Journal of Applied Mathematics

Volume 2012, Article ID 838397, 13 pages

http://dx.doi.org/10.1155/2012/838397

## The Sum and Difference of Two Lognormal Random Variables

Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Hong Kong

Received 15 May 2012; Accepted 19 July 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 C. F. 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 presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of *N* lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

#### 1. Introduction

“*Given two correlated lognormal stochastic variables, what is the stochastic dynamics 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 long-standing problem has wide applications in many fields such as telecommunication studies [1–6], financial modelling [7–9], actuarial science [10–12], biosciences [13], physics [14], and so forth. Although the lognormal distribution is well known in the literature [15, 16], yet almost nothing is known of the probability distribution of the sum or difference of two correlated lognormal variables. However, it is commonly agreed that the distribution of either the sum or difference is neither normal nor lognormal.

The aforesaid problem can be formulated as follows. Given two lognormal stochastic variables and obeying the following stochastic differential equations: where , denotes a standard Weiner process associated with , and the two Weiner processes are correlated as , the time evolution of the joint probability distribution function of the two correlated lognormal variables is governed by the backward Kolmogorov equation where subject to the boundary condition This joint probability distribution function tells us how probable the two lognormal variables assume the values and at time , provided that their values at are given by and . Since is known in closed form as follows: the probability distribution of the sum or difference, namely , of the two correlated lognormal variables can be obtained by evaluating the integral Despite that many methods have been developed to address the problem, a closed-form representation for the probability distribution of the sum or difference is still missing. Hence, we must resort to numerical methods to perform the integration. Nevertheless, the numerically exact solution does not provide any information about the stochastic dynamics of the sum or difference explicitly.

In the lack of knowledge about the probability distribution of the sum or difference of two correlated lognormal variables, several analytical approximation methods which focus on finding a good approximation for the desired probability distribution have been proposed in the literature [1–6, 8, 17–27]. Essentially, these analytical approximations assume a specific distribution that the sum or difference of the two correlated lognormal variables follow, and then use a variety of methods to identify the parameters for that specific distribution. However, no mathematical justification for the specific distribution was apparently given. In spite of this shortcoming, these approximations attract considerable attention and have been extended to tackle the algebraic sums of correlated lognormal variables, too.

In this communication, we apply the Lie-Trotter operator splitting method [28] to derive an approximation for the dynamics of the sum or difference of two correlated lognormal variables. It is shown that both the sum and difference can be described by a shifted lognormal stochastic process. Approximate probability distributions of both the sum and difference of the lognormal variables are determined in closed form, and illustrative numerical examples are presented to demonstrate the accuracy of these approximate distributions. Unlike previous studies which treat the sum and difference in a separate manner, our proposed method thus provides a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. In addition, in terms of the approximate solutions, we are able to obtain an analytical series expansion of the exact solutions, which can allow us to improve the approximation systematically. Moreover, we believe that this new approach can be extended to study both the algebraic sum of lognormals, and the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

#### 2. Lie-Trotter Operator Splitting Method

It is observed that the probability distribution of the sum or difference of the two correlated lognormal variables, that is, , also satisfies the same backward Kolmogorov equation given in (1.2), but with a different boundary condition To solve for , we first rewrite the backward Kolmogorov equation in terms of the new variables as where The corresponding boundary condition now becomes Accordingly, the formal solution of (2.2) is given by

Since the exponential operator is difficult to evaluate, we apply the Lie-Trotter operator splitting method [28] to approximate the operator by (see the appendix) and obtain an approximation to the formal solution , namely where the relation is utilized. For , which is normally valid unless and are both close to zero, the operators and can be approximately expressed as in terms of the two new variables: where and. Without loss of generality, we assume that . Obviously, both and are lognormal (LN) random variables defined by the stochastic differential equations and their closed-form probability distribution functions are given by for . As a result, it can be inferred that within the Lie-Trotter splitting approximation both and are governed by a shifted lognormal process. It should be noted that for the Lie-Trotter splitting approximation to be valid, needs to be small.

Alternatively, we can also approximate the operator by and It is not difficult to recognize that follows the square-root (SR) stochastic process defined by the stochastic differential equation and has the closed-form probability distribution function for , where is the modified Bessel function of the first kind of order one. Accordingly, we have shown that within the Lie-Trotter splitting approximation, which requires to be small, can be described by a shifted square-root process, too.

Moreover, in terms of the approximate solutions , we can express the exact solutions in the following form: where The integrals over the temporal variables can be evaluated analytically. If we keep terms up to the order of , then can be approximated by This analytical series expansion can allow us to improve the approximate solutions systematically.

#### 3. Illustrative Numerical Examples

In Figure 1 we plot the approximate closed-form probability distribution function of the sum given in (2.11) for different values of the input parameters. We start with , , , and in Figure 1(a). Then, in order to examine the effect of , we decrease its value to in Figure 1(b) and to in Figure 1(c). In Figures 1(d), 1(e), and 1(f) we repeat the same investigation with a new set of values for and , namely and . Without loss of generality, we set for simplicity. The (numerically) exact results which are obtained by numerical integrations are also included for comparison. It is clear that the proposed approximation can provide accurate estimates for the exact values. Moreover, 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 correlation parameter decreases from to . It is also observed that the errors increase with the ratio as expected but they seem to be not very sensitive to the changes in and .

Next, we apply the same sequence of analysis to the two approximate closed-form probability distribution functions of the difference given in (2.11) and (2.15). Similar observations about the accuracy of the proposed approximation can be made for the difference , too (see Figures 3 and 4). However, contrary to the case of , the estimation performs better for positive correlation. Of the two different approximation schemes for the , the shifted LN process seems to have a comparatively better performance than the shifted SR process, as evidenced by the numerical results.

#### 4. Conclusion

In this paper we have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both the algebraic sum of lognormals, and the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

#### Appendix

#### Lie-Trotter Splitting Approximation

Suppose that one needs to exponentiate an operator which can be split into two different parts, namely and . For simplicity, let us assume that , where the exponential operator is difficult to evaluate but and are either solvable or easy to deal with. Under such circumstances, the exponential operator , with being a small parameter, can be approximated by the Lie-Trotter splitting formula [28]: This can be seen as the approximation to the solution at of the equation by a composition of the exact solutions of the equations and at time .

#### References

- L. Fenton, “The sum of lognormal probability distributions in scatter transmission systems,”
*IRE Transactions on Communications Systems*, vol. 8, no. 1, pp. 57–67, 1960. View at Publisher · View at Google Scholar - J. I. Naus, “The distribution of the logarithm of the sum of two lognormal variates,”
*Journal of the American Statistical Association*, vol. 64, no. 326, pp. 655–659, 1969. View at Publisher · View at Google Scholar - M. A. Hamdan, “The logarithm of the sum of two correlated lognormal variates,”
*Journal of the American Statistical Association*, vol. 66, no. 333, pp. 105–106, 1971. View at Publisher · View at Google Scholar - C. L. Ho, “Calculating the mean and variance of power sums with two lognormal components,”
*IEEE Transactions on Vehicular Technology*, vol. 44, no. 4, pp. 756–762, 1995. View at Publisher · View at Google Scholar · View at Scopus - J. Wu, N. B. Mehta, and J. Zhang, “Flexible lognormal sum approximation method,” in
*Proceedings of IEEE Global Telecommunications Conference (GLOBECOM '05)*, vol. 6, pp. 3413–3417, St. Louis, Mo, USA, December 2005. View at Publisher · View at Google Scholar · View at Scopus - X. Gao, H. Xu, and D. Ye, “Asymptotic behavior of tail density for sum of correlated lognormal variables,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2009, Article ID 630857, 28 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Milevsky and S. E. Posner, “Asian options, the sum of lognormals, and the reciprocal gamma distribution,”
*Journal of Financial and Quantitative Analysis*, vol. 33, no. 3, pp. 409–422, 1998. View at Google Scholar · View at Scopus - R. Carmona and V. Durrleman, “Pricing and hedging spread options,”
*SIAM Review*, vol. 45, no. 4, pp. 627–685, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Dufresne, “The log-normal approximation in financial and other computations,”
*Advances in Applied Probability*, vol. 36, no. 3, pp. 747–773, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas, and D. Vyncke, “The concept of comonotonicity in actuarial science and finance: applications,”
*Insurance: Mathematics and Economics*, vol. 31, no. 2, pp. 133–161, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Vanduffel, T. Hoedemakers, and J. Dhaene, “Comparing approximations for risk measures of sums of non-independent lognormal random variables,”
*North American Actuarial Journal*, vol. 9, no. 4, pp. 71–82, 2005. View at Google Scholar · View at Zentralblatt MATH - A. Kukush and M. Pupashenko, “Bounds for a sum of random variables under a mixture of normals,”
*Theory of Stochastic Processes*, vol. 13 (29), no. 4, pp. 82–97, 2007. View at Google Scholar · View at Zentralblatt MATH - J. H. Graham, K. Shimizu, J. M. Emlen, D. C. Freeman, and J. Merkel, “Growth models and the expected distribution of fluctuating asymmetry,”
*Biological Journal of the Linnean Society*, vol. 80, no. 1, pp. 57–65, 2003. View at Publisher · View at Google Scholar · View at Scopus - M. Romeo, V. Da Costa, and F. Bardou, “Broad distribution effects in sums of lognormal random variables,”
*European Physical Journal B*, vol. 32, no. 4, pp. 513–525, 2003. View at Publisher · View at Google Scholar · View at Scopus - E. L. Crow and K. Shimizu,
*Lognormal Distributions: Theory and Applications*, Marcel Dekker, New York, NY, USA, 1988. - E. Limpert, W. A. Stahel, and M. Abbt, “Log-normal distributions across the sciences: keys and clues,”
*BioScience*, vol. 51, no. 5, pp. 341–352, 2001. View at Google Scholar · View at Scopus - N. C. Beaulieu and F. Rajwani, “Highly accurate simple closed-form approximations to lognormal sum distributions and densities,”
*IEEE Communications Letters*, vol. 8, no. 12, pp. 709–711, 2004. View at Publisher · View at Google Scholar · View at Scopus - Q. T. Zhang and S. H. Song, “A systematic procedure for accurately approximating lognormal-sum distributions,”
*IEEE Transactions on Vehicular Technology*, vol. 57, no. 1, pp. 663–666, 2008. View at Publisher · View at Google Scholar · View at Scopus - C. L. J. Lam and T. Le-Ngoc, “Estimation of typical sum of lognormal random variables using log shifted gamma approximation,”
*IEEE Communications Letters*, vol. 10, no. 4, pp. 234–235, 2006. View at Publisher · View at Google Scholar · View at Scopus - L. Zhao and J. Ding, “Least squares approximations to lognormal sum distributions,”
*IEEE Transactions on Vehicular Technology*, vol. 56, no. 2, pp. 991–997, 2007. View at Publisher · View at Google Scholar · View at Scopus - N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,”
*IEEE Transactions on Wireless Communications*, vol. 6, no. 7, pp. 2690–2699, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Borovkova, F. J. Permana, and H. V. D. Weide, “A closed form approach to the valuation and hedging of basket and spread options,”
*The Journal of Derivatives*, vol. 14, no. 4, pp. 8–24, 2007. View at Publisher · View at Google Scholar - D. Dufresne, “Sums of lognormals,” in
*Proceedings of the 43rd Actuarial Research Conference*, University of Regina, Regina, Canada, August 2008. - T. R. Hurd and Z. Zhou, “A Fourier transform method for spread option pricing,”
*SIAM Journal on Financial Mathematics*, vol. 1, pp. 142–157, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Li, V. D. Chakravarthy, Z. Wu et al., “A low-complexity approximation to lognormal sum distributions via transformed log skew normal distribution,”
*IEEE Transactions on Vehicular Technology*, vol. 60, no. 8, pp. 4040–4045, 2011. View at Publisher · View at Google Scholar - N. C. Beaulieu, “An extended limit theorem for correlated lognormal sums,”
*IEEE Transactions on Communications*, vol. 60, no. 1, pp. 23–26, 2012. View at Publisher · View at Google Scholar - J. J. Chang, S. N. Chen, and T. P. Wu, “A note to enhance the BPW model for the pricing of basket and spread options,”
*The Journal of Derivatives*, vol. 19, no. 3, pp. 77–82, 2012. View at Publisher · View at Google Scholar - H. F. Trotter, “On the product of semi-groups of operators,”
*Proceedings of the American Mathematical Society*, vol. 10, no. 4, pp. 545–551, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH