The Sum and Difference of Two Lognormal Random Variables

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]  Tho Le-Ngoc,et al.  Estimation of typical sum of lognormal random variables using log shifted gamma approximation , 2006, IEEE Communications Letters.

[2]  M. Romeo,et al.  Broad distribution effects in sums of lognormal random variables , 2002, physics/0211065.

[3]  H. Trotter.  On the product of semi-groups of operators , 1959 .

[4]  M. A. Hamdan.  The Logarithm of the Sum of Two Correlated Log-Normal Variates , 1971 .

[5]  P. Holgate.  Lognormal Distributions: Theory and Applications , 1989 .

[6]  Zhiqiang Wu,et al.  A Low-Complexity Approximation to Lognormal Sum Distributions via Transformed Log Skew Normal Distribution , 2011, IEEE Transactions on Vehicular Technology.

[7]  Thomas R. Hurd,et al.  A Fourier Transform Method for Spread Option Pricing , 2009, SIAM J. Financial Math..

[8]  Norman C. Beaulieu,et al.  Highly accurate simple closed-form approximations to lognormal sum distributions and densities , 2004, IEEE Communications Letters.

[9]  Keith Q. T. Zhang,et al.  A Systematic Procedure for Accurately Approximating Lognormal-Sum Distributions , 2008, IEEE Transactions on Vehicular Technology.

[10]  S. Borovkova,et al.  A Closed Form Approach to the Valuation and Hedging of Basket and Spread Option , 2007 .

[11]  W. Stahel,et al.  Log-normal Distributions across the Sciences: Keys and Clues , 2001 .

[12]  D. C. Freeman,et al.  Growth models and the expected distribution of fluctuating asymmetry , 2003 .

[13]  Jingxian Wu,et al.  Approximating a Sum of Random Variables with a Lognormal , 2007, IEEE Transactions on Wireless Communications.

[14]  D. Dufresne.  The log-normal approximation in financial and other computations , 2004, Advances in Applied Probability.

[15]  S. Posner,et al.  Asian Options, The Sum Of Lognormals, And The Reciprocal Gamma Distribution , 1998 .

[16]  Norman C. Beaulieu.  An Extended Limit Theorem for Correlated Lognormal Sums , 2012, IEEE Transactions on Communications.

[17]  C. Ho.  Calculating the mean and variance of power sums with two log-normal components , 1995 .

[18]  A Note to Enhance the BPW Model for the Pricing of Basket and Spread Options , 2012, The Journal of Derivatives.

[19]  Hong Xu,et al.  Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables , 2009, Int. J. Math. Math. Sci..

[20]  PUPASHENKO BOUNDS FOR A SUM OF RANDOM VARIABLES UNDER A MIXTURE OF NORMALS In two papers : , 2008 .

[21]  J. Naus.  The Distribution of the Logarithm of the Sum of Two Log-Normal Variates , 1969 .

[22]  D. Dufresne.  SUMS OF LOGNORMALS , 2008 .

[23]  L. Fenton.  The Sum of Log-Normal Probability Distributions in Scatter Transmission Systems , 1960 .

[24]  Jan Dhaene,et al.  Comparing Approximations for Risk Measures of Sums of Nonindependent Lognormal Random Variables , 2005 .

[25]  Jingxian Wu,et al.  Flexible lognormal sum approximation method , 2005, GLOBECOM '05. IEEE Global Telecommunications Conference, 2005..

[26]  Lian Zhao,et al.  Least Squares Approximations to Lognormal Sum Distributions , 2007, IEEE Transactions on Vehicular Technology.

[27]  René Carmona,et al.  Pricing and Hedging Spread Options , 2003, SIAM Rev..