Applied Mathematics, 2013, 4, 1503-1511
Published Online November 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.411203
Open Access AM
The Sum and Difference of Two Constant Elasticity of
Variance Stochastic Variables
Chi-Fai Lo
Department of Physics, Institute of Theoretical Physics,
The Chinese University of Hong Kong, Hong Kong, China
Email: cflo@phy.cuhk.edu.hk
Received August 20, 2013; revised September 20, 2013; accepted September 27, 2013
Copyright © 2013 Chi-Fai Lo. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We have applied the Lie-Trotter operator splitting method to model the d ynamics of both the sum and difference of two
correlated constant elasticity of variance (CEV) stochastic variables. Within the Lie-Trotter splitting approximation,
both the sum and difference are shown to follow a shifted CEV stochastic process, and approximate probability distri-
butions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and
accuracy of these approximate distributions. These approximate probability distributions can be used to valuate
two-asset options, e.g. spread options and basket options, where the CEV variables represent the forward prices of the
underlying assets. Moreover, we believe that this new approach can be extended to study the algebraic sum of N CEV
variables with potential applications in pricing multi-asset options.
Keywords: Constant Elasticity of Variance Stochastic Variables; Probability Distribution Functions; Backward
Kolmogorov Equation ; Lie-Trotter Splitting Approximation
1. Introduction
Recently Lo [1] proposed a new simple approach to ta c k le
the long-standing problem: “Given two correlated log-
normal stochastic variables, what is the stochastic dy-
namics of the sum or difference of the two variables?”; or
equivalently, “What is the probability distribution of the
sum or difference of two correlated lognormal stochastic
variables?” The solution to this problem has wide appli-
cations in many fields including financial modelling, act u-
arial sciences, telecommunications, biosciences and phys-
ics [2-15]. By means of the Lie-Trotter op erator splitting
method [16], Lo showed that both the sum and difference
of two correlated lognormal stochastic processes could
be approximated by a shifted lognormal stochastic proc-
ess, and approximate probability distributions were de-
termined in closed form. Unlike previous studies which
treat the sum and difference in a separate manner [2-5,
8,13,15,17-27], Lo’s method provides a new unified ap-
proach to model the dynamics of both the sum and dif-
ference of the two stochastic variables. In addition, in
terms of the approximate solution s, Lo presented an ana-
lytical series expansion of the exact solutions, which can
allow us to improve the appro ximation systematically.
In this communication we extend Lo’s approach to
study the dynamics of both the sum and difference of two
correlated constant elasticity of variance (CEV) stochas-
tic processes. The CEV process was first proposed by
Cox and Ross [28] as an alternative to the lognormal
stochastic movements of stock prices in the Black-
Scholes model. The CEV process, which is defined by
the stochastic differential equation
2
ddfor0SSZ
2
 (1)
with dZ being a standard Weiner process, has been
receiving much attention because it has the ability to give
rise to a volatility skew and to explain th e volatility s mile
[29-41]. In the limiting case 2
, the CEV model
returns to the conventional Black-Scholes model, whilst
it is reduced to the Orn stein-Uhlen beck mod el in the case
of 0
. By the Lie-Trotter operator splitting method,
we show that both the sum and difference of two corre-
lated CEV processes can be modelled by a shifted CEV
process. Approximate probability distributions of both the
sum and difference are determined in closed form, and
illustrative numerical examples are presented to demon-
strate the validity and accuracy of these approximate dis-
tributions.
C.-F. LO
1504
2. Lie-Trotter Operator Splitting Method
We consider two CEV stochastic variables 1 and 2,
which are described by the stochastic differential equa-
tions:
S S
2
dd ,
iii i
SSZi
1,2
2
(2)
for 0
. Here di
Z
denotes a standard Weiner
process associated with , and the two Weiner proc-
esses are correlated as 12
i
S
dd d
Z
Zt
1
. Without loss of
generality, we also assume that 2
. The joint prob-
ability distribution function
, ,S t
1 2
,,PSS t1020 0 of the
two correlated CEV variables obeys the backward Kol-
mogorov equation [42-45]
;S

121020 0
0
ˆ,,; ,,0LPSStSS t
t



 (3)
where
22
222
1101210 20
210 20
102
2
220 2
20
1
ˆ2
1,
2
LS SS
SS
S
SS



 
(4)
subject to the bound ary condition

121020 0110220
,,; ,,PSS tSSttSSSS

 

.
(5)
This joint probability distribution function tells us how
probable the two CEV variables assume the values
and 2 at time 0, provided that their values at
are given by 10 and 20 . Once
is found, the probability distribution of the sum or dif-
ference, namely 12
, of the two correlated
CEV variables can be ob taine d by evaluati ng the integ ra l
1
S
0
t
0
,t
Stt
S S
SS
1 21020
,,; ,PS StSS
S


1020 0
1 21210 20012
00
,; ,,
dd, ,;,,.
PStSS t
SSPSStSS tSSS



(6)
Unfortunately, the joint probability distribution function
is not available in closed form, except for the case of
0
. Hence, we must resort to numerical methods, e.g.
the finite-difference method or Monte Carlo simulation.
Nevertheless, the numerically exact solution does not
provide any information about the stochastic dynamics of
the sum or difference explicitly.
It is observed that the probability distribution of the
sum or difference of the two correlated CEV variables,
i.e.
1020 0
,; ,,PStSS t
, also satisfies the same back-
ward Kolmogorov equation given in Equation (3), but
with a different boundary condition [43]

1020 01020
,; ,,.
P
StS S ttSSS

 (7)
To solve for
1020 0
,; ,,

000
0
ˆˆˆ,;,,0LLLPStSSt
t



0
 

 (8)
where
2
2
2
00 0
112
00
2
20
22
00
1
ˆ121
22
1
SS S
LSS
S
SS


 



 


 

 





(9)
2
2
2
00 0
112
00
2
20
22
00
1
ˆ121
22
1
SS S
LSS
S
SS


 



 


 

 





(10)
2
22
00 00
01 200
ˆ22
SS SS
LSS


 











(11)
22
12 12
2.
 
 (12)
The corresponding boundary condition now becomes

000 0
,; , ,.PStSSttS S
 
 
(13)
Accordingly, the formal solution of Equation (8) is
readily given by




000
00 0
,; , ,
ˆˆˆ
exp .
PStSSt
tt LL LSS



 
(14)
Since the exponential op erator

00
ˆˆˆ
exp tt LLL

is difficult to evaluate, we
apply the Lie-Trotter operator splitting method1 to ap-
*Suppose that one needs to exponentiate an operator which can be
ˆ
C
split into two different parts, namely ˆ
A
and ˆ
B
. For simplicity, let us
assume that ˆˆˆ
CAB
, where the exponential operator
ˆ
exp Cis
difficult to evaluate but
ˆ
exp
A
and

ˆ
exp
B
are either solvable or
easy to deal with. Under such circumstances the exponential operator
ˆ
exp C
, with
being a small parameter, can be approximated by
the Lie-Trotter splitting formula (Trotter, 1959):


2
ˆˆˆ
expexp exp.CABO


This can be seen as the approximation to the solution at t
of the
equation
ˆ
ˆˆˆ
dd
Yt ABY by a composition of the exact solutions
of the equations ˆ
ˆˆ
ddYtAY and ˆˆ
ddYtBY
P
StS S t
, we first rewrite the
backward Kolmogorov equation in terms of the new
variables as
0102
SSS
 ˆ
at time t
.
0
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C.-F. LO 1505
proximate the operator by [16,46-49]




000
ˆˆˆ
ˆ
exp exp,
LT
OttLttLL

 
(15)
and obtain an approximation to the formal solution

000
,; , ,
P
StSSt

, namely





000
00
,; , ,
ˆˆ
exp
LT
LT
PStSSt
OSSttLSS

 

 
0
(16)
where the relation




00 00
ˆˆ
exp tt L LSSSS

 
 
is util-
ized. For

2
00 1SS
, which is normally valid unless
10 and 20 are both close to zero, the operators S Sˆ
L
and can be approximately expressed as
ˆ
L
2
202
0
1
ˆ2
LS
S

(17)
in terms of the two new variables:
22
12
00 2
SS S






0
(18)
2
00 0
22
12
.SS S






(19)
Here the parameters
and
are defined by
2
2
(20)
2
22
112 .
2






(21)
Obviously, both and
S
S
are CEV random vari-
ables defined by the stochastic differential equations
2
dSSZ

d,

(22)
and their closed-form probability distribution functions
are given b y














CEV 00
1212222
00
12
22
2
00
22
0
220
,; ,
24
22
2
exp 2
fStSt
SS SS
I
tt tt
SS
tt

 


 


 











 

(23)
for 0, where denotes the modified Bessel
function of order
tt

I
. As a result, it can be inferred that
within the Lie-Trotter splitting approximation both S
and S
are governed by a shifted CEV process. It
should be noted that for the Lie-Trotter splitting ap-
proximation to be valid, needs to be small.
20
tt
3. Illustrative Numerical Examples
In Figure 1 we plot the approximate closed-form prob-
ability distribution function of the sum of two un-
correlated CEV variables ( i.e. S
0
) given in Equation
(23) for different values of the input parameters. We start
with 10110S
, 20 105S
, 10.5
and 20.3
in
Figure 1(a). Then, in order to examine the effect of 20,
we decrease its value to in
Figure 1(b) and to 65 in
Figure 1(c). In Figures 1(d)-(f) we repeat the same
investigation with a new set of values for 1
S
85
and 2
,
namely 10.3
and 20.2
. Without loss of gener-
ality, we set 01tt
for simplicity. The (numerically)
exact results which are obtained by numerical integra-
tions are also included for comparison. It is clear that the
proposed approximation can provide accurate estimates
for the exact values.
In order to have a clearer picture of the accuracy, we
plot the corresponding errors of the estimation in Figure
2. We can easily see that major discrepancies appear
around the peak of the probability distribution function
and that the estimation deteriorates as the elasticity pa-
rameter
increases. It should be noted that the pro-
posed approximation is exact in the special case of
0
, i.e. the Ornstein-Uhlenbeck model. We also
observed that the errors increase with the ratio 00
SS
and 2
(or equivalently, 1
and 2
) as expected.
Next, we apply the same sequence of analysis to the
approximate closed-form probability distribution func-
tion of the difference S
given in Equation (23). Simi-
lar observations about the accuracy of the proposed ap-
proximation can be made for the difference S
, too (see
Figures 3 and 4).
4. Conclusion
In this communication we have applied a new unified
approach proposed by Lo (2012) to model the dynamics
of both the sum and difference of two correlated CEV
stochastic variables. By the Lie-Trotter operator splitting
method, both the sum and difference are shown to follow
a shifted CEV stochastic process, and approximate
probability distributions are determined in closed form.
Illustrative numerical examples are presented to demon-
strate the validity and accuracy of these approximate
distributions. These approx imate probability distributio ns
can be used to valuate two-asset options, e.g. the spread
options, where the CEV variables represent the forward
prices of the underlying assets. Moreover, we believe
that this new approach can be extended to study the
Open Access AM
C.-F. LO
Open Access AM
1506
(a) (b)
(c) (d)
(e) (f)
Figure 1. Probability density vs. S1 + S2: the dotted lines denote the distributions of the approximate shifted CEV process, and
the solid lines show the exact results. (a) S10 = 110, S20 = 105, σ1 = 0.5 and σ2 = 0.3; (b) S10 = 110, S20 = 85, σ1 = 0.5 and σ2 = 0.3;
(c) S10 = 110, S20 = 65, σ1 = 0.5 and σ2 = 0.3; (d) S10 = 110, S20 = 105, σ1 = 0.3 and σ2 = 0.2; (e) S10 = 110, S20 = 85, σ1 = 0.3 and σ2
= 0.2; (f) S10 = 110, S20 = 65, σ1 = 0.3 and σ2 = 0.2.
C.-F. LO 1507
(a) (b)
(c) (d)
(e) (f)
Figure 2. Error vs. S1 + S2: the error is calculated by subtracting the approximate estimate from the exact result. (a) S10 =
110, S20 = 105, σ1 = 0.5 and σ2 = 0.3; (b) S10 = 110, S20 = 85, σ1 = 0.5 and σ2 = 0.3; (c) S10 = 110, S20 = 65, σ1 = 0.5 and σ2 = 0.3; (d)
S10 = 110, S20 = 105, σ1 = 0.3 and σ2 = 0.2; (e) S10 = 110, S20 = 85, σ1 = 0.3 and σ2 = 0.2; (f) S10 = 110, S20 = 65, σ1 = 0.3 and σ2 =
0.2.
Open Access AM
C.-F. LO
1508
(a) (b)
(c) (d)
(e) (f)
Figure 3. Probability density vs. S1 S2: the dotted lines denote the distributions of the approximate shifted CEV process, and
the solid lines show the exact results. (a) S10 = 110, S20 = 105, σ1 = 0.5 and σ2 = 0.3; (b) S10 = 110, S20 = 85, σ1 = 0.5 and σ2 = 0.3;
(c) S10 = 110, S20 = 65, σ1 = 0.5 and σ2 = 0.3; (d) S10 = 110, S20 = 105, σ1 = 0.3 and σ2 = 0.2; (e) S10 = 110, S20 = 85, σ1 = 0.3 and σ2
= 0.2; (f) S10 = 110, S20 = 65, σ1 = 0.3 and σ2 = 0.2.
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C.-F. LO 1509
(a) (b)
(c) (d)
(e) (f)
Figure 4. Error vs. S1 S2: the error is calculated by subtracting the approximate estimate from the exact result. (a) S10 =
110, S20 = 105, σ1 = 0.5 and σ2 = 0.3; (b) S10 = 110, S20 = 85, σ1 = 0.5 and σ2 = 0.3; (c) S10 = 110, S20 = 65, σ1 = 0.5 and σ2 = 0.3; (d)
S10 = 110, S20 = 105, σ1 = 0.3 and σ2 = 0.2; (e) S10 = 110, S20 = 85, σ1 = 0.3 and σ2 = 0.2; (f) S10 = 110, S20 = 65, σ1 = 0.3 and σ2 =
0.2.
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C.-F. LO
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algebraic sum of N CEV variables with potential applica-
tions in pricing multi-asset options.
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