期权期货及其他衍生品Ch32HullOFOD10thEdition

Chapter

32No-ArbitrageModelsof

theShort

RateOptions,

Futures,

and

Other

Derivatives,

10thEdition,

Copyright

©

John

C.

Hull

20171No-arbitrage

Term

Structure

ModelsOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

20172A

no-arbitrage

model

is

a

model

designedto

fit

today’s

term

structure

of

interest

raDeveloping

No-ArbitrageModel

for

rOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

20173Amodel

for

r

can

be

made

to

fit

theinitial

term

structure

by

including

afunction

of

time

in

the

driftHo-Lee

ModelOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

20174dr

=

q(t)dt

+

sdzMany

analytic

results

for

bond

prices

andoption

pricesInterest

rates

normally

distributedOne

volatility

parameter,

sAll

forward

rates

have

thesame

standarddeviationDiagrammatic

Representation

of

Ho-

Lee

(Figure

32.1,

page

716)Options,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

20175ShortRaterrrrTimeHull-White

ModelOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

20176dr

=

[q(t

)

–

ar

]dt

+

sdz

Many

analytic

results

for

bond

prices

andoption

pricesTwo

volatility

parameters,

a

and

sInterest

rates

normally

distributed

Standard

deviation

of

a

forward

rate

is

adeclining

function

of

its

maturityDiagrammatic

Representation

of

Hull

and

White

(Figure

32.2,

page

717)ShortRaterrrrTimeForward

RateCurveOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

20177Black-Karasinski

Model

(equation

32.9)Future

value

of

r

is

lognormalVery

little

analytic

tractabilityOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

20178Options

on

Zero-Coupon

Bonds(equation

32.10,

page

719-720)In

Vasicek

and

Hull-White

model,

price

of

call

maturing

at

T

on

azero-coupon

bond

lasting

to

s

isLP(0,s)N(h)−KP(0,T)N(h−sP)Price

of

put

isKP(0,T)N(−h+sP)−LP(0,s)N(h)whereOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

20179Options

on

Coupon-Bearing

BondsOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201710In

a

one-factor

model

a

European

option

on

acoupon-bearing

bond

can

be

expressed

as

aportfolio

of

options

on

zero-coupon

bonds.We

first

calculate

the

critical

interest

rate

athe

option

maturity

for

which

the

coupon-

bearingbond

price

equals

the

strike

price

atmaturityThe

strike

price

for

each

zero-coupon

bond

isset

equal

to

its

value

when

the

interest

rateequals

this

critical

valueInterest

Rate

Trees

vs

Stock

Price

TreesOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201711The

variable

at

each

node

in

an

interestrate

tree

is

the

Dt-period

rateInterest

rate

trees

work

similarly

to

stockprice

trees

except

that

the

discount

rateused

varies

from

node

to

nodeTwo-Step

Tree

Example

(Figure

32.4,

page

722)Payoff

after

2

years

is

MAX[100(r

–

0.11),

0]10%0.35**Options,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

2017121.11*10%0.238%0.00pu=0.25;

pm=0.5;

pd=0.25;

Time

step=1yr14%312%12%110%08%06%0*:

(0.25×3

+

0.50×1

+

0.25×0)e–0.12×1**:

(0.25×1.11

+

0.50×0.23

+0.25×0)e–0.10×1Alternative

Branching

Processes

in

a

Trinomial

Tree

(Figure

32.5,page

723)(a)Options,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201713(b)(c)Procedure

for

Building

TreeOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201714dr

=

[q(t

)

–

ar

]dt

+

sdzAssume

q(t

)

=

0

and

r

(0)

=

0Draw

a

trinomial

tree

for

r

to

match

themean

andstandard

deviation

of

the

processfor

rDetermine

q(t

)

one

step

at

a

time

so

thatthe

treematchesthe

initial

term

structureExample

(page723

to

728)Options,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201715s

=

0.01a

=

0.1Dt

=1

yearMaturityZero

Rate0.53.43013.8241.54.18324.5122.54.81235.086Building

the

First

Tree

for

the

Dt

rate

RSet

vertical

spacing:Change

branching

when

jmax

nodes

frommiddle

where

jmax

is

smallest

integer

greatethan

0.184/(aDt)Chooseprobabilities

on

branches

so

thatmean

change

in

R

is

-aRDt

and

S.D.

of

change

isOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201716The

First

Tree(Figure

32.6,

page

724)ABCDEFGHINodeOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201717ABCDEFGHIR0.000%1.732%0.000%-1.732%3.464%1.732%0.000%-1.732%-3.464%p

u0.16670.12170.16670.22170.88670.12170.16670.22170.0867p

m0.66660.65660.66660.65660.02660.65660.66660.65660.0266p

d0.16670.22170.16670.12170.08670.22170.16670.12170.8867Shifting

NodesOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201718Work

forward

through

treeRemember

Qij

the

value

of

a

derivativeproviding

a

$1

payoff

at

node

j

at

time

iDtShift

nodes

at

time

iDt

by

ai

so

that

the

(i+1)Dbond

is

correctly

pricedThe

Final

TreeA(Figure

31.7,

Page

727)BCDEFGHINode

AOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201719BCDEFGHIR3.824%6.937%5.205%3.473%9.716%7.984%6.252%4.520%2.788%p

u0.16670.12170.16670.22170.88670.12170.16670.22170.0867p

m0.66660.65660.66660.65660.02660.65660.66660.65660.0266p

d0.16670.22170.16670.12170.08670.22170.16670.12170.8867ExtensionsOptions,

Futures,

and

Other

Derivatives,

10th

Edition,Copyright

©

John

C.

Hull

201720The

tree

building

procedure

can

be

extendedto

cover

more

general

models

of

the

form:dƒ(r

)

=

[q(t

)

–

a

ƒ(r

)]dt

+

sdzWe

set

x=f(r)

and

proce