Applied Mathematical Sciences, Vol. 7, 2013, no. 128, 6355 - 6367HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2013.39511

WKB Approximation for the Sum of

Two Correlated Lognormal Random Variables

C.F. Lo

Institute of Theoretical Physics and Department of PhysicsThe Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

cflo@phy.cuhk.edu.hk

Copyright c© 2013 C.F. Lo. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduc-

tion in any medium, provided the original work is properly cited.

Abstract

In this paper we apply the idea of the WKB method to derive aneffective single lognormal approximation for the probability distributionof the sum of two correlated lognormal variables. An approximate prob-ability distribution of the sum is determined in closed form, and illus-trative numerical examples are presented to demonstrate the validityand accuracy of the approximate distribution. Our analysis shows thatthe proposed method is able to provide a simple, efficient and accurateapproximation to this probability distribution of the sum of two corre-lated lognormal variables. We also discuss how this new approach canbe straightforwardly extended to study the sum of N lognormals.

Keywords: Lognormal random variables, probability distribution func-tions, backward Kolmogorov equation, Lie-Trotter splitting approximation,WKB approximation

1 Introduction

The probability distribution of the sum of two correlated lognormal stochas-tic variables has many important applications in various fields such as telecom-munication studies [1-6], financial modelling [7-9], actuarial science [10-12],biosciences [13], physics [14,15], etc. Although the lognormal distribution iswell known in the literature [16,17], yet almost nothing is known of the prob-ability distribution of the sum of two correlated lognormal variables. Thus,

6356 C.F. Lo

much effort has been made to look for good analytical approximations for thedesired probability distribution [1-6,8,18-27]. Essentially, these analytical ap-proximations assume a specific distribution that the sum of the two correlatedlognormal variables follows, and then use a variety of methods, e.g. momentmatching, moment generating function matching, least squares fitting, etc.,to identify the parameters for that specific distribution. The most commonlyused approximate distribution for the sum is a single lognormal distributionbecause numerical computations have shown that the probability distributionof the sum is a distribution which bears some resemblance to the lognormaldistribution [1]. However, no explicit mathematical justification for the specificdistribution has been given. In spite of this shortcoming, these approximationsattract considerable attention and have been extended to approximate the sumsof N correlated lognormal variables, too.

Recently, by means of the Lie-Trotter operator splitting method (Trot-ter, 1959), Lo [28] showed that both the sum and difference of two correlatedlognormal stochastic processes could be approximated by a shifted lognormalstochastic process, and approximate probability distributions were determinedin closed form. Unlike previous studies which treat the sum and difference in aseparate manner [1-6,8,18-27], Lo’s method provides a new unified approach toaccurately model the dynamics of both the sum and difference of two lognormalvariables. In this communication, based upon the Lie-Trotter operator splittingapproximation proposed by Lo, we apply the idea of the WKB method [29] toderive an effective single lognormal approximation for the dynamics of the sumof two correlated lognormal variables. An approximate probability distributionof the sum is determined in closed form, and illustrative numerical examplesare presented to demonstrate the validity and accuracy of the approximate dis-tribution. In accordance with the analysis, the proposed method is not onlyable to provide a simple, efficient and accurate approximation to the probabil-ity distribution of the sum of two correlated lognormal variables, but also ithas a better performance than Lo’s approximation. Furthermore, we discusshow this new approach can be extended to study the sum of N lognormals aswell.

2 Effective Single Lognormal Approximation

Given two lognormal stochastic variables S1 and S2 obeying the stochasticdifferential equations:

dSi

Si

= σidZi , i = 1, 2 (1)

where σ2i =Var(lnSi), dZi denotes a standard Weiner process associated with

Si, and the two Weiner processes are correlated as dZ1dZ2 = ρdt, the probability

Sum of two correlated lognormals 6357

distribution function of the sum of two correlated lognormal variables, i.e.P+ (S+, t;S10, S20, t0), satisfies the backward Kolmogorov equation [30-33]{

∂

∂t0+ L

}P+ (S+, t;S10, S20, t0) = 0 (2)

for t > t0 where

L =1

2σ21S

210

∂2

∂S210

+ ρσ1σ2S10S20∂2

∂S10∂S20

+1

2σ22S

220

∂2

∂S220

,

subject to the boundary condition [31]

P+ (S+, t;S10, S20, t0 → t) = δ (S10 + S20 − S+) . (3)

To solve for P+ (S+, t;S10, S20, t0), the backward Kolmogorov equation is firstre-written in terms of the new variables S±0 ≡ S10 ± S20 as{

∂

∂t0+ L+ + LR

}P+ (S+, t;S+0, S−0, t0) = 0 (4)

where

L+ =1

8

[σ2+ (S+0)

2 + 2(σ21 − σ2

2

)S+0S−0 + σ2

− (S−0)2] ∂2

∂S2+0

(5)

LR =1

4

{(σ21 − σ2

2

) [(S+0)

2 + (S−0)2]+

(σ21 + σ2

2

)S+0S−0

} ∂2

∂S+0∂S−0

+

1

8

[σ2+ (S−0)

2 + 2(σ21 − σ2

2

)S+0S−0 + σ2

− (S+0)2] ∂2

∂S2−0

(6)

σ± =√σ21 + σ2

2 ± 2ρσ1σ2 . (7)

The corresponding boundary condition now becomes

P+ (S+, t;S+0, S−0, t0 → t) = δ (S+0 − S+) . (8)

Accordingly, the formal solution of Eq.(5) is given by

P+ (S+, t;S+0, S−0, t0) = exp{

(t− t0)(L+ + LR

)}δ (S+0 − S+) . (9)

Since the exponential operator in Eq.(10) is difficult to evaluate, the Lie-Trotter operator splitting method [34] can be applied to approximate the op-

6358 C.F. Lo

erator by1

OLT+ = exp

{(t− t0) L+

}exp

{(t− t0) LR

}, (10)

and obtain an approximation to the formal solution P+ (S+, t;S+0, S−0, t0),namely

PLT+ (S+, t;S+0, S−0, t0) = OLT

+ δ (S+0 − S+)

= exp{

(t− t0) L+

}δ (S+0 − S+) (11)

where the relation exp{

(t− t0) LR

}δ (S+0 − S+) = δ (S+0 − S+) is utilized.

It is apparent that the approximate solution PLT+ (S+, t;S+0, S−0, t0) satisfies

the backward Kolmogorov equation{∂

∂t0+

1

2σ2effS

2+0

∂2

∂S2+0

}PLT+ (S+, t;S+0, S−0, t0) = 0 (12)

with the boundary condition

PLT+ (S+, t;S+0, S−0, t0 → t) = δ (S+0 − S+) , (13)

where

σ2eff =

1

4

[σ2+ + 2

(σ21 − σ2

2

) S−0

S+0

+ σ2−

(S−0

S+0

)2]

= σ21

(S10

S+0

)2

+ 2ρσ1σ2

(S10

S+0

)(S20

S+0

)+ σ2

2

(S20

S+0

)2

. (14)

As the volatility σeff is a function of S+0, the solution PLT+ (S+, t;S+0, S−0, t0)

cannot be a lognormal distribution. Then, here comes the question: “Is itpossible to derive an accurate single lognormal approximation to the solution

1Suppose that one needs to exponentiate an operator C which can be split into twodifferent parts, namely A and B. For simplicity, let us assume that C = A + B, where the

exponential operator exp(C)

is difficult to evaluate but exp(A)

and exp(B)

are either

solvable or easy to deal with. Under such circumstances the exponential operator exp(εC)

,

with ε being a small parameter, can be approximated by the Lie-Trotter splitting formula:

exp(εC)

= exp(εA)

exp(εB)

+O(ε2)

.

This can be seen as the approximation to the solution at t = ε of the equation dY /dt =(A + B

)Y by a composition of the exact solutions of the equations dY /dt = AY and

dY /dt = BY at time t = ε. Details of the Lie-Trotter splitting approximation can be foundin [34-39].

Sum of two correlated lognormals 6359

PLT+ (S+, t;S+0, S−0, t0)?” To answer this question, we may apply the idea of

the WKB method which is a powerful tool for obtaining a global approximationto the solution of a linear ordinary differential equation.2

Proposition :

If σeff is a slowly-varying function of S+0, i.e.

S+0

σeff

∣∣∣∣∂σeff∂S+0

∣∣∣∣� 1 , (15)

then the solution PLT+ (S+, t;S+0, S−0, t0) can be approximated by

PLNeff (S+, t;S+0, S−0, t0) =

1

S+

√2πσ2

eff (t− t0)×

exp

{−[ln (S+/S+0) + 1

2σ2eff (t− t0)

]22σ2

eff (t− t0)

}(16)

which resembles the lognormal distribution very closely.

Proof :

First of all, it is not difficult to show that

S+0

σeff

∣∣∣∣∂σeff∂S+0

∣∣∣∣ =S+0

2σ2eff

∣∣∣∣∂σ2eff

∂S+0

∣∣∣∣=

∣∣∣∣∣{

(σ21 − σ2

2) + σ2− [S−0/S+0]

σ2+ + 2 (σ2

1 − σ22) [S−0/S+0] + σ2

− [S−0/S+0]2

}

× S−0

S+0

∣∣∣∣ � 1 (17)

2The WKB method provides approximate solutions of differential equations of the form

d2y (x)

dx2+ k (x)

2y (x) = 0 ,

provided that k (x) is slowly varying, i.e.∣∣∣∣ 1

k (x)

dk (x)

dx

∣∣∣∣� 1 .

The completed approximate solution is given by

y (x) ≈ 1√k (x)

exp

{±i∫

k (x) dx

}.

It is obvious that the approximate solution will be reduced to the usual plane-wave solutionif k (x) is replaced by a constant. Details of the method can be found in [29,40,41].

6360 C.F. Lo

provided (σ21 − σ2

2) /σ2+ 6 1 (or equivalently, ρ > −σ2/σ1) and |S−0/S+0| � 1.

Similarly, we can also show that

S2+0

4σ2eff

∣∣∣∣∂2σ2eff

∂S2+0

∣∣∣∣ =

∣∣∣∣∣{

(σ21 − σ2

2) + 32σ2− [S−0/S+0]

σ2+ + 2 (σ2

1 − σ22) [S−0/S+0] + σ2

− [S−0/S+0]2

}

× S−0

S+0

∣∣∣∣ � 1 . (18)

Then, substituting PLNeff (S+, t;S+0, t0) into the left-hand side (L.H.S.) of Eq.(13),

we obtain, after simplification,

L.H.S. = − S+0

2 (t− t0)

[ln

(S+

S+0

)+

1

2σ2eff (t− t0)

](∂σ2

eff

∂S+0

)∂PLN

eff

∂(σ2eff

) −1

2σ2effS

2+0

[(∂2σ2

eff

∂S2+0

)∂PLN

eff

∂(σ2eff

) +

(∂σ2

eff

∂S+0

)2∂2PLN

eff

∂(σ2eff

)2]

(19)

where

∂PLNeff

∂(σ2eff

) =1

2σ2eff

{1

σ2eff (t− t0)

[ln

(S+

S+0

)+

1

2σ2eff (t− t0)

]2−[

ln

(S+

S+0

)+

1

2σ2eff (t− t0)

]− 1

}PLNeff (20)

and

∂2PLNeff

∂(σ2eff

)2 =1

2(σ2eff

)2 {3

2+ 3 ln

(S+

S+0

)+ σ2

eff (t− t0) +(1

2− 3

σ2eff (t− t0)

)[ln

(S+

S+0

)+

1

2σ2eff (t− t0)

]2−

1

σ2eff (t− t0)

[ln

(S+

S+0

)+

1

2σ2eff (t− t0)

]3+

1

2(σ2eff

)2(t− t0)2

[ln

(S+

S+0

)+

1

2σ2eff (t− t0)

]4}PLNeff .(21)

Since σeff is a slowly-varying function of S+0 as shown in Eq.(18) and Eq.(19),and

|σ1S10 − σ2S20|S+0

< σeff <σ1S10 + σ2S20

S+0

, (22)

it can be inferred that L.H.S. ≈ 0 in Eq.(20) and PLNeff (S+, t;S+0, S−0, t0) can

be a good approximate solution of Eq.(13). (Q.E.D.)

Sum of two correlated lognormals 6361

As a consequence, we have succeeded in deriving an effective single lognor-mal approximation PLN

eff (S+, t;S+0, S−0, t0) to the probability distribution ofthe sum of two correlated lognormal variables, namely P+ (S+, t;S+0, S−0, t0).The essence of the proposed approximation can be summarised as follows:

1. We first assume that the sum S+ of the two correlated lognormal variablesis governed by the lognormal process:

dS+ = σS+dZ+ (23)

for some constant parameter σ. Then the probability distribution func-tion of the sum S+ is given by the lognormal distribution

f (S+, t;S+0, t0)

=1

S+

√2πσ2 (t− t0)

exp

{−[ln (S+/S+0) + 1

2σ2 (t− t0)

]22σ2 (t− t0)

}(24)

for t > t0.

2. Next, we replace the constant σ of the lognormal distribution f (S+, t;S+0, t0)by σeff which is a function of the values of the two correlated lognormalvariables S1 and S2 at t = t0. The resultant function PLN

eff (S+, t;S+0, S−0, t0)gives a good approximation of the probability distribution of the sum ofthe two correlated lognormal variables, provided that σeff is a slowly-varying function of S+.

3 Illustrative Numerical Results

In Figure 1 we plot the approximate closed-form probability distributionfunction of the sum S+ given in Eq.(17) for different values of the input pa-rameters. We start with S10 = 110, S20 = 100, σ1 = 0.25 and σ2 = 0.15in Figure 1(a). Then, in order to examine the effect of S20, we decrease itsvalue to 70 in Figure 1(b) and to 40 in Figure 1(c). In Figures 1(d)-(f) werepeat the same investigation with a new set of values for σ1 and σ2, namelyσ1 = 0.5 and σ2 = 0.3. Without loss of generality, we set t− t0 = 1 for simplic-ity. Both the (numerically) exact results obtained by numerical integrationsand the approximate probability distribution function of Lo’s approximation[28] are also included for comparison. It is clear that the proposed approxi-mation can provide accurate estimates for the exact values and have a betterperformance than Lo’s approximation. Moreover, to have a clearer pictureof the accuracy, we plot the corresponding errors of the estimation in Figure2. We can easily see that major discrepancies appear around the peak of theprobability distribution function and that the estimation deteriorates as the

6362 C.F. Lo

correlation parameter ρ decreases from 0.5 to −0.5. It is also observed thatthe errors increase with the ratio S−0/S+0 as expected but they seem to be lesssensitive to the changes in σ1 and σ2.

4 Conclusion

Based upon the Lie-Trotter operator splitting approximation proposed byLo [28], we have applied the idea of the WKB method to derive an effectivesingle lognormal approximation for the dynamics of the sum of two correlatedlognormal stochastic variables. An approximate probability distribution ofthe sum is determined in closed form, and illustrative numerical examples arepresented to demonstrate the validity and accuracy of these approximate dis-tributions. The analysis shows that the proposed method provides a simple,efficient and accurate approximation to the probability distribution of the sumof two correlated lognormal variables.

Moreover, this new approach can be straightforwardly extended to studythe sum of N lognormals by simply re-defining the σ2

eff in Eq.(15) as

σ2eff =

N∑i,j=1

ρijσiσj

(Si0

S+0

)(Sj0

S+0

)where ρii = 1 and ρij = ρji. With this revised σeff , the probability distributionof the sum of N lognormals can be accurately approximated by the distributionfunction PLN

eff (S+, t;S+0, S−0, t0) in Eq.(17). The proof can be outlined asfollows:

1. As in the case of two lognormals, we first defineN new stochastic variablesin terms of the N lognormals, one of which represents the sum S+.

2. Then we write down the backward Kolmogorov equation for the sum S+

as in Section II.

3. Next, applying the Lie-Trotter splitting approximation, we can derivean approximate backward Kolmogorov equation for the sum S+. Thisbackward equation is the same as Eq.(13) except that the revised σeff isused instead.

4. Finally, we apply the idea of the WKB method to show that the dis-tribution function PLN

eff (S+, t;S+0, S−0, t0) with the revised σeff providesa good approximation to the probability distribution of the sum of Nlognormals.

Sum of two correlated lognormals 6363

Figure 1: Probability density vs. S1 + S2. The solid lines denote the distri-butions of the effective single lognormal approximation, the dotted lines showthe results of Lo’s approximation [28], and the dash lines represent the (nu-merically) exact results. (a) S10 = 110, S20 = 100, σ1 = 0.25 and σ2 = 0.15;(b) S10 = 110, S20 = 70, σ1 = 0.25 and σ2 = 0.15; (c) S10 = 110, S20 = 40,σ1 = 0.25 and σ2 = 0.15; (d) S10 = 110, S20 = 100, σ1 = 0.5 and σ2 = 0.3; (e)S10 = 110, S20 = 70, σ1 = 0.5 and σ2 = 0.3; (f) S10 = 110, S20 = 40, σ1 = 0.5and σ2 = 0.3.

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6364 C.F. Lo

Figure 2: Error vs. S1 + S2. The solid lines denote the errors of the effectivesingle lognormal approximation and the dotted lines show the errors of Lo’sapproximation [28]. (a) S10 = 110, S20 = 100, σ1 = 0.25 and σ2 = 0.15;(b) S10 = 110, S20 = 70, σ1 = 0.25 and σ2 = 0.15; (c) S10 = 110, S20 = 40,σ1 = 0.25 and σ2 = 0.15; (d) S10 = 110, S20 = 100, σ1 = 0.5 and σ2 = 0.3; (e)S10 = 110, S20 = 70, σ1 = 0.5 and σ2 = 0.3; (f) S10 = 110, S20 = 40, σ1 = 0.5and σ2 = 0.3.

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Received: September 16, 2013