complete binomial models_張森林教授講稿

8/8/2019 Complete Binomial Models_

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Binomial ModelsDr. San-Lin Chung

Department of FinanceNational Taiwan University


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In this lecture, I will cover the following topics:

1. Brief Review of Binomial Model

2. Extensions of the binomial models in the

literature

3. Fast and accurate binomial option models

4. Binomial models for pricing exotic options5. Binomial models for other distributions or

processes


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1. Brief Review of Binomial Trees

Binomial trees are frequently used to

approximate the movements in the price

of a stock or other asset In each small interval of time the stock

price is assumed to move up by a

proportional amount u or to move down

by a proportional amount d


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The main idea of binomial option pricing theoryis pricing by arbitrage. If one can formulate a

portfolio to replicate the payoff of an option,

then the option price should equal to the price

of the replicating portfolio if the market has noarbitrage opportunity.

Binomial model is a complete market model, i.e.

options can be replicated using stock and risk-free bond (two states next period, two assets).

On the other hand, trinomial model is not a

complete market model.


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Generalization (Figure 10.2, page 202)

A derivative lasts for time Tand is

dependent on a stock

S0 u

u

S0d

d

S0


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Generalization

(continued)

Consider the portfolio that is long ( shares and short

1 derivative

The portfolio is riskless when S0u(

u=

S0d(

d or

( !

u df

u d0 0

S0 u( u

S0d( d

S0

f


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Generalization(continued)

Substituting for( we obtain

= [p u

+ (1 p )d

]erT

where

p e du d

rT

!


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Risk-Neutral Valuation

= [p u + (1 p )d]e-rT

The variablesp and (1 p ) can be interpreted as the

risk-neutral probabilities of up and down movements

The value of a derivative is its expected payoff in arisk-neutral world discounted at the risk-free rate

S0u

u

S0d

d

S0


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Irrelevance of Stocks Expected

Return

When we are valuing an option in terms

of the underlying stock the expected

return on the stock is irrelevant


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Tree Parameters for a

Nondividend Paying Stock

We choose the tree parametersp, u, and d

so that the tree gives correct values for themean & standard deviation of the stock

price changes in a risk-neutral world

erHt=pu + (1p )d

W2Ht=pu 2 + (1p )d2 [pu + (1p )d]2

A further condition often imposed is u = 1/ d


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2. Tree Parameters for a

Nondividend Paying Stock(Equations 18.4 to 18.7)

When Htis small, a solution to the equations is

tr

t

t

ea

du

dap

ed

eu

H

HW

HW

!

!

!

!


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The Complete Tree

(Figure 18.2)

S0

S0u

S0dS0 S0

S0u2

S0d2

S0u

2

S0u3 S0u

4

S0d2

S0u

S0d

S0d4

S0d3


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Backwards Induction

We know the value of the option

at the final nodes

We work back through the tree

using risk-neutral valuation to

calculate the value of the option

at each node, testing for earlyexercise when appropriate


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Example: Put Option

S0 = 50; X= 50; r=10%; W = 40%;

T=

5 months=

0.4167;Ht=1 month =0.0833

The parameters imply

u = 1.1224; d= 0.8909;a = 1.0084;p = 0.5076


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Example (continued)

Figure 18.3 89.00.00

9.35

0.00

0. 0 0. 0

0.00 0.00

6 .99 6 .99

0.6 0.0056.12 56.12 56.12

2.16 1.30 0.00

50.00 50.00 50.00

. 9 3. 2.66

.55 44.55 44.55

6.96 6.38 5.45

39.69 39.69

10.36 10.31

35.36 35.36

14.64 14.64

31.50

18.50

28.0

21.93


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Trees and Dividend Yields

When a stock price pays continuous dividends atrate q we construct the tree in the same way butset a =e(r q )Ht

As with Black-Scholes: For options on stock indices, q equals the

dividend yield on the index

For options on a foreign currency, q equals

the foreign risk-free rate For options on futures contracts q = r


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Binomial Tree for Dividend

Paying Stock

Procedure:

Draw the tree for the stock price lessthe present value of the dividends

Create a new tree by adding

the present value of the dividends at

each node

This ensures that the tree recombines and

makes assumptions similar to those when the

Black-Scholes model is used


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There have been many extensions of the CRRmodel. The extensions can be classified into fivedirections.

The first direction consists in modifying the lattice toimprove the accuracy and computational efficiency.

Boyle (1 88)

Breen (1 1)

Broadie and Detemple (1 6)

Figlewski and Gao (1 )

Heston and Zhou (2000)

II. Literature Review (1/5)


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The third direction of extensions consists in showingthe convergence property of the binomial OPM.

Cox, Ross, and Rubinstein (1 79)

Amin and Khanna (1994)

He (1990)

Nelson and Ramaswamy (1990)



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The fourth direction of the literature generalizes the

binomial model to price options under stochastic

volatility and/or stochastic interest rates.Stochastic interest rate: Black, Derman, and Toy

(1990), Nelson and Ramaswamy (1990), Hull and

White (1994), and others.

Stochastic volatility: Amin (1991) and Ho, Stapleton,and Subrahmanyam (1995) Ritchken and Trevor

(1999)



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The fifth extension of the CRR model focus on

adjusting the standard multiplicative-binomial

model to price exotic options, especiallypath-dependent options.

Asian options: Hull and White (1993) and Daiand Lyuu (2002).

Barrier options: Boyle and Lau(1994),Ritchken (1995), Boyle and Tian (1999), and

others.



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3. Alternative Binomial Tree

3.1 Jarrow and Rudd (1982)Instead of setting u = 1/dwe can set

each of the 2 probabilities to 0.5 and

ttr

ttr

ed

eu

HWHW

HWHW

!

!

)2/(

)2/(

2

2


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3.2 Trinomial Tree (Page 409)

11

1

1

11

1

1

2

322

2

2

!

!

!

!

!

!

(

(

uu

MuMMup

PPp

uu

MMMup

ed

meu

d

dum

u

t

t

PW

PW

S S

Sd

Su

pu

pm

pd


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3.3 Adaptive Mesh Model

This is a way of grafting a high resolution

tree on to a low resolution tree

We need high resolution in the region ofthe tree close to the strike price and option

maturity


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3.4 BBS and BBSR

The Binomial Black & Scholes (BBS) method is

proposed by Broadie and Detemple (1996). The

BBS method is identical to the CRR method, except

that atthetimestep just before option maturity the

Black and Scholesformula replaces at all the nodes.


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3.5 Tian (1999)(1/3)Thesecond method was putforward by Tian(1999) termed

flexible binomial model. To constructtheso-called

flexible binomial model, thefollowing specification is

proposed:

where is an arbitrary constant, called the tilt

parameter. It is an extra degree offreedom overthe

standard binomial model. In orderto have nonnegative

probability, thetilt parameter mustsatisfy the inequality (8)

after jumps, u and d, are redefined.

7,,22

dudepedeu

tr

tttt

!!!(

(((( PWWPWW


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3.6 Heston and Zhou (2000)

Heston and Zhou (2000) show thatthe accuracy or rate ofconvergence ofbinomial method depend, crucially on the

smoothness ofthe payofffunction. They have given an

approachthat isto smooththe payofffunction. Intuitively,

ifthe payofffunction atsingular points can besmoothing,

the binomial recursion might be more accurate.Hencethey

letG(x) bethesmoothed one;

whereg(x) isthe actual payofffunction.

(

( (!x

xx

dyyxgXG2

1


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3.7 Leisen and Reimer (1996)


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3.7 Leisen and Reimer (1996)


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3.8 WAND (2002, JFM)


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3.8 WAND (2002, JFM)

WAND (2002) showed thatthe binomial option pricing errors

are related to the node positioning and they defined a ratio for

node positioning.


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3.8 WAND (2002, JFM)

The relationship between theerrors and node positioning.


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Theorem 1. In the GCRR model, the three parameters are as

follows:

where is a stretch parameter which determines the shape of

the binomial tree. Moreover, when , i.e., the number of time

steps n grows to infinity, the GCRR binomial prices will converge

to the Black-Scholes formulae for European options.

Obviously the CRR model is a special case ofour GCRR

model when .

We can easily allocatethestrike price at one ofthefinal nodes.

1

,

,

,

t

t

r t

u e

d e

e dp u d

PW

WP

(

(

(

!

!

!

RP0p(t

1!P

3.9 GCRR model


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4. Binomial models for exotic

options

Topics:

1. Path dependent options using trees

Lookback options Barrier options

2. Options where there are two stochastic

variables (exchange option, maximumoption, etc.)


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Path Dependence:

The Traditional View

Backwards induction works well for

American options. It cannot be used forpath-dependent options

Monte Carlo simulation works well for

path-dependent options; it cannot be

used for American options


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Extension of Backwards

Induction Backwards induction can be used for some

path-dependent options

We will first illustrate the methodology usinglookback options and then show how it can

be used for Asian options


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Lookback Example (Page 462)

Consider an American lookback put on a stock where

S = 50, W = 40%, r =10%, Ht =1 month & the life of

the option is 3 months

Payoff is Smax-ST

We can value the deal by considering all possible

values of the maximum stock price at each node

(This example is presented to illustrate the methodology. A more efficient

ways of handling American lookbacks is in Section 20.6.)


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Example: An American Lookback Put

Option (Figure 20.2, page 463)

S0 = 50, W = 40%, r =10%, Ht =1 month,

56.12

56.12

4.68

44.55

50.00

6.38

62.99

62.99

3.36

50.00

56.12 50.00

6.12 2.66

36.69

50.00

10.31

70.70

70.70

0.00

62.99 56.126.87 0.00

56.12

56.12 50.00

11.57 5.45

44.55

35.36

50.00

14.64

50.00

5.47A


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Why the Approach WorksThis approach works for lookback options because

The payoff depends on just 1 function of the path followed

by the stock price. (We will refer to this as a path

function) The value of the path function at a node can be calculated

from the stock price at the node & from the value of the

function at the immediately preceding node

The number of different values of the path function at anode does not grow too fast as we increase the number of

time steps on the tree


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Extensions of the

Approach

The approach can be extended so thatthere are no limits on the number of

alternative values of the path function at anode

The basic idea is that it is not necessary toconsider every possible value of the path

function It is sufficient to consider a relatively small

number of representative values of thefunction at each node


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Working Forward

First work forwards through the treecalculating the max and min values of

the path function at each node

Next choose representative values ofthe path function that span the range

between the min and the max

Simplest approach: choose the min, the

max, and N equally spaced values

between the min and max


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Backwards Induction

We work backwards through the tree in the

usual way carrying out calculations for each

of the alternative values of the path functionthat are considered at a node

When we require the value of the derivative

at a node for a value of the path functionthat is not explicitly considered at that node,

we use linear or quadratic interpolation


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Part of Tree to Calculate Value of

an Option on the Arithmetic

Average (continued)

Consider Node X when the average of 5

observations is 51.44

Node Y: If this is reached, the average becomes51.98. The option price is interpolated as 8.247

Node Z: If this is reached, the average becomes

50.49. The option price is interpolated as 4.182

Node X: value is(0.50568.247 + 0.49444.182)e0.10.05 =6.206


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A More Efficient Approach for

Lookbacks (Section 20.6, page 465)

y !

y

y

efine

here is the MAX stock price

onstr ct a tree for ( )

se the tree to a e the ookback

option in "stock price nits" rather

than o ars

Y tF t

S t

F t

Y t

( )( )

( )

( )


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True Barrier vs Tree Barrier for a

Knockout Option: The Binomial Tree Case

Barrier assumed by tree

True barrier


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True Barrier vs Tree Barrier for a Knockout

Option: The Trinomial Tree Case

Barrier assumed by tree

True barrier


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BumpingUp Against the Barrierwith the

BinomialMethod,JD,Boyle andLau (1994)


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2 2

2, 1,2,3,

log

m T F m m

S

H

W! !

L

m F(m)1 21.38

2 85.52

3 192.42

4 342.08

5 534.51


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On PricingBarrierOptions,

JD, Ritchken (1995)


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2

2

2

with probability

0 with probability

with probability

where 1 and , , and are

1

2 2

11

1

2 2

u

a

m

d

u m d

u

m

d

t P

t P

t P

P P P

tP

P

tP

PW

\

PW

P

QP PW

P

QP PW

(

!

(u

(!

!

(!


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ComplexBarrier

OptionsCheuk and Vorst (1996)

Time varying barrier

Double barriers


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log*

log*

log*

t i

t i

i

u i u i e

d i d i e

m i m i e

PW L

PW L

L

LL

L

(

( ! !! !

! !


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Alternative Solutions

to the Problem Ensure that nodes always lie on the

barriers

Adjust for the fact that nodes do notlie on the barriers

Use adaptive mesh

In all cases a trinomial tree is

preferable to a binomial tree


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Multi-Asset Case

Reference: Boyle, P. P., J. Evnine, and S. Gibbs, 1989, Numerical Evaluation of Multivariate

Contingent Claims, The Review of Financial Studies, 2, 241-250.

Chen, R. R., S. L. Chung, and T. T. Yang, 2002, Option Pricing in a Multi-Asset,

Complete Market Economy, Journal of Financial and Quantitative Analysis, 37, 649-666.

Ho, T. S., R. C. Stapleton, and M. G. Subrahmanyam, 1995, Multivariate Binomial

Approximations for Asset Prices with Nonstationary Variance and Covariance

Characteristics, The Review of Financial Studies, 8, 1125-1152.

Kamrad, B., and P. Ritchken, 1991, Multinomial Approximating Models for Options

with kState Variables, Management Science, 37, 1640-1652.

Madan, D. B., F. Milne, and H. Shefrin, 1989, The Multinomial Option Pricing Model

and Its Brownian and Poisson Limits, The Review of Financial Studies, 2, 251-265.


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Modeling Two Correlated Variables

Consider a two-asset case:

Under the first approach: Transform variables so that they are notcorrelated & build the tree in the transformed variables

11 1

1

2

2 2

2

1 2

( ) ,

( ) ,

cov( , ) .

tt

t

tt

t

t t

dSr q dt dZ

S

dSr q dt dZ

S

dZ dZ d t

W

W

V

!

!

!

2

1 1 1 1 1

2

2 2 2 2 2

ln / 2

ln / 2

t t

t t

d S r q dt dZ

d S r q dt dZ

W WW W

!

!


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Modeling Two Correlated Variables

Take the correlation into account by adjusting the probabilities

21 1 1 1

22 2 2 2

1( )

21 1

1( )

22 2

r q t tZ

t t t

r q t tZ

t t t

S S e

S S e

W W

W W

( (

(

( (

(

!

!

2 2

1 1 2 2

1 2

2 2

1 1 2 2

1 2

1 2

2 2

1 1 2 2

1 2

1 1

1 2 2(1,1) 14

1 11 2 2(1, 1) 14

( , )1 1

1 2 2( 1,1) 14

r q r qp t

r q r q

p t

Z Z

r q r q

p t

W WV

W W

W W

VW W

W W

VW W

! (

-

! ( -

!

! (

2 2

1 1 2 2

1 2

1 11 2 2( 1, 1) 14

r q r q

p t

W W

VW W

-

! ( -

Multi Asset tree model under complete


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Multi-Asset tree model under complete

market economy

Chen, R. R., S. L. Chung, and T. T. Yang, 2002, Option Pricing in aMulti-Asset, Complete-Market Economy, JournalofFinancialandQuantitative Analysis, Vol. 37, No. 4, 649-666.

With two uncorrelated Brownian motions with equal variances, thethree points,A, B, and C, are best to be equally apart from eachother. This can be achieved most easily by choosing 3 points,

located 120 degrees from each other, on the circumference of acircle, as shown in Exhibit 2.

x axis

axis

A

C B

C

B

A


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market economy

Proposition 2The rotation of the axes is defined as follows:

where J is the rotationangle ofthe x-axiscounterclockwise andy-axis

clockwise. Afterrotation, we have:

the meansandthe variancesof the rotatedellipse remainunchangedand

the correlationisafunctionofthe rotationdegree J:

*22

*

*

tan1

1

tan1

tan

tan1

tan

tan1

1

22

22

X=!

-

v

-

!

-

j

j

j

j

y

x

y

x

JJ

J

J

J

J

2

sin 1 VJ

!



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market economy

Finally, for any given time, t, the next period stock prices are:

-

-

((v

-

!

-

((

((

((

33

22

11

2,22,1

,3,2,3,1

,2,2,2,1

,1,2,1,1

)(ln)(ln

1

11

lnln

lnln

lnln

2

2

2

1

yrxr

yrxryrxr

trStrS

SS

SS

SS

yx

yx

yx

tt

tttt

tttt

tttt

WW


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5. Binomial models for

other processes


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Time-varying volatility

processes

how to construct a recombined


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how to construct a recombinedbinomial/trinomial tree under time-

varying volatility?1. Amin (1991) suggested changing the number of steps

(or dt) such that the tree is recombined.

2. Ho, Stapleton, and Subrahmanyam (1995) suggested

using two steps to match the conditional and

unconditional volatility and unconditional mean.

3. Using the trinomial tree of Boyle (1988) or Ritchken

(1995). See next page.

Ref : Amin (1995, pp.39-40) has a very nice discussion onthis issue.

Amin, 1995, Option Pricing Trees, Journal of Derivatives,

34-46.


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Amin (1991)(1/2)

Assume that the underlying asset price follows

dS=rSdt+ W(t)Sdz

then the annual variance of the asset price over the period

[0, T] is

Let Nbe the number of time steps desired, then .

The time step for each period is denoted as h((t),

h(2(t), , h(n(t). Amin let

!T

TdttV 0

2

W

N

Tt!(

ttnh

tnthttht

(!(

(!!((!(( 222 22 WWW .


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Following Boyle (1988), the asset price, at any given time,

can move into three possible states, up, down, or middle, in

the next period. IfSdenotes the asset price at time t, then at

time t+ dt, the prices will be Su, Sd, orSm. The parameters

are defined as follows

andwhere P u 1, the dispersion parameter, is chosen freely as

long as the resulting probabilities are positive. Let Wi

dt

dt

ed

eu

PW

PW

!

!

1m !


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represent the instantaneous volatility at time ti, then we can

set

In this case the tree is recombining and the probability ofeach branch is of course time varying.

jieedtdt jjii ,! WPWP


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To guarantee that the resulting probabilities are positive, we

must carefully choose dtand P. Roughly speaking, dtmust

be small enough such that

ForP, as discussed in Boyle (1988), its values must be larger

than 1. Denote the maximum and minimum of the

instantaneous volatility for the period from time 0 to TasWmax and Wmin. Then

PW

W

)5.0( 2v

rdt

dtdtee minminmaxmax

WPWP !


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We can arbitrarily set Pmax as 1.1 and then all otherPi will be

larger than 1 automatically.

Ref :

Boyle, P. (1988), A Lattice Framework for Option Pricing with

Two State Variables, JournalofFinancialandQuantitative

Analysis, 23, 1-12.


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Stochastic volatility

stochastic interest rate

processes


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Reference:1. Hilliard, J. E., and A. Schwartz, Pricing Options on Traded

Assets under Stochastic Interest Rates and Volatility: A Binomial

Approach, Journal of Financial Engineering, 6, 281-305.2. Hilliard, J. E., A. L. Schwartz, and A. L. Tucker, 1996, Bivariate

Binomial Options Pricing with Generalized Interest Rate

Processes, Journal of Financial Research, 14, 585-602.

3. Nelson, D. B., and K. Ramaswamy, 1990, Simple Binomial

Processes as Diffusion Approximations in Financial Models,

Review of Financial Studies, 3, 393-430.4. Ritchken, P., and R. Trevor, 1999, Pricing Option under

Generalized GARCH and Stochastic Volatility Processes, Journal

of Finance, 54, 377-402.

5. Hillard, J. E., and A. Schwartz, 2005, Pricing European and

American Derivatives under a Jump-Diffusion Process: A Bivariate

Tree Approach, Jour

nalofFinancialandQuantitative Analysis, 40,671-691.

6. Camara, A., and S. L. Chung, 2006, Option Pricing for the

Transformed-Binomial Class, JournalofFutures Markets, Vol. 26,

No. 8, 759-788.


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Nelson and Ramaswamy (1990) proposed a general treemethod to approximate diffusion processes.

Generally a binomial ortrinomial tree is not recombinedbecausethe volatility is not a constant. Nelson andRamaswamy (1990) suggested a transformation ofthevariablesuchthatthetransformed variablehas a constantvolatility.



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Nelson and Ramaswamy (1990)For example, under the CEV model:


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Nelson and Ramaswamy (1990)For example, under the CIR model:

Hilliard-Schwartz: Stochastic


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Hilliard-Schwartz: Stochastic

VolatilityThe asset price and return volatility are assumed to follow:

dS= msdt+f(S)h(V)dZsdV= mvdt+ bVdZv (1)

UnderQ measuremsS(r- d).

First ofall, makethefollowing transformation to obtain aunit variance variableY:

b

Yln

!

vv dZdtb

bV

mdY

! 5.0


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Next, makethefollowing transformation to obtain a constant

variance variableH:

(4)

where

hhh

hVVSS

dZdtm

dtmbVdZHdZSVHdH

W

U

!

!

svsvvvssvvsshVbVSHVbHVSHmHmHm VU ! 222 5.05.0

? A 5.0225.01 bHbHSVh

! VW


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Then make anothertransformation to obtain a unit variance

variable Q.

(5)

where

hq dZdtmdQ

!

25.0 hhhh

hq Q

mm W

W!


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Binomial treeforYand trinomial treeforQ

T/n =h : 0 1 2

Y0, 0

Y1, -1

Y1, 1

Y2, -2

Y2, 2

Y2, 0

Q1, -1

Q1, 0

Q1, 1

Q2, -2

Q2, -1

Q2, 0

Q2, 1

Q2, 2

Q0, 0

O ti P i i d GARCH


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Option Pricing underGARCH

Ritchken, P., and R. Trevor, 1999, Pricing Option underGeneralized

GARCH and Stochastic Volatility Processes, Journal of Finance, 54,

377-402.

GARCH model:

The main idea is to keep the spanning of the tree flexible, i.e. the size

of up or down movements can be adjusted to match the conditional

variance.


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Option Pricing for the

Transformed-Binomial Class

Antnio Cmara1 and San-Lin Chung2

January 2004

1 School of Management, University of Michigan-Flint, 3118 William S. White

Building, Flint, MI 48502-1950. Tel: (810) 762-3268, Fax: (810) 762-3282,

Email: [email protected]

2 Department of Finance, The Management School, National Taiwan University,

Taipei 106, Taiwan. Tel:886-2-23676909, Fax:886-2-23660764, Email:

[email protected].

Ab t t


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This paper generalizes the seminal Cox-Ross-Rubinstein (1979) binomial option pricing model(OPM) to all members of the class of transformed-binomial pricing processes. Our investigationaddresses issues related with asset pricing

modeling, hedging strategies, and option pricing.We derive explicit formulae for (1) replicating orhedging portfolios; (2) risk-neutral transformed-binomial probabilities; (3) limiting transformed-normal distributions; and (4) the value of contingent

claims. We also study the properties of thetransformed-binomial class of asset pricing rocesses.We illustrate the results of the paper with severalexamples.

Abstract


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multiplicative-binomial option pricingmodel: Cox, Ross, and Rubinstein (1979),Rendlemen and Bartter (1979), and Sharpe(1978)

pricing by arbitrage: According to this rule,when there are no arbitrage opportunities, if

a portfolio of stocks and bonds replicates thepayoffs of an option then the option musthave the same current price as its replicatingportfolio.

I. Introduction (1/7)


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Multiplicative-binomial (hereafter, M-binomial) model:

This M-binomial model assumes that u = 2 and d=0.5.


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The SL-binomial model with a lower bound E at

maturity:

For example, ifr=1.25 and E =10 then this SL-

binomial model assumes that u = 2.1429 and d=

0.3571.


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The following SU-binomial model:

In this SU-binomial model, it is assumed that u =1.4107 and d=0.7106.


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The following SB-binomial model with a threshold F:

For example, ifF =300 then this SB-binomial model

assumes that u = 2.5 and d=0.4545.


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The SL-binomialmodel

Following Johnson (1949), the transformation for the

SL-binomial is defined as the following in this article:

, ,ln t i ti k i k g S S rE (!


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The SU-binomialmodel

The transformation for the SU-binomial model is

defined as:

1, , , ,sinh ln 1i k i k i k i k g S S S S ! !


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The SB-binomialmodel

The third example considered in this paper is the SB-

binomial model, corresponding to the SB-normal model

of Johnson (1949). The transformation for the SB-

binomial model is as follows:

,

,

,

lni k

i k

i k

Sg S

SF

!

Figure 1: The convergence ofthe S_L binomial priceto its closed form

solution


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solution

13.3

13.35

13.4

13.45

13.5

13.55

13.6

20 40 60 80 100 120 140 160 180 200

number oftimes teps

p

rice

S_L b inomial

price

closed-form

This figure shows the convergence pattern resulting from option pricecalculations with the SL-binomial model. We use the following selection of

parameters: S=100, K=100, r=0.1, E = 20, t=1.0, W =0.25.

Figure 2: The convergence of the S_U binomial price to its closed form

solution


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solution

23.75

23.8

23.85

23.9

23.95

24

24.05

24.1

24.15

24.2

24.25

24.3

20 40 60 80 100 120 140 160 180 200

number of time steps

price

S_U binomial

price

closed-form

This figure shows the convergence pattern resulting from option price

calculations with the SU-binomial model. We use the following selection of

parameters: S=100, K=100, r=0.1, t=1.0, W =0.25.

Figure 3: The convergence ofthe S_B binomial price


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12.08

12.1

12.12

12.14

12.16

12.18

12.2

12.22

12.24

20 40 60 80 100 120 140 160 180 200

number oftimes teps

p

rice

S_B b inomial

price

This figure shows the convergence pattern resulting fromoption price calculations with the S

B-binomial model. We

use the following selection of parameters: S=100, K=100, r=0.1, F =

300, t=1.0, W =0.25.

complete binomial models_張森林教授講稿

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