the yen as the base currency.

The value of h is 1/3 since the intervals are 1/3 of a year. The values of u and d are:

$

$

( ) (0.01 0.06)(1 / 3 ) 0.11 1 / 3

( ) (0.01 0.06)(1 / 3 ) 0.11 1 / 3

1.04796

0.92295

s

s

- + - +

- - - -= = =

= = =

r r h h

r r h h

u e e

d e e


$( ) (0.01 0.06)(1/ 3) 0.92295* 0.48413

1.04796 0.92295

- -- -= = =- -

r r he d e p

u d

The tree of prices for a dollar and the tree of option prices are below. Both trees areexpressed in yen:

Dollar 135.8037 American Put 0.0000129.5891 0.0000123.6588 119.6048 3.3472 0.0000

118.0000 114.1315 8.2741 6.5101108.9086 105.3382 12.9513 12.6618

100.5177 19.080092.7733 25.2267

It is not rational to exercise this American put option early.

The value of the American put option is:

[ ]0.01(1 / 3) (0.48413)(3.3472) (1 0.48413)(12.9513) 8.2741-

+ - =e

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Solution 37

A Chapter 10, Options on Currencies

The pound call is an option to buy a pound. Since it is franc-denominated, the strike priceis denominated in francs.

Since the call is franc-denominated, we treat the franc as the base currency. For this problem, the subscript f denotes franc.


( ) ( 0.05 0.07)(1) 0.12 1

( ) ( 0.05 0.07 )(1) 0.12 1

1.10517

0.86936

s

s

- + - +

- - - -

= = =

= = =

f

f

r r h h

r r h h

u e e

d e e


( ) (0.05 0.07)(1) 0.86936* 0.47004

1.10517 0.86936

- -- -= = =- - f r r he d e

pu d

The tree of prices for a pound and the tree of option prices are below. Both trees areexpressed in francs:

Pound American Call2.9069 0.5469

2.6303 0.27032.3800 2.2867 0.1209 0.0000

2.0691 0.00001.7988 0.0000

The node that is in bold type in the American call option tree above indicates that early

exercise is optimal at that node since the option exercise value exceeds the optionexpected present value at that node.


[ ]0.05(1) (0.47004)(0.2703) (1 0.47004)(0.000) 0.1209- + - =e

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Solution 38

D Chapter 10, Options on Futures Contracts

Since there is only one period, it doesnt make any difference whether the call option is aEuropean option or an American option.

The lease rate was thrown in as a red herring. It could be used to calculate the current price of gold, but we dont need to know the current price of gold. But if you are curious,it is:

d - - - -= = =( r )T (0.07 0.04)(1)0 0 ,T S F e 600e 582.27

The values of F u and F d are:

0.12 1

0.12 1

1.12750

0.88692

s

s - -= = =

= = =

hF

hF

u e e

d e e


1 1 0.88692* 0.47004

1.12750 0.88692- -= = =- -

F

F F

d p

u d

The futures price tree and the call option tree are below:

0,1F 1,1F Call Option

676.4981 56.4981600.0000 24.7608

532.1523 0.0000

If the futures price rises to $676.4981, then the call option has a payoff of:676.4981 620 56.4981- =

The price of the call option is:

0.07(1) (0.47004)(56.4981) (1 0.47004)(0.0000) 24.7608e

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Solution 39

A Chapter 10, Options on Futures Contracts

The value of h is 1/3 since the intervals are 1/3 of a year. The values of F u and F d are:

0.30 1 / 3

0.30 1 / 31.18911

0.84097

s

s - -= = == = =

hF

hF

u e ed e e


1 1 0.84097* 0.45681

1.18911 0.84097- -= = =- -

F

F F

d p

u d


0,1F 13 ,1

F 23 ,1

F 1,1F American Call

1,681.3806 681.38061,413.9825 413.9825

1,189.1099 1,189.1099 229.5336 189.10991,000.0000 1,000.0000 122.4206 84.3943

840.9651 840.9651 37.6628 0.0000707.2224 0.0000

594.7493 0.0000

If the futures price reaches $1,413.9825, then it is optimal to exercise the call option earlysince the option exercise value exceeds the expected present value of the option at thatnode.

The value of the American option is:

0.07(1 / 3) (0.45681)(229.5336) (1 0.45681)(37.6628) 122.4206e

Solution 40


The values of F u and F d are:

0.30 1 / 3

0.30 1 / 3

1.18911

0.84097

s

s - -

= = =

= = =

hF

hF

u e e

d e e


1 1 0.84097* 0.45681

1.18911 0.84097- -= = =- -

F

F F

d p

u d

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0,1F 13 ,1

F 23 ,1

F 1,1F American Call

1,681.3806 681.38061,413.9825 413.9825

1,189.1099 1,189.1099 229.5336 189.10991,000.0000 1,000.0000 122.4206 84.3943840.9651 840.9651 37.6628 0.0000

707.2224 0.0000594.7493 0.0000

If the futures price reaches $1,413.9825, then it is optimal to exercise the call option earlysince the option exercise value exceeds the expected present value of the option at thatnode.

The number of futures contracts that the investor must be long is:

229.5336 37.6628 0.5511( ) 1,189.1099 840.9651- -D = = =- -u dF F

V V F u d

Solution 41

E Chapter 10, Two-Period Binomial Model

Since the stock does not pay dividends, the price of the American call option is equal tothe price of an otherwise equivalent European call option.

The stock price tree and the associated option payoffs at the end of 2 years are:

Call Payoff75.6877 27.687758.3605

45.0000 50.7386 2.738639.1230

34.0135 0.0000


( ) (0.06 0.00)1 0.8694* 0.45014

1.2969 0.8694

r he d e p

u d

d - -- -= = =- -

The value of the call option is:

( ) ( )0 0

00.06(2) 2

( , ,0) ( *) (1 *) ( , , )

(0.45014) (27.6877) 2(0.45014)(1 0.45014)(2.7386) 0

6.1783

nr hn n i i n i i

i


i

e

- - -

=-

= - = + - +

=

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Solution 42

E Chapter 10, Three-Period Binomial Model for Currency

When the type of binomial model is not specified in the question, we should assume thatthat we are to use the textbooks standard binomial model. Question 5 on the SOAssample exam is an example of a question where we are expected to assume that thetextbooks standard binomial model applies.


( ) (0.10 0.04)0.25 0.32 0.25

( ) (0.10 0.04)0.25 0.32 0.25

1.19125

0.86502

f

f

r r h h

r r h h

u e e

d e e

s

s

- + - +

- - - -

= = =

= = =


( ) (0.10 0.04)(0.25) 0.86502* 0.46009

1.19125 0.86502

r he d e p

u d

d - -- -= = =- -

The tree of pound prices and the tree of option prices are below:

Pound 3.2119 American Put 0.00002.6962 0.0000

2.2634 2.3323 0.0988 0.00001.9000 1.9579 0.2629 0.1877

1.6435 1.6936 0.4151 0.35641.4217 0.6283

1.2298 0.8202

If the exchange rate falls to 1.4217 at the end of 6 months, then early exercise is optimal.

The value of the American put option at time 0 is 0.2629.

Solution 43

D Chapter 10, Risk-Neutral Probability

When the type of binomial model is not specified in the question, then we should assumethat that we are to use the textbooks standard binomial model.


( ) (0.085 0.022)(0.25) 0.28 0.25

( ) (0.085 0.022)(0.25) 0.28 0.25

1.16853

0.88316

r h h

r h h

u e e

d e e


( ) (0.085 0.022)(0.25) 0.88316* 0.46506

1.16853 0.88316

r he d e p

u d

d - -- -= = =- -

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The risk-neutral probability of a downward movement is:

(1 *) 1 0.46506 0.53494 p- = - =

Solution 44

D Chapter 10, Replication

The call option pays 11 in the up-state and 0 in the down-state:

[38 27,0] 11

[21 27,0] 0u

d

V Max

V Max

The dividend rate is zero, so the number of shares needed to replicate the option is:

0 1 11 0 0.6471( ) 38 21

h hu d u dV V V V e e eS u d Su Sd

Solution 45

B Chapter 10, One-Period Binomial Tree

A straddle consists of a long call and a long put on the same stock, where both optionshave the same strike price and the same expiration date. The question describes the

payoff of a straddle, so it is not necessary to know the definition of a straddle to answerthis question.

If the stock price moves up, then the straddle pays $25. If the stock price moves down,then the straddle pays $2:

95 25 = -70 9575 V 68 2 = -70 68

In this case, 95/75u and 68/ 75d .


( ) (0.07 0.00)1 68/75* 0.46067

95 /75 68 / 75

r he d e p

u d

The value of the straddle is:

[ ]- -= + - = + - =

0.07(1)( *) (1 *) 0.46067(25) (1 0.46067)(2)

11.74

rh u dV e p V p V e

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The cost now of replicating the payoffs resulting from buying the put and selling the callis equal to the cost of establishing a position consisting of A shares of Stock A, B shares ofStock B, and C lent at the risk-free rate:

+ + = - + - + =50 50 50 ( 0.9) 50 ( 0.155) 57.2332 4.48 A B C

Solution 47

D Chapter 10, Replication

The end-of-year payoffs of the call and put options in each scenario are shown in the tablebelow. The rightmost column is the payoff resulting from buying the call option andselling the put option.

End of YearPrice ofStock A

End of YearPrice ofStock B

(45) A P

Payoff

(40) BC

Payoff-(40) (45) B AC P

Payoff

Scenario 1 $30 $0 15 0 15

Scenario 2 $40 $50 5 10 5

Scenario 3 $50 $40 0 0 0

We need to determine the cost of replicating the payoffs in the rightmost column above.We can replicate those payoffs by determining the proper amount of Stock A, Stock B, andthe risk-free asset to purchase.

Lets define the following variables:

===

Number of shares of Stock A to purchaseNumber of shares of Stock B to purchase Amount to lend at the risk-free rate

A BC

We have 3 equations and 3 unknown variables:

+ + = -+ + =+ + =

0.05

0.05

0.05

Scenario 1: 30 0 15

Scenario 2: 40 50 5

Scenario 3: 50 40 0

A B Ce

A B Ce

A B Ce

The two equations below are obtained by subtracting the first equation above from thesecond and third equations above:

+ =+ =

10 50 2020 40 15

A B A B

We can solve for B by subtracting twice the first equation above from the second equation:

- = -

=

60 255

12 B

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We can put this value of B into one of the two equations found earlier in order to find thevalue of A :

+ =

+ =

= -

10 50 205

10 50 2012

112

B

A

A

We can now find the value of C using one of the 3 original equations:

+ + = -- + + = -

= -

0.05

0.05

30 0 151

30 0 151211.89

A B Ce

Ce

C

Since C is negative, the payoffs resulting from buying the call and selling the put are

replicated by borrowing $11.89, which is equivalent to lending $11.89.The cost now of replicating the payoffs resulting from buying the call and selling the putis equal to the cost of establishing a position consisting of A shares of Stock A, B shares ofStock B, and C lent at the risk-free rate. Since the price of Stock A is $40 and the price ofStock B is $30, we have:

-+ + = + - = -1 540 30 40 30 11.89 2.7212 12

A B C

Solution 48

E Chapter 10, Replication

The risk-free asset can be replicated with the three stocks. Lets find the cost ofreplicating an asset that is certain to pay a fixed amount at time 1. We can choose anyfixed amount, and in this solution we use $100. We begin by defining the followingvariables:

===

Number of shares of Stock A to purchaseNumber of shares of Stock B to purchaseNumber of shares of Stock C to purchase

A BC

We have 3 equations and 3 unknown variables. The $100 on the right side of theequations below is the fixed amount certain to be paid by our risk-free asset:

+ + =+ + =

+ + =

Scenario 1: 200 0 0 100Scenario 2: 50 0 100 100Scenario 3: 0 300 50 100

B C A B C

A B C

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The solution to these equations is:

==

=

0.50.755

24

AC

B

The cost now of replicating an asset that is certain to pay $100 at time 1 is equal to thecost of establishing a position consisting of A shares of Stock A, B shares of Stock B, and C shares of Stock C. Since the price of Stock A is $100, the price of Stock B is $50, and theprice of Stock C is $40, the cost of the risk-free asset is:

+ + = + + =5100 50 40 100 0.5 50 40 0.75 90.416724

A B C

Since t is one year, we can now solve for r :

==

90.4167 1000.1007

rter

Solution 49

E Chapter 10, Replication

The end-of-year payoffs of the call and put options in each scenario are shown in the tablebelow. The rightmost column is the payoff resulting from buying the put option andselling the call option.

Scenario

End of Year

Price of Stock A

End of Year

Price ofStock B

(100) A P

Payoff

(200) BC

Payoff

-(100) (200) A B P C Payoff

1 $200 $0 0 0 0

2 $50 $0 50 0 50

3 $25 $300 75 100 25

We need to determine the cost of replicating the payoffs in the rightmost column above.We can replicate those payoffs by determining the proper amount of Stock A, Stock B, andthe risk-free asset to purchase.

Lets define the following variables:===

Number of shares of Stock A to purchaseNumber of shares of Stock B to purchase Amount to lend at the risk-free rate

A BC

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Since Stock A pays a $10 dividend at time 0.5, each share of Stock A that is purchasedprovides its holder with the final price of Stock A at time 1 plus the accumulated value of$10. Since Stock B pays continuously compounded dividends of 5%, each share of stock B

purchased at time 0 grows to 0.05e shares of Stock B at time 1.

We have 3 equations and 3 unknown variables:

+ + + = + + + = + + + = -

0.08(0.5) 0.08

0.08(0.5) 0.08

0.08(0.5) 0.05 0.08

Scenario 1: 200 10 0 0

Scenario 2: 50 10 0 50

Scenario 3: 25 10 300 25

e A B Ce

e A B Ce

e A e B Ce

Subtracting the first equation from the second equation allows us to solve for A :

+ - + =

= = -+ - -

0.08(0.5) 0.08(0.5)

0.04 0.04

50 10 200 10 50

50 1350 10 200 10

e A e A

A e e

The equation associated with Scenario 1 can now be used to find C :

0.08(0.5) 0.08

0.08(0.5) 0.08 0.08(0.5) 0.08

200 10 0 0

1200 10 200 10 64.7437

3

e A B Ce

C e Ae e e

We use the equation associated with Scenario 3 to find B :

+ + + = - - + + + = - = -

0.08(0.5) 0.05 0.08

0.08(0.5) 0.05 0.08

25 10 300 25

125 10 300 64.7437 25

30.2642

e A e B Ce

e e B e

The cost now of replicating the payoffs resulting from buying the put and selling the callis equal to the cost of establishing a position consisting of A shares of Stock A, B shares ofStock B, and C lent at the risk-free rate. Since the price of Stock A is $100 and the priceof Stock B is $75, we have:

-+ + = + - + =1100 75 100 75 ( 0.2642) 64.7437 11.59313

A B C

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Solution 50

A Chapter 10, Delta

The delta of the put option is 0.3275. The delta of the corresponding call option satisfiesthe following formula:

d --

D - D =D - - =D =

0.07(0.75)( 0.3275)

0.6214

T Call Put

Call

Call

e

e

Solution 51

A Chapter 10, Replication

Since we are given the value of the put option and the delta of the put option, we can findthe amount that must be lent to replicate the put option:

= D += - +=

0

3.89 52( 0.3275)

20.92

Put Put Put

Put

Put

V S B

B

B

The put option is replicated by lending $20.92, so =20.92 X . We can now solve for Call B :

-

-- =- == -

0.10(0.75)20.92 50

25.47

rT Put Call

Call

Call

B B Ke

B e

B

Therefore $25.47 must be borrowed to replicate the call option.

Solution 52

E Chapter 10, Greeks in the Binomial Model


d s

d s

- + - +

- - - -= = =

= = =

( ) (0.11 0.04)(1) 0.32 1

( ) (0.11 0.04)(1) 0.32 1

1.47698

0.77880

r h h

r h h

u e e

d e e

The risk-neutral probability of an upward movement is:d - -- -= = =- -

( ) (0.11 0.04)(1) 0.77880* 0.42068

1.47698 0.77880

r he d e p

u d

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Stock American Put87.2589 0.0000

59.0792 0.000040.0000 46.0110 5.6299 0.0000

31.1520 10.848024.2612 17.7388

If the stock price initially moves down, then the resulting put price is $10.8480. Thisprice is in bold type above to indicate that it is optimal to exercise early at this node:

- =42 31.1520 10.8480The exercise value of 10.8480 is greater than the value of holding the option, which is:

[ ]- + - =0.11(1) (0.42068)0.0000 (1 0.42068)(17.7388) 9.2060e

The current value of the American option is:[ ]- + - =0.11(1) (0.42068)0.0000 (1 0.42068)(10.8480) 5.6299e

We need to calculate 3 values of delta:

d

d

d

- -

- -

- -

- -D = = = -- -- -D = = =--- -D = = = ---

0.04 1

0.04 12

0.04 12

0 10.8480( ,0) 0.3732

59.0792 31.15200.0000 0.0000

( , ) 0.000087.2589 46.0110

0.0000 17.7388( , ) 0.

46.0110 24.2612

h u d

h uu ud

h ud dd

V V S e e

Su SdV V

Su h e eSu SudV V

Sd h e eSud Sd

7836

We can now calculate gamma:

D - D - -G G = = =- -( , ) ( , ) 0 ( 0.7836)

( ,0) ( , ) 0.028159.0792 31.1520h

Su h Sd hS S h

Su Sd

We can now estimate theta:

q

-- - - D - G =

-- - - - -=

= -

2

2

( )( ) ( ,0) ( ,0)

2( ,0)2

(46.0110 40.0000)

0 5.6299 (46.0110 40.0000)( 0.3732) (0.0281)22 1

1.9467

udSud S

V V Sud S S S S

h

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Solution 53

B Chapter 10, Greeks in the Binomial Model

The question does not tell us when the call option expires, so we cannot assume that itexpires in 6 months.

The risk-neutral probability of an upward movement is:d - -- -= = =- -

( ) (0.10 0.03)0.25 61.60/70* 0.4637

82.38/70 61.60/ 70

r he d e p

u d

The time 0 value of the option is:

[ ]- - = + - = + - =

0.10 0.250 ( *) (1 *) [0.4637 14.91 (1 0.4637) 2.72]

8.1658

rhu dC e p C p C e

The delta-gamma-theta approximation is:

2

0

2

( )( ) ( ,0) ( ,0) 2 ( ,0)2

(72.49 70)8.1658 (72.49 70)(0.5822) (0.0233) 2(0.25)( 7.3361)

26.0197

udSud S

C C Sud S S S h S

Solution 54

B Chapter 10, Greeks in the Binomial Model


d - -- -= = =- -( ) (0.10 0.03)0.25 61.60/70* 0.4637

82.38/70 61.60/ 70r he d e p

u d

The time 0 value of the option is:

[ ]- - = + - = + - =

0.10 0.250 ( *) (1 *) [0.4637 14.91 (1 0.4637) 2.72]

8.1658

rhu dC e p C p C e

The delta-gamma-theta approximation is:

2

0

2

( )( ) ( ,0) ( ,0) ( ,0)

2

(82.38 70)8.1658 (82.38 70)(0.5822) (0.0233) (0.25)( 7.3361)

215.32

uSu S

C C Su S S S h S

The difference between the estimate and the actual value of $14.91 (which can be readfrom the tree of option prices) is:

- =15.32 14.91 0.41

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Solution 55

E Chapter 10, Three-Period Binomial Tree


d - - --= = =- -

( 0.10 0.065) 1 210( )

300375 210300 300

* 0.6102

r h ee d p u d

The tree of prices for the American call option is shown below:

American Call 285.9375168.7500

98.6519 28.125057.4945 15.5292

8.5744 0.00000.0000

0.0000

Early exercise is optimal if the stock price increases to $468.75 at the end of 2 years. Thisis indicated by the bolding of that node in the tree above.

The price of the option is:

0.10 1 0.6102 98.6519 (1 0.6102) 8.5744 57.4945e

Solution 56

A Chapter 10, Greeks in the Binomial Model

We need to calculate the two possible values of delta at the end of 1 year:

0.065 12

0.065 12

168.75 15.5292( , ) 0.6961

468.75 262.50

15.5292 0.0000( , ) 0.1260

262.50 147.00

h uu ud

h ud dd

V V Su h e e

Su SudV V

Sd h e eSud Sd

We can now calculate gamma:

( , ) ( , ) 0.6961 0.1260( ,0) ( , ) 0.003455

375 210hSu h Sd h

S S hSu Sd

Solution 57



d - - --= = =- -

( 0.10 0.065) 1 210( )300

375 210300 300

* 0.6102r h ee d

pu d

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The tree of prices for the American put option is shown below:

American put 0.00000.0000

13.8384 0.000037.6195 39.2367

85.0000 111.2500148.0000

192.1000

It is not necessary to calculate all of the prices in the tree above to answer this question,but we included the full tree for the sake of completeness.

Early exercise is optimal if the stock price falls to $210 at the end of 1 year or falls to $147at the end of 2 years. This is indicated by the bolding of those two nodes in the treeabove.

The only year-2 node at which early exercise occurs is the bottom node. But to reach thatnode, the stock must first pass through the year-1 bottom node. Since it will be exercisedat the end of year 1 if the stock price reaches $210, there is no possibility of the stockbeing exercised at the end of year 2. Therefore, =2 0 p .

Solution 58


We are given that the ratio of the factors applicable to the futures price is:

= 43

F

F

ud

The formula for the risk-neutral probability of an up move can be used to find F u and

F d :

-= --

=-

-=

-=

= =

1

1

43

1*

*

11

3 10.9

41.2

3

F

F F

F F

F F

F

F

F F d

d du dd d

d

F

F F

d p

u d

p

d

u d

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The tree of futures prices is therefore:

Futures Prices 115.200096.0000

80.0000 86.4000

72.0000 64.8000

The tree of prices for the European put option is:

European Put 0.00000.0000

8.5399 0.000013.1342

20.2000

The price of the European put is:

0.05 0.5 (1/3)0.000 (2/ 3)13.1342 8.5399e The tree of prices for the American put option is:

American Put 0.00000.0000

8.5399 0.000013.1342

20.2000

The price of the American put is:

0.05 0.5

(1/3)0.0000 (2/3)13.1342 8.5399e

Early exercise of the American option is never optimal, so the prices of the Europeanoption and the American option are the same.

The price of the American put option exceeds the price of the European put option by:

8.5399 8.5399 0.0000

Solution 59

C Chapter 10, Options on Futures Contracts

We are given that the ratio of the factors applicable to the futures price is:

=0.6F F

du

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The price of the American call option exceeds the price of the European call option by:

- =10.3283 10.1370 0.1912

Solution 60

C Chapter 10, American Put Option


( ) (0.06 0.00)0.5 0.35 0.5

( ) (0.06 0.00)0.5 0.35 0.5

1.3198

0.8045

r h h

r h h

u e e

d e e

The risk-free probability of an upward movement is:

( ) (0.06 0.00)0.5 0.8045* 0.4384

1.3198 0.8045

r he d e p

u d

The stock price tree is:98.99

75

60.34

Lets use trial and error. We begin with the middle strike price of $93. The value ofimmediate exercise is:

- =93 75 18The value of holding on to the option is:

- - - =0.06 0.5(93 60.34) (1 0.4384) 17.80e

Since 17.25 17.00 , the investor will not exercise immediately if the strike price is $92.If the strike price is less than $92, then the option is even less in-the-money, meaningthat it is even less likely to be exercised early.

Therefore, the smallest integer-valued strike price for which the investor will exercise theput option at the beginning of the period is $93.

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Does the solution above suggest that the correct answer is $57? No, because theinequality above is only valid if K 60.34 , which is obviously contradicted by a solution ofK 57.235 = . If the strike price is less than $60.34, then both 6-month nodes produce a

payoff, and the inequality becomes:

0.06(0.5 ) 0.06(0.5 )75 K (98.99 K ) 0.4384e (60.34 K ) (1 0.4384)e75 K (98.99 K ) 0.4255 (60.34 K ) 0.545075 K 42.1168 0.4255K 32.8832 0.5450K K 0

- -- - + - -- - + - - - + -

Solution 62

E Chapter 10, American Put Option


( ) (0.06 0.08)0.5 0.35 0.5

( ) (0.06 0.08)0.5 0.35 0.5

1.2681

0.7730

r h h

r h h

u e e

d e e

The risk-free probability of an upward movement is:

( ) (0.06 0.08)0.5 0.7730* 0.4384

1.2681 0.7730

r he d e p

u d

The stock price tree is:

95.10

75

57.97

Lets use trial and error. We begin with the middle strike price of $96. The value ofimmediate exercise is:

- =93 75 21 The value of holding on to the option is:

- - - + - - =0.06 0.5 0.06 0.5(96 95.10) 0.4384 (96 57.97) (1 0.4384) 21.10e e

Since

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Since 25.00 24.99 , the option is exercised immediately if the strike price is $100.Therefore, the smallest integer-valued strike price for which the investor will exercise theput option at the beginning of the period is $100.

Alternate Solution

The strike price K , for which an investor will exercise the put option at the beginning ofthe period must be at least $75, since otherwise the payoff to immediate exercise would be

zero. Since we are seeking the lowest strike price that results in immediate exercise, letsbegin by determining whether there is a strike price that is greater than $75 but less than$95.10 that results in immediate exercise.

If there is strike price that is less than $95.10, then the value of exercising now mustexceed the value of holding the option:

0.06(0.5)

0.06(0.5)

75 ( 57.97)(1 *)

75 ( 57.97)(1 0.4384)75 0.5450 31.59

0.4550 43.41

95.39

K K p e

K K e K K

K

K

The inequality above suggests that the strike price must be greater than 95.39, but theoriginal inequality was predicated on the assumption that

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Solution 63

B Chapter 10, Arbitrage in the Binomial Model

If the stock price moves up, then the option pays Max[0, 110 100] = $10. If the stockprice moves down, then the option pays Max[0, 80 100] = $0:

110 10100 V

80 0

Lets use the replication method to find the value of the option:

0.10(1)

0.04(1)

10 00.3016

( ) 110 80110(0) 80(10)

25.6211110 80

h u d

rh rhd u d u

V V e e

S u duV dV SuV SdV

B e e eu d Su Sd

If no arbitrage is available, then the price of the option is:

= D + = - =0 100(0.3016) 25.6211 4.5402V S B The actual option price of $4.00 is less than $4.5402, so the options price is too low.

Arbitrage is obtained by buying the option (buy low) and also replicating the sale of theoption. The sale of the option is replicated by selling D shares and borrowing B . In thiscase, B is negative, so borrowing B is equivalent to lending 25.6211.

To summarize, arbitrage is obtained by:

buying the option for $4.00

selling 0.3016 shares of stock

lending $25.6211 at the risk-free rate.

These actions are described in Choice B.

Solution 64

C Chapters 10 and 13, Theta in the Binomial Model

The formula for theta is:

q

-- - - D - G =

2( )( ) ( ,0) ( ,0)

2( ,0)2

udSud S

V V Sud S S S S

h

Since we have = =120S Sud , this simplifies to:q

-=( ,0)2

udV V S h

The risk-neutral probability of an upward move is:

-= =-0.0860 48

* 0.629575 48e

p

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Since the put option is an American option, we solve for its value from right to left. Thecompleted tree is shown below.

0.00000.0000

1.4035 0.00004.9195 4.1039

12.0000 12.000021.6000

29.2800

The bolded entries in the table above indicate where early exercise is optimal.

The value of theta can now be found:

q - -= = = -

4.1039 4.9195( ,0) 0.4078

2 2 1udV V S

h

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