ch 12 hull fundamentals 8 the d

7/25/2019 Ch 12 Hull Fundamentals 8 the d

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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull 2!1"

Introduction to BinomialTrees

Chapter 12

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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"

A Simple Binomial Model

A stock price is currently $20 In three months it will be either $22 or $18

Stock Price = $22

Stock Price = $18Stock price = $20

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Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=

A Call Option (Figure 12.1, page 27!

A !"month call option on the stock has a strike price o#21

3

%p&o'e

(own&o'e


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Setting "p a #is$less %ort&olio

)or a port#olio that is lon* shares an+ ashort 1 call option 'alues are

Port#olio is riskless when 22, 1

= 18 or = 02-


4

22

, 1

18

%p

&o'e

(own&o'e


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'aluing te %ort&olio(#is$)Free #ate is 12*!

.he riskless port#olio is/

lon* 02- shares short 1

call option.he 'alue o# the port#olio in ! months is

22 02- , 1 = -0

.he 'alue o# the port#olio to+ay is-e, 01202- = !0

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'aluing te Option

.he port#olio that is

lon* 02- shares short 1option

is worth !.he 'alue o# the shares is

-000 3= 02- 20 4.he 'alue o# the option is there#ore

0!! 3= -000 , ! 4

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+eneraliation (Figure 12.2, page 27-!

A +eri'ati'e lasts #or time Tan+ is+epen+ent on a stock

Suu

Sdd

S

7

%p&o'e

(own&o'e


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+eneraliation (continued!

5alue o# a port#olio that is lon* shares an+ short 1+eri'ati'e/

.he port#olio is riskless when S0u, u = S0d, d or


8

S0u, u

S0d, d

%p&o'e

(own&o'e

dSuS

fdu

00

=

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+eneraliation(continued!

5alue o# the port#olio at time TisSu, u

5alue o# the port#olio to+ay is(Su u)e

rT

Another e7pression #or the

port#olio 'alue to+ay is S,fence = S,

3Su, u)erT

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+eneraliation(continued!

Substitutin* #or we obtain

= [pu+ (1 p)d]erT

where


10

p e du d

rT

=


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pas a %roailit/

It is natural to interpretpan+ 19pas the probabilitieso# up an+ +own mo'ements

.he 'alue o# a +eri'ati'e is then its e7pecte+ payo## in

+iscounte+ at the risk"#ree rate


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S0u

u

S0d

d

S0


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#is$)0eutral 'aluation

:hen the probability o# an up an+ +ownmo'ements arepan+ 1"pthe e7pecte+ stockprice at time Tis S0e

rT

.his shows that the stock price earns the risk"#ree rate

;inomial trees illustrate the *eneral result that to'alue a +eri'ati'e we can assume that the

e7pecte+ return on the un+erlyin* asset is therisk"#ree rate an+ +iscount at the risk"#ree rate

.his is known as usin* risk"neutral 'aluation


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Irreleance o& Stoc$s 34pected

#eturn

:hen we are 'aluin* an option in terms o#

the un+erlyin* stock the e7pecte+ returnon the stock is irrele'ant

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Original 34ample #eisited

Sincepis a risk"neutral probability 20e01202- = 22p+ 18(1 p);p= 0-2!

Alternati'ely< we can use the #ormula

Su= 22u= 1

Sd= 18

d= 0

S

p

31

,p4

14

6523.09.01.1

9.00.250.12

=

=

=

e

du

dep

rT


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'aluing te Option "sing #is$)

0eutral 'aluation

.he 'alue o# the option is

e,01202- 0-2!1 > 0!0?

= 0!!

Su= 22u= 1

Sd= 18d= 0

S

0$1-2!

0$!22

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A T5o)Step 34ampleFigure 12.6, page 28

@ach time step is ! monthsK=21< r =12

20

22

18

22

1B8

12

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'aluing a Call OptionFigure 12., page 28

5alue at no+e ; = e,01202-30-2!!2 > 0!04 = 202-

5alue at no+e A = e,01202-30-2!202- > 0!04

= 1282!

201282!

22

18

22

!2

1B800

1200

202-

00

A

;

C

(

@

)

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A %ut Option 34ampleFigure 12.7, page 26

K = -2< time step=1yr

r = -< u =1!2< d = 08


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9at :appens 9en te %ut

Option is American (Figure 12., page 2!


19

-0-08B

0

0

20

8

!2

20

11

120C.he American #eature

increases the 'alue at no+e

C #rom B! to 120000

.his increases the 'alue o#the option #rom 1B2! to-08B


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;elta

(elta 34 is the ratio o# the chan*ein the price o# a stock option to thechan*e in the price o# theun+erlyin* stock

.he 'alue o# 'aries #rom no+e tono+e

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Coosing u and d

One way o# matchin* the 'olatility is to set

where is the 'olatility an+tis the len*th

o# the time step .his is the approach use+by Co7< oss< an+ ubinstein 31BB4

21

t

t

eud

eu

==

=

1


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Assets Oter tan 0on);iidend

%a/ing Stoc$s

)or options on stock in+ices< currencies

an+ #utures the basic proce+ure #orconstructin* the tree is the same e7cept#or the calculation o#p


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Te %roailit/ o& an "p Moe


23

contract#uturesa#or1

rate#ree"risk#orei*ntheisherecurrency wa#or

in+e7theonyiel+

+i'i+en+theisein+e7 wherstocka#or

stockpayin*+non+i'i+ena#or

=

=

=

=

=

a

rea

qea

ea

du

dap

f

trr

tqr

tr

f )(

)(


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Increasing te Time Steps

In practice at least !0 time steps arenecessary to *i'e *oo+ option 'alues

(eri'aDem allows up to -00 time steps tobe use+


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Te Blac$)Scoles)Merton Model

.he ;S& mo+el can be +eri'e+ by lookin*at what happens to the price o# a@uropean call option as the time stepten+s to Eero

See Appen+i7 to Chapter 12


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ch 12 hull fundamentals 8 the d

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