Download - Complete Binomial Models_張森林教授講稿
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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)
II. Literature Review (3/5)
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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)
II. Literature Review (4/5)
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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.
II. Literature Review (5/5)
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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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Multi Asset tree model under complete
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Multi-Asset tree model under complete
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
!
Multi Asset tree model under complete
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Multi-Asset tree model under complete
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.
Nelson and Ramaswamy (1990)
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Nelson and Ramaswamy (1990)
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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:
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.