WILLOW
TREE
By
A n d y C . T . Ho
B . Sc. (Mathematics) S i m o n Fraser University
M . Sc. (Mathematics) University of B r i t i s h C o l u m b i a
P h . D . (Mathematics) University of B r i t i s h C o l u m b i a
A
THESIS
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M A S T E R
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STUDIES
MATHEMATICS
We accept this thesis as conforming
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T H E
UNIVERSITY
O F BRITISH
COLUMBIA
M a y 2000
© A n d y C . T . H o , 2000
O F
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....
Department o f
The U n i v e r s i t y o f B r i t i s h C o l u m b i a
V a n c o u v e r , Canada
—
Date
,
.N.i,U|
M
|
LOOP
http://www.library.ubc.ca/spcoll/thesauth.html
2000/07/21
Abstract
We present a tree algorithm, called the willow tree, for financial derivative pricing. The
setup of the tree uses a fixed number of spatial nodes at each time step. The transition
probabilities are determine by solving linear programming problems. The willow tree
method is radically superior in numerical performance when compared to the binomial
tree method.
ii
Table of Contents
Abstract
ii
List of Figures
v
List of Tables
vi
Acknowledgment
1
2
3
4
vii
Introduction
1
B i n o m i a l Tree
3
W i l l o w Tree
8
3.1
Basic Setup
8
3.2
R a n k of constraint m a t r i x
12
3.3
Convergence of the willow tree
17
Option Pricing
21
4.1
European Option
21
4.2
C a l i b r a t i o n of
22
4.3
N u m e r i c a l Results for European Options
26
4.4
N u m e r i c a l Results for A m e r i c a n O p t i o n
27
iii
(
5
Extension
31
5.1
Extension to other models
31
5.2
Conclusion
33
Bibliography
35
iv
List of Figures
3.1
A discrete approximation of the standard normal distribution
4.2
Small variation of
\Zi(.l) - zj(l)| ~ 10"
v
3
20
28
List of Tables
4.1
European Call ;
4.2
E u r o p e a n C a l l : Percentage of
4.3
E u r o p e a n P u t : Percentage of
4.4
A m e r i c a n P u t : Percentage of
4.5
A m e r i c a n P u t : Average Percentage of
4.6
A m e r i c a n P u t : 50 nodes 20 time steps willow tree vs 200 time steps
s
^ g
•
26
\V°-V°A
"
\v°-v° I
1
28
c / l
y U
1
\
^
29
^"Jf^ !
29
0 0
binomial tree
^"J^ "
0
1
29
30
vi
Acknowledgment
I would like to acknowledge Professor U l r i c h Haussmann for his advice and support
through out my M a t h e m a t i c a l Finance P r o g r a m i n U B C . I would also like to thank m y
parents for their support and encouragement.
vii
Chapter 1
Introduction
Since the celebrated option pricing model introduced by Black a n d Scholes [2] i n 1973,
M a t h e m a t i c a l O p t i o n pricing theory has undergone tremendous revolutionary development.
T h e basis of many of these pricing models stems from incorporating random
diffusion processes generated from B r o w n i a n motion to describe the stochastic dynamics
of various quantities i n the financial market.
For many of these models, the theoret-
ical price of an option can be determined by either solving a P D E or computing an
expectation w i t h respect to a random process. For most situations, because of the unavailability of closed form solutions, option prices can only be determined numerically.
A m o n g the many numerical approaches for option pricing models, a popular one is the
binomial re-combining tree algorithms. T h i s algorithm was first developed by C o x , Ross,
and Rubinstein [4] i n 1979.
T h e binomial tree generates branches of paths of prices
by bifurcating the prices recursively from one time step to the next. T h e value of the
option can then be estimated by computing the risk neutral expectation of the payoff
function on the security recursively backward from maturity. Because of its simple setup,
this algorithm is widely used i n evaluating many non-path-dependent
options i n the fi-
nance industry. Since the first introduction of the binomial tree, there have been many
successful variations of the algorithm applied to many different area of option pricing!
In this paper, we would like to present a new type of re-combining tree algorithm,
called the willow tree, which was first developed by Michael C u r r a n [3] i n 1998. T h e setup
1
2
Chapter 1. Introduction
of the willow tree is based on estimating the standard Brownian motion b y a discrete
M a r k o v process. T h e transition probabilities are determined by solving a sequence of
linear programming problems. These transition probabilities only need to be computed
once a n d then they can be stored for future useage.
F r o m the M a r k o v process, other
more general processes can be constructed from suitable mappings. O n e distinct feature
of the willow tree is that the number of nodes at each time step is constant. T h i s is i n
contrast to the binomial tree where the number of nodes grows as 0(M )
2
where M is
the number of time steps. This feature of the willow tree allows the computation process
to be very efficient even w i t h a large number of time steps a n d this becomes especially
advantageous for multi-factor models.
T h e layout of this paper is as follows.
Chapter 2 describes briefly the b i n o m i a l
tree algorithm. W e show that under the setup for computing E u r o p e a n Options, i t is
equivalent to a n explicit finite difference method. Chapter 3 describes the setup of the
willow tree. Chapter 4 demonstrates how to apply the willow tree algorithm to compute
prices for E u r o p e a n and A m e r i c a n vanilla options. Here, we will only concentrate our
study on vanilla options, but the result can easily be extended to many other types
of options. W e use the result computed from the binomial tree as a bench mark for
comparison. It turns out that the computational time for the willow tree is at least ten
times less t h a n that for the binomial tree for a given accuracy. Chapter 5 summarizes
our results and discusses the various extensions of the basic setup of the willow tree.
Chapter 2
B i n o m i a l Tree
We would like to show that the binomial tree algorithm for computing E u r o p e a n option
prices is equivalent to an explicit finite difference method under suitable setups.
Let
us first briefly describe the binomial tree setup for computing a E u r o p e a n option price.
Please see Chapter 10 of [5] for more details. Let
denote the b i n o m i a l tree discrete
process where m is the time index and n is the node index. Starting w i t h an i n i t i a l stock
price SQ, allow each stock price S™ to change into two possible prices
cm+l
°n+l
qm+1
„. cm
—
U
O
n )
°n
~
rlQ
m
a o
n
where u > 1 is the up move factor, d < 1 is the down move factor, 0 < m < M and
0 < n < m.
C o n t i n u i n g this process, we obtain a tree lattice of the price distribution.
Because this a recombining tree,
S™ = S° u d - .
n
(1)
m n
Q
A t each node of the tree, we let p to be the probability of an up move and 1 - p of a
down move.
m*1
S
n+1
Chapter 2.
Binomial Tree
4
B y choosing the appropriate parameters and letting the time step size 6t —> 0 , i t is
possible to show that the tree process converges to the Black Scholes (geometric B r o w n i a n
motion) continuous process
dS = S (rdt + adB ),
(2)
t
where r and a are the (risk-neutral) drift and instantaneous volatility respectively, and
B
t
is the standard B r o w n i a n motion process. We match the first two moments of the
discrete process to that of the continuous process. B y matching the (risk-neutral) mean
and variance of the two processes, we get
pu+(l-p)d
= e
pu + (l-p)d
2
(3)
rSt
= ( ° .
2
2r+
2)5t
e
(4)
Because there are only two equations here, there is still one degree of freedom left i n
determining a choice of the parameters p, u, d. B y imposing the condition d = l/u, and
the two conditions (3) and (4), we find
e
rdt
d
- d
= A-y/T^T,
u = A+y/A^l,
(5)
where
A = \ (e~
r5t
+ ° )
{r+
2)5t
e
.
(6)
Let F(S, T) be the contingent payoff function at expiry time T for an European option
and V
m
n
be values of the option on the nodes of a binomial tree w i t h parameters p, u, d.
A t time T = M6t,
V
M
n
= F(S?).
(7)
Chapter 2.
5
Binomial Tree
T h e subsequent V™ are computed by iterating the formula
fStyrn
pV™? + (1 -
=
)V™
+1
P
"i/m+l , "
e
u — d *n+l
'
r6t
_^ T/m+1
u—~
d "
V
r5t
0
0
i
u—d
(8)
^ n
u — d
T h i s setup of the binomial tree is actually equivalent to an explicit finite difference
method. M o r e explictly, let us consider the Black Scholes P D E of a European option
w i t h variables t, x = In S
SV
St
T
,
1 SV
(
2
2
1 \ SV
.
2
(9)
where t is the time and S is the spot price of the underlying stock. B y w r i t i n g
SV
6e- V
H
T /
rt
rV — e
St
St '
(9) becomes
,Se~ V
St
1 SV
rt
(
2
2
1.
sv
(10)
Sx
We convert (10) into a finite difference equation by approximating the derivatives w i t h
finite differences (See Chapter 10 of [5]). T h e corresponding finite difference equation for
(9) is
•,-rU
1+1
Km+1 _
grt ym
m
(11)
t
ym+1 _ ym+1
1
2
n+l
n
v
x +\
T/m+1 _ ym+
n
n—lm+1
y
+ (r - -cr )
2
7 1 + 1
_
n
v
Xr
2-n+l
n
Z
v
1
X
n
—\
= 0,
/ 3^n+l %n—\
where
*
m
= mft,
atftf = logfiEff,
C
+ 1
= log^
+ 1
>
<-x = log S T
(12)
Chapter
2.
Binomial
Tree
We would like to express the x's i n terms of u, d. F r o m (12) and (12),
x
n + 1
-x
n
Xn-Xn-!
x x-x -x
n+
n
tm+1
tin
=
loguS™ - l o g 5 ^ = l o g u
=
log S™ - log dS™ = - log d
=
log u/d
=
&t
and
<ft
logw/d \
logw
logd
Further rearrangment gives
where
i
n
° &t
{r - jo ) 6t
logw/dlogd
\ogu/d
2
cm+
=
1
3
\ogu/d \\ogu
7 1 + 1
logu/dlogd
logw/d
F r o m comparing (13) w i t h (8), we wish to show that
p
r < K
—
log a y
d
Chapter 2.
7
Binomial Tree
Let us expand A, u - 1, d - 1 i n terms of 8t .
F r o m (5-6),
1/2
r
2
2
2
M - 1
=
.4 - 1 + yfW^l
= o8t
+ °—8t +
d-l
=
A - l - V ^ l
= -a8t /
1/2
0{8t ' )
3 2
2
1
+ ^-8t
2
+
0(8t ^ ).
3
Using the above and the expansion logo; — (x — 1) — (x — l ) / 2 -\
, we have
2
log u — log d
=
u-d
+
0(8t )
logw
=
a8t
+
0(8t )
logd
=
-a8t
1/2
2
3/2
3/2
+
1/2
0(8t ^ ).
3
2
Subsituting the above into (16) yields
™ i
+
oHt
=
n + 1
( « - d ) ( l + 0(c5t))crc5i /2(i
1 + r«ft - (1 - aSt '
1 2
+
(r ~
(u-d)(l
(8t))
1
0
+ %6t) +
k
2
)
^
+ 0(8t))
Q{8t^ )
2
u —d
e
r5t
-d
+
0(8t / )
3 2
u —d
Similarly, it is straight forward to show that
= 1 ^ (
C ™ ?
=
1
+
0
^
1 / 2
))
0{8t ' ).
3 2
We conclude the chapter by stating the equivalence of the two algorithms i n the
following theorem.
T h e o r e m 2.1 In calculating the price of a Black Scholes European option model, the
binomial
tree algorithm for the case d = 1/u is equivalent
applied to the corresponding
Black Scholes differential
to the forward
equation.
Euler
Scheme
Chapter 3
Willow Tree
3.1
Basic Setup
In setting u p a willow tree lattice for modelling option prices, we first need to set u p
a discrete M a r k o v process that converges to a B r o w n i a n motion i n the l i m i t .
A more
general process of a security can then be modelled from the M a r k o v process under suitable mappings (see Chapter 4). M o r e specifically, let z\, z%, • • •, z be the representative
n
normal variates w i t h corresponding probabilities qi, c^, • • •, q , i-e.,
n
<&)} is a discrete
approximation of the standard normal density function where Oj = P(Z = Zi). For
example, C u r r a n suggests a choice (see Figure 3.1):
^
=
N-\(i-0.5)/n)
ft
=
1/n
(21)
where N(x) is the cumulative standard normal distribution function.
Let Y
tk
be a discrete M a r k o v process w i t h time nodes tk = £ j = i hj, for some
w i t h each hj > 0. Y
tk
takes on the value y/hZi
hi,h,2,-"
when the embedded M a r k o v chain X is
i n state i, 1 < i < n. W e can view each realized sequence of Y
k
tk
as a p a t h traversed on a
tree lattice consisting of parallel layers of nodes. T h e nodes at a given layer correspond
to the possible states {xk} at time tk. If Xk = i we assign transition probabilities p-j. that
Xk+\ = j. A t the initial time t = 0, there is only one node corresponding to state 0 w i t h
the transition probabilities being p§i = Qi-
8
9
Chapter 3. Willow Tree
Sample paths on a willow tree
We wish to make Y
tk
converge to a B r o w n i a n motion B i n the limit n —> co and
t
hk —>• 0. W e first impose the following conditions on p^:
IZP% =
1
V
j=i
f^Pij^k
j=i
*
(22)
+ hkZj = yftkZi V i
£ p% (t + h )z] - t z\ = h
j=i
k
k
E?4 = ^
i=l
k
k
(23)
Vi
V j
(24)
(25)
P & > 0 V t , j .
(26)
T h e first condition (22) is just the statement that sum of the conditional probabilities at
a node should be unity. T h e second condition (23) constrains the conditional mean
E[Y
\Y }=Y
tk+hk
tk
\/t ,hk
tk
k
(27)
which corresponds to the first moment of dB being zero. Similarly, the t h i r d condition
t
(24) constrains the conditional variance
Var[Y
\Y ]=h
tk+hk
tk
k
Vt ,h
k
k
(28)
10
Chapter 3. Willow Tree
which corresponds to the second moment of dB equal dt. W e impose the condition (25)
t
so that the unconditional probability of ending at node i at time tk is q i.e., b y i n d u c t i o n
iy
and (25)
jlJ2—jk-l
jk-i
Hence {q } is a stationary distribution, i.e. {X }
t
is stationary.
k
In order to further narrow down the choice of the transition matrices
ensure that the convergence is sufficiently fast, for each k, the
a n c
i to
are determined from
the following linear programming problem.
Subject to
^EEpJ-lA + ^ i - vW
EPij = W
(29)
(30)
1
j=i
^ P t f V*fc + h Zj = yftkZi Vz
k
(31)
j=l
f; P y (** + h )z] - t «? = fc V i
j=i
k
fc
E?iP6 = ^ y 7
fc
(32)
(33)
t=i
rfj>0Vx,j.
(34)
Essentially, we match the first two moments of dB and we require the t h i r d power of
t
the distance between transitions to be as small as possible so that we get the proper
convergence.
We would like to recast the L P problem so that we can easily see that the choice of
the distribution {zi,qi}
must have mean = 0 and variance = 1. F o r convenience, let us
drop the fc's and introduce the following notation:
Chapter
3.
Willow
Tree
• v is the transpose of a m a t r i x v
1
• P is the n x n transition m a t r i x w i t h entries Pij
• F is the m a t r i x w i t h entries Fji = \ZJ - bzi\
3
a
=
h
t
,
1
o =
Vl+a
In terms of the above notation, the L P becomes
nun trace
(PF)
Subject to :
Pu
= u
Pz
= bz
p
= br
r
+ (1 - b )
2
qP
=
l
p^ >
2
(f
0
A c c o r d i n g to constraints (37) and (39),
qz
t
= q Pz
l
=
bq z.
l
Since 6 ^ 0 , this implies that z has mean = 0, i.e.,
12
Chapter 3. Willow Tree
Similarly, from (36), (38), a n d (39)
qr = b qr +
t
2
t
(l-b ).
2
Since a ^ 0, this implies that z has variance = 1, i.e.
gV = 1.
The
(42)
choice of q a n d z suggested i n (21) meets the mean = 0 requirement (41) b u t
fails the variance = 1 requirement (42). However, there are at least two modifications
for allowing the suggested q a n d z to satisfy both requirements. T h e first one is to use a
non-standard normal distribution AT (0,1 + e) w i t h mean = 0 a n d variance = 1 + e for
£
some appropriate e so that Zi = A r ( ( z - 0.5)/n) satisfies (42). T h e second one is to
-1
e
modify only the two end points z\ a n d z
n
as z\ — 8 a n d z + S so that (42) is true for an
n
appropriate choice of S.
3.2
Rank of constraint matrix
Let us consider the L P i n the original setup but express the constraint i n terms of a
m a t r i x equation. L e t us index the entries of P as a vector v = {vi,V2,V3, . . . . i v } where
the first n entries of v correspond to the first column of P a n d the second n entries
correspond to second column a n d so on. In terms of v, the L P can be stated as
min
V
fv
(43)
Subject to :
where / is a vector w i t h positive entries.
Mv = w
(44)
Vi > 0 V i
(45)
Chapter
3.
Willow
13
Tree
Let {zi, qi} be an approximation of the standard normal distribution w i t h mean = 0
and variance = 1 such that a = {u, z, r} as described i n previous section is a linearly
independent set. We show that the rank of M is 4 n - 3 . We start by choosing a convenient
basis 0 = {0i,02 - - • 0n} of l R where 0i = u, 02 = z, 03 = r. In selecting 0, we require
n
that Si = span of {040s, 0e, • • • ,0n} is orthogonal to 52 = span of {0i,02,0s}-
Let T be
the m a t r i x that transforms 0 into the standard basis E = { e i , • • •, e „ } and S =
T~ PT.
l
We w i l l show that S has n — (An — 3) degree of freedom i n choosing its entries and hence
2
the rank of M must be in — 3.
£2 can be constructed explicitly i n the following way. Let a = {0i, • • • ,0 }
m
be a
linear independent set and
A =[0i
02 •••
(46)
0]
m
be the m a t r i x consisting of the vectors i n a. F r o m a QR-decomposition,
C
A = Q
(47)
0
where C is a m x m full ranked upper triangular matrix. Using the m a t r i x Q, we let
B
=
0
Q
(48)
In—m
C"
0
1
0
where I n
m
(49)
Q-
In-
is a (n - m) x (n - m) identity matrix. We check that B and T have the
desired properties T [ A B ] = / „ and Si _L S2
T[AB]
C"
1
0
C
Q-
1
0
In—rn
Q
0
Q
0
ln—m
Chapter 3. Willow Tree
14
0
= In0
In-m _
For the orthogonality property, we use the fact that Q is hermitian, i.e., Q
-
1
= Q*. L e t
cij € a a n d bi be the columns of B. T h e n
<L+iCj = 0 j <m
=
w h e r e Cj i s t h e i
t / l
c o l u m n vector of
C
0
We now show that S — TPT
1
has n - (An - 3) degrees of freedom. U s i n g the basis
2
(3, the L P (35-40) can be restated as the following problem:
m i n trace (SG)
(50)
Se\ = e i
(51)
Sea = be2
(52)
Subject to
Se = b e + (1 2
3
3
b)
f
X
ei
(53)
(54)
q'S = q
(T- ST)..
2
>0 Vi,j
(55)
Chapter 3.
15
Willow Tree
where G = TFT"
1
and q =
(T )V
_1
1 0
F r o m the above, S must be of the form
(1 - 6 )
*
* • •
*
* *
2
*
0
b
0
*
* • •
0
0
b
2
*
* • • *
*
0
0
0
*
* • • *
*
0
0
0
*
* • • *
*
* • • *
*
'•
0
0
0
*
* • • *
*
0
0
0
*
* • • *
*
(56)
where the entries of the first three columns are determined and remaining entries subject
to the constrains (54) and (55).
We determine q by computing its components.
{u, z,r}.
We first suppose that q G Span of
F r o m the definition of q
ft = e\q =
ftT-yq
= (T" *)^ =
1
fi .
q
(57)
Hence, by our choice of /?,
q* = ( a i , a , a , 0 , 0 , 0 , •••,())
2
3
(58)
Chapter 3. Willow Tree
16
where for 1 < i < 3, Oj = (3\q. Thus S must be of the form
1
0
(1-6 )
0
6
0
S24
0
0
b
S34 535
0
0
0
0
0
0
2
2
5
1 4
Sis
Sl(n-l)
S25
S2(n-1) S^n
:
*
0
0
0
*
*
0
0
0
*
*
-Sin
53(n-l)
Sz
*
*
n
(59)
where for 4 < j < n , a\S\j + a^S2j + C23S3J = 0. This implies there are n? — (An — 3) degree
of freedoms i n choosing the unknowns. Thus the rank of M must be An - 3. F o r the
case q not i n the span of {u,z,r},
we may choose /?4 so that q is i n the span {u, z,r,p4}.
Then
g* = ( 1 , 0 , 1 , ^ f t , 0 , 0 . - - - , 0 ) ,
and we have sij + S3j + q ^^^
1
(60)
= 0 i f j > A and Su + «34 + <7*/?4S44 = <7*/?4 i f j = A. Result
follows. W e summarize our discussion on the rank of the m a t r i x M of the L P problem
by the following proposition.
P r o p o s i t i o n 3.1 The rank of the matrix M in the LP problem specified by (43-45)
is An - 3.
Note that i n the L P problem, there is a further nonnegative constraint (45) on v i n
addition to the linear equation (44). T h u s a n immediate collorary to the proposition is
that the solution of the L P problem is a "corner point" of the feasible region constrained
by (44) a n d (45) w i t h only An - 3 non-trivial values. Thus at each node, on average,
Chapter
3.
17
Willow Tree
there are only 4 moves that have non-trivial probability. T h e low density of the transition
matrices gives a very high efficiency i n calculating expectations w i t h respect to these
transition probabilities.
3.3
Convergence of the willow tree
W e don't have a proof for the convergence of Y
tk
to a B r o w n i a n motion. Nevertheless, we
outline a plausible strategy for showing the convergence. We need the following Central
L i m i t Theorem i n our strategy.
T h e o r e m 3.1
Let X\,
Xi,
Central Limit
be independent variables
= 0,var(Xi)
EXj
and following
Theorem
satisfying
2
T?\v2\
= a],E\X]\
< oo
:
1
M
Y,E\X?\->0
asM-+oo
where
M
Then
1
M
5>j->Jv(o i)
f
in distribution
where N(p, cr ) is the normal distribution
2
with mean // and variance cr
2
Chapter
Willow
3.
Tree
18
T h e criterion for a process being a B r o w n i a n motion is the following (page 176 of [6]):
1. It is (almost surely) continuous,
2. the increment Y
s+t
—Y
s
is distributed as a normal N(0, t) and is independent of T ,
B
the history up to s, s > 0.
To show that Y
tk
converges to B r o w n i a n motion, we would like to show that i n the
l i m i t , Y satisfy the above two conditions. For the first condition, we require (Kolmogorov)
E\Y +h
s
— Y\
s
3
< Kh
1+e
for some K < oo and e > 0. A s for the second condition, we would
t r y to use the C e n t r a l L i m i t theorem to show that Y
s+t
- Y
s
is normal and independent
of Ts.
Let us write Y
s+t
—Y
s
as a telescoping sum
Now let us compute the mean and variance for each of the
(30-31)
E\Y (j+i)&t
s+
= 0,
—
Y jAt]
s+
I^+Q+IJA*
—
Y j& s+
t
From
Chapter
3.
19
Willow Tree
F r o m (30-33) and the fact that the mean is zero,
var(Y i)&
s+{j+
-
t
Ys+jAt)
E[(Ys+(j+i)&t -
E
<?iPij [Zj\/s
Ys+jAt) ]
2
+ (J +
= zZQi{ i(s + jAt)
i
2z
-
z^s+jAt
+ At-2zUs
+
jAt))
= *E&
A
i
=
At.
T h i s implies that a(M)
=
t .
1/2
T h e last t h i n g we need to show is that E[\Y ( )
3+
j+1
- Y jAt\ ]
3
At
s+
< K\At\
1+£
as M ->• oo
and n, the number of spatial nodes, -> oo. This suffices for condition 1 a n d it also implies
M-l
1
CT(M)
3
(61)
E
i=0
E[\Y
Since (29) is just E[\Y ^ ^
3+
+
At
possible values of E[\Y (j i)At
3+
- Y
s+u+1)At
+
\}
3
3+jAt
- Y
< Kt-^\At\
e
-+ 0.
\ ],
so the minimization produces the smallest
— Y j& \ ]-
Hence we expect the following conjecture to
3
s+jAt
3
3+
t
be true.
C o n j e c t u r e 3.1 Let {zi(n),qi(n)}
mations of the normal distribution
ofY (n,M)
tk
be some "reasonable" choices of discrete
such that the corresponding transition matricespfj(n,
satisfy the LPs specified by (35-40).
the limit M —>• oo and n(M)
approxi-
—>• oo, Y
tk
Then there exists n(M)
converges to Brownian
motion.
M)
such that in
Chapter
3.
Willow
Tree
A cfscreta approximation to Die standard normal distribution
Each point * has probability 0.1
Figure 3.1: A discrete approximation of the standard normal distribution
Chapter 4
Option
4.1
Pricing
European Option
A n equity following the Black Scholes continuous process defined i n (2), under a m a r t i n gale measure (see Chapter 3 of [1]), has the same spot price distributions as
S(t)
= S(0)e -£ ° .
{r
(62)
)tk+ Bt
Because we conjecture that the tree distribution Y
tk
constructed i n Chapter 3 converges
to the B r o w n i a n motion B , the distribution of equity prices
t
should converge to the distribution of (62) i n the limit. A standard technique for calculating an European option on S(t)
given by (62) w i t h payoff F(S(T))
at terminal time
T uses the conditional expectation w i t h respect to the distribution (62). Let V be the
f
value of the option at time t. T h e n
V
1
= E[e- F(S(T))\S(t)].
rT
(64)
In a willow tree setup, as i n a binomial tree estimation, V° is estimated by iteratively
taking conditional expectations w i t h respect to the transition probabilities. T h e terminal
condition at expiry T is
= ns^21
(65)
Chapter 4.
Option Pricing
22
T h e subsequent V^ are calculated iteratively using the formula
k
ytk
=
e
-r(t
M
- t ) J2 P^V- ,
n > 1.
k+1
k + 1
f c
(66)
j=i
A t the home node,
N
o
V
=
-n j2
1
e
q j
v^.
(67)
F r o m (39)
V°
=
=
- ^i
r
e
t k + 1
-
t k
q'P P ---P F(S )
1
2
M
T
e- q F(S )
rT
J
(68)
T
where q denotes the tranpose of q and F(S )
1
T
denotes the vector
(F(ST), • • •
,^(5^)).
Thus the price calculated from a multi-time step setup is equivalent to the price calculated
from a one time step setup. T h i s is not so surprising since the distribution of the equity
price is chosen v i a (62) and so it is not necessary to construct it v i a the tree.
4.2
C a l i b r a t i o n of
{zi,qi}
There are many ways of choosing {zi,qi} i n a willow tree setup. W e calibrate a choice
by comparing (68) for a call w i t h its corresponding closed form solution for (64). Let us
consider a n European call w i t h strike K.
T h e closed form solution for the price of the
call is
V? = ^-r= f
f
( V -1r *
r
)rWT
l5
- K) e-^dz
(69)
where
s._hsS-(r-flr
Csff
(70)
Chapter 4. Option
23
Pricing
For a given {zi, <&}, the willow tree solution (68) is then
K
=
e- J2Qi( ° ~^
rT
i=i*
=
£_
s
e(r
-)
)T+<TVTzi K
^
W
'
* ( 0 (r-^)TWT
5
e
2
i
_
K
\
e
- ^
d
z
( 7 1
)
where i* is the m i n i m u m of the index i such that Z{ > z* a n d
qt = —== /
e
2
dz.
C o m p a r i n g (69) a n d (71), it is easy to see the error V° - V° is
f
where
E
=
o
Ei
f
eh*') e~$dz
=
(f
0 =
aVT.
Zi
- e^ ) e-^dz
(72)
z
Note that except for E , the other Ei have no dependence on the strike K.
0
We would like to pick the discrete approximation {zi,qi}
between V ° a n d
c
so the absolute difference
is small. L e t a > 0 be the a number so that
(•-a+Az
-a-t-/32
/
-oo
,2
e? e~dz
z
=
o(l/N )
3
e?'e-*rdz = o(l/JV )
/
3
Ja-Az
and Az = 2a/N. W e set the ft i n the following manner.
c-o+Az
-oo
ftv = /
Ja—Az
R
2
e~~*dz
e 2 dz.
Chapter 4. Option Pricing
24
For 2 < i < N—l, we partition [—a+Az, a—Az] into intervals A = [—a + iAz, — a + (i + l)Az]
{
of equal length Az and let
be
-a+(i+l)Az
/
J2
e-^dz.
(73)
•a+iAz
Now for 2 < i < N — 1, the z% is just any point i n the interval A- For i = 1, iV we choose
2i so that A^(^i) = c?j/2 where N{x) is the standard normal distribution function.
B y expanding e@ as
Zi
eP*+ {3e (z -z)
+ 0(e (Az) )
/3z
IBz
i
(74)
2
and b y (72), for i* < i < N - 1, we have
\Ei\
= J (pe (z -z)
+
0z
A
<
i
O(p e (Az) ))e-^dz
2
(pAz + 0((1 (Az) ))
2
f
2
l3z
2
e? e-$dz.
z
T h e same argument works for \E \. T h i s implies that
0
„-/3 /2c0
-oo
2
|J571 <
_
V27T
(pAz + 0(P (Az) ))
2
2
V
/
' Jz*
2
e? e-*dz
z
+ 0(l/iV )
3
and
|£|
=
cf
_
\E\
_
^ J~ (so <r-4)rWfc _ ^
V
e
^ff_
e
-$
d z
(PAz + Q ( / ? ( A * ) ) ) / ~ e ^ d s
„C!/o
.2
"I"
2
2
0 ( 1 / i V
3
)
Cf
Q(l/N )
3
<
((3Az + 0(p (Az) ))
2
2
+
Hence if
IK/I > l / i V
3 / 2
(75)
Chapter 4. Option
25
Pricing
then
(76)
"cf
For example, N = 50, a = 5, 0 = .1, ^ fi
v
~ .01.
]
Actually, we can get a tighter bound for the case where Zi{0) vary slowly for a range
of 0 . F r o m (72), we can solve for a Zi(0) so that Ei = 0.
Zi{0) = ^ l o g
(77)
z2
(
1
S'(eP'e-4dz\
2TT
1
± l cr
0
0
°
(ePPfJ
_
\V2t
e
a
* »
d
f»
Qi
£ + log ^ w - f l + N ( « { - f l j
B y choosing the
Zj(/?)
for a particular
jf
0, it is easy to see that (72) can be written as
( e ^ - e ^ ) e - ^ d z .
E x p a n d i n g the Zi(0) at 0,
Hence, i f \zi{0) - Zi(0) \ is small, we can get a tighter bound. For example, for .1 < 0 < 1
and N = 50, i n Figure 4.2 the plot shows that |ZJ(.1) - ^ ( 1 ) | ~ 1 0 . In fact, for the
- 3
given range of parameters, \zi(0) — Zi(0)\ ~ 1 0 . T h i s implies that for \V^\ > .003, the
- 3
error should be less than 1%.
Note that the above analysis also shows that for pricing European options, the choice
{Zi, qi) based on the equal partition of q i n (21) is only accurate when the corresponding
Chapter
4.
Option
Pricing
26
z's of the non-zero payoff values are concentrated near z — 0, while our choice based
on equal p a r t i t i o n of a n interval of z w i l l be accurated for a much wider range of payoff
functions.
4.3
N u m e r i c a l Results for E u r o p e a n Options
We compare the results of European and A m e r i c a n vanilla options computed from the
willow tree and computed from the binomial tree. Here we use a 50 spatial nodes setup
i n the willow tree method.
We first compare the values V of European O p t i o n C a l l and P u t computed from the
w
willow tree setup to the values V f obtained from the closed form solution. In Table 4.2C
4.3 we list the percentage error of several test cases relative from the closed form solution.
In Table 4.2, for the cases where the errors are greater t h a n 1%, the corresponding closed
form solution of the calls is unrealistically small w i t h value < 1 0 . For a l l the other
- 5
values, the error is always w i t h i n 1%.
In Table 4.3, since the range of parameters used
Table 4.1: European C a l l :
T = l , r=0.6, M = Moneyness(=
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.0000
0.0000
0.0000
0.0006
0.0049
0.0202
0.0543
0.1099
0.3
0.0000
0.0001
0.0013
0.0065
0.0206
0.0478
0.0902
0.1472
0.4
0.0001
0.0014
0.0065
0.0193
0.0427
0.0783
0.1260
0.1847
0.5
0.0010
0.0053
0.0164
0.0369
0.0681
0.1099
0.1616
0.2221
0.6
0.0033
0.0123
0.0300
0.0575
0.0950
0.1418
0.1969
0.2592
cr = V o l a t i l i t y
0.7
0.0076
0.0220
0.0461
0.0798
0.1227
0.1736
0.2316
0.2957
0.8
0.0138
0.0339
0.0639
0.1032
0.1506
0.2051
0.9
0.0216
0.0474
0.0829
0.1269
0.1784
0.2362
0.2658
0.3317
0.2993
0.3670
1
0.0308
0.0619
0.1024
0.1509
0.2060
0.2667
0.3321
0.4015
for the puts are identical to the calls, the closed form values are never too small. Hence
the relative errors are always very small.
Chapter
4.4
4.
Option
27
Pricing
Numerical Results for American Option
The formula for calculating a vanilla A m e r i c a n option is only a slight variation of its
European counterpart, taking account of allowing early exercise of the option.
The
terminal condition at expiry T is
(78)
V f = F(S?).
The subsequent V£ are calculated iteratively using the formula
k
yt
k
=
-r(t -t
e
k+1
k)
M
A
X
^F(S<*), £
P%v} ^
k+1
,
n > 1.
(79)
A t the home node,
y ° = e " ^ max ( F ( S ° ) , | j q ^ j .
(80)
We bench mark the results of the willow tree and binomial tree calculation against
the 1000 time step binomial tree calculation. Here we use a 50 nodes setup and only
consider A m e r i c a n put. In Table 4.4, we see that other t h a n the values corresponding
to put w i t h unrealistic value 0.023, all the other 5 0 time steps and 20 time steps values
are comparable to the 200 time steps binomial tree calculation. In Table 4.5, we list the
average relative errors on the range of parameters : S p o t = 2 0 0 , r = 0 . 5 , a = 0 . 1 , 0 . 2 , • • • 0.5,
K = 1 0 0 , 1 2 0 , • • •, 300, T = 1,2. We see that on average, the 5 0 time steps willow tree is
more accurate t h a n the 200 time steps binomial tree.
L a s t l y we compare the speed performance of two methods. In Table 4.6, by comparing
400 american put calculations from a 50 nodes 20 time steps willow tree setup a n d a 200
time steps binomial tree setup, we see that the willow tree is more than 10 times faster.
Chapter
4.
Option
Pricing
28
Figure 4.2: Small variation of Zi(J3): | ^ ( . l ) - 2,(1)1 ~ I O
- 3
\v°-v°
Table 4.2: European C a l l : Percentage of
M/CT
0.2
0.3
0.4
100
3.9274
0.5
0.6
0.7
0.8
0.9
1
0.5701
0.0065
0.6478
0.4463
0.1314
0.1116
T=l,
0.3
1.5927
1.7901
0.0853
0.1393
0.0614
0.5237
0.3782
0.0332
r=0.6, M = Moneyness,
0.4
0.5
0.6
0.0696 0.2202 0.9789
0.0941 0.4644 0.3897
0.6927 0.6301 0.1908
0.3119 0.1243 0.0780
0.0096 0.1412 0.2847
0.2607 0.1997 0.1715
0.2327 0.0081 0.1465
0.0316 0.0636 0.1417
w
c }
v 6
a = Volatility
0.7
0.8
0.7266 0.5118
0.4410 0.2592
0.0460 0.3746
0.3299 0.1823
0.2095 0.1839
0.1945 0.2017
0.1609 0.0017
0.0382 0.0948
0.9
0.4824
0.4446
0.1224
0.0798
0.2040
0.1355
0.3202
0.1314
1
0.2242
0.1702
0.1018
0.0748
0.2199
0.0434
0.0204
0.0698
Chapter 4. Option Pricing
29
Table 4.3: European P u t : Percentage of
M/cr
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.2
6.E-09
2.E-08
2.E-07
2.E-06
3.E-05
9.E-05
7.E-05
l.E-04
T = l , r=0.6, M = Moneyness,
0.4
0.5
0.6
0.3
6.E-08 4.E-08 2.E-06 3.E-05
2.E-06 l . E - 0 6 2.E-05 5.E-05
l . E - 0 6 5.E-05 l . E - 0 4 6.E-05
9.E-06 6.E-05 5.E-05 5.E-05
l . E - 0 5 4.E-06 l . E - 0 4 3.E-04
3.E-04 2.E-04 2.E-04 2.E-04
3.E-04 3.E-04 l . E - 0 5 3.E-04
5.E-05 6.E-05 l . E - 0 4 4.E-04
\V°—V°
\
c
a
'
a = Volatility
0.7
0.8
6.E-05 7.E-05
l . E - 0 4 9.E-05
2.E-05 2.E-04
3.E-04 2.E-04
3.E-04 3.E-04
3.E-04 4.E-04
4.E-04 5.E-06
l . E - 0 4 3.E-04
1
7.E-05
l.E-04
l.E-04
l.E-04
5.E-04
l.E-04
7.E-05
3.E-04
0.9
l.E-04
2.E-04
l.E-04
l.E-04
4.E-04
3.E-04
l.E-03
5.E-04
Table 4.4: A m e r i c a n P u t : Percentage of '^JfooQ '
00
a
0.5
0.3
0.2
0.5
0.2
0.4
0.2
0.1
0.3
0.2
Spot=200, r=0.5, a = Volatility, K = Strike, T = M a t u r i t y i n years,
Vw N = Value of P u t using W i l l o w Tree w i t h N time steps
VB N = Value of P u t using B i n o m i a l Tree w i t h N time steps
Ew N = % error relative to the 1000 steps binomial tree calculation
EB N = % error relative to the 1000 steps binomial tree calculation
K
T V 1000 V 200
V 50
V 20 E 200 E 50 E 20
1 106.931
300
106.911 106.958 106.993
0.0256
0.0579
0.0188
1
260
61.787
61.779
61.783
61.772
0.0064
0.0252
0.0136
1
23.942
0.0462
220
23.931
23.936
23.926
0.0239
0.0681
1
24.632
180
24.560
24.623
24.669
0.2915
0.0383 0.1507
0.022
120 1
0.023
0.023
0.022
4.6582
4.6582
2.1329
2
89.852
280
89.848
89.843
89.850
0.0057
0.0049
0.0025
41.354
41.211
240 2
41.363
41.310
0.0216
0.1283 0.3677
200 2
5.755
5.760
5.720
5.665
0.0986
0.6105 1.5540
180 2
16.679
16.663
16.680
16.701
0.0939
0.0088 0.1346
140 2
1.089
1.086
0.2494
1.086
1.088
0.0867
0.2239
B
B
w
w
B
w
Table 4.5: A m e r i c a n P u t : Average Percentage of ^
V
w
^°° ^
0
v
Spot=200, r=0.5, a = 0.1,0.2, • • • 0.5, K = 100,120, • • •, 300, T = 1,2
average of E 200 average of Ew 50
average of E 20
0.37
0.46
0.51
B
w
Chapter
4.
Option
Pricing
30
Table 4.6: A m e r i c a n P u t : 50 nodes 20 time steps willow tree vs 200 time steps b i n o m i a l
tree
C P U time for 400 calculations
cycle per sec 1000000
W i l l o w Tree
C P U cycles
start load time W T
0
load time
end load time W T
140000
140000
start time W T
2550000
load time & calculations
end time W T
C P U cycles
B i n o m i a l Tree
start time B T
end time B T
4330000
31880000
calculations
C P U time (sec)
0.1400
2.5500
C P U time (sec)
27.5500
Chapter 5
Extension
5.1
Extension to other models
T h e setup of willow tree for pricing vanilla options can be easily extended to models on
equity w i t h dividend payments and some other type of options. We give a list of some
of the modifications.
• E q u i t y w i t h continuous dividend payments :
Adjust the spot values at each node by the discount factor e~
qAt
where q is the
dividend rate.
• E q u i t y w i t h discret dividend payments :
Adjust the spot values by the dividend payments at the nodes correponding to the
payment schedule.
• Barrier Options :
Use a n interpolation on the lower and upper boundaries expections. For example,
consider a single down and out barrier w i t h barrier B.
S*£.\ and S£j. < B < S £ j .
+ 1
A t tk+i, let S^
+1
For z < ra*, we compute
ra—1
J=l
TO
i=i
We define Vl
k
by an interpolation between
31
and V^£ .
p
< B <
Chapter
5.
32
Extension
• B u r m u d a n Options :
In the willow tree setup, the set of discrete time nodes used must contain the a l lowable exercise dates as a subset. In the iterative procedure for calculating V f ,
fc
apply (79) on the allowable exercise dates and apply (66) on the non-allowable exercise dates. Note that between two time nodes which are adjacent to the allowable
exercise dates and have no allowable exercise dates i n between, the iterative procedure is E u r o p e a n and only one time step calculation is needed. It is an interesting
problem to determine the m i n i m u m number of non-uniform time steps needed for
the calculation when using transition matrices calculated from uniform time steps
w i t h A t = 1 day.
• T w o factor models :
In a two factor stochastic model, the B r o w n i a n motion is two dimensional. Let p
be the instantaneous correlation between the components Bi(t)
a n d E>2(t). T h e
second component B<i(t) can be constructed from two independent one dimensional
B r o w n i a n motions Bi(t)
and B$(t)
where
B {t)=pB (t)-Jl^fiB {t).
2
1
z
(81)
Thus to set up the corresponding W i l l o w Tree for the B r o w n i a n motion, one can
use a three dimensional grid where at each time node tk, the two dimensional
grid (y/Uz ,py/tkZ
m
- \ / l - p' yftk~Zm) is a discretized version of a two dimensional
2
m
B r o w n i a n motion. Similar constructions can be used for higher dimensional willow
tree setups.
• Deterministic time dependent instantaneous volatility :
Suppose the instantaneous volatility o(t) is deterministic and time dependent. T h e
willow tree setup accommodates a(t) by replacing Yk = y/h z w i t h Yk = yJ~L{t) z
33
Chapter 5. Extension
where
E(t) = f
a (s)ds.
2
Js=0
Stochastic volatility :
A more challenging extension of the willow tree setup is to construct a discrete
process that incorporates implied volatility smile. Using the idea introduced by
K l a u s Said [7], we can consider the two factor model
S
dY
X (K,T)
=
rdt+
^/x(S,t)YdB
1
= ct{Y - Y)dt +
(VYdB
2
2
E[Y (T)\S(T) = K)
2
K
dK '
2
where CK,T is the current value of a call o n S w i t h strike K a n d m a t u r i t y T. In
a willow tree setup, i f one is able to solve for X(S,t)
recursively on a grid, then a
tree distribution of S can be determined. A s a result, after appropriate calibration
on the parameter £, option prices can be determined from the tree distribution of
S.
5.2
Conclusion
We have presented the setup of a willow tree algorithm for calculating prices of vanilla
options. F o r better accuracy on a larger class of options, a good choice of the discrete
approximation of the standard normal distribution used i n the setup is calibrated by
using a n equal partition of on an interval of the variate z.
F r o m the results of the
numerical calculation, the speed performance of the willow tree is much superior to
the binomial tree. T h e superority will even be more extensive for multi-factor models.
J
Chapter
5.
Extension
34
F u r t h e r extensions of the setup can be applied to other options. In particular, m o r e
is r e q u i r e d if o n e w o u l d like t o c o n s i d e r a m o d e l t h a t i n c o r p o r a t e s s t o c h a s t i c
a n d is c o n s i s t e n t w i t h t h e m a r k e t v o l a t i l i t y s m i l e .
work
volatility
Bibliography
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[3] M i c h a e l C u r r a n , June 1, 1998, W i l l o w Power, Riskcare, L i m i t e d , L o n d o n , E n g l a n d .
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35