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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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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-
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
4
22
, 1
18
%p
&o'e
(own&o'e
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
'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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
'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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
+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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
8
S0u, u
S0d, d
%p&o'e
(own&o'e
dSuS
fdu
00
=
6
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
+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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
11
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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
Irreleance o& Stoc$s 34pected
#eturn
:hen we are 'aluin* an option in terms o#
the un+erlyin* stock the e7pecte+ returnon the stock is irrele'ant
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
Original 34ample #eisited
Sincepis a risk"neutral probability 20e01202- = 22p+ 18(1 p);p= 0-2!
Alternati'ely< we can use the #ormula
Su= 22u= 1
Sd= 18
d= 0
S
p
31
,p4
14
6523.09.01.1
9.00.250.12
=
=
=
e
du
dep
rT
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
'aluing te Option "sing #is$)
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
'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!
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
;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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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$s
)or options on stock in+ices< currencies
an+ #utures the basic proce+ure #orconstructin* the tree is the same e7cept#or the calculation o#p
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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Te %roailit/ o& an "p Moe
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
23
contract#uturesa#or1
rate#ree"risk#orei*ntheisherecurrency wa#or
in+e7theonyiel+
+i'i+en+theisein+e7 wherstocka#or
stockpayin*+non+i'i+ena#or
=
=
=
=
=
a
rea
qea
ea
du
dap
f
trr
tqr
tr
f )(
)(
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Increasing te Time Steps
In practice at least !0 time steps arenecessary to *i'e *oo+ option 'alues
(eri'aDem allows up to -00 time steps tobe use+
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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Te Blac$)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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. ull2!1"
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