Journal of Statistical Planning and
Inference 136 (2006) 1349 – 1359
www.elsevier.com/locate/jspi
Moments and properties of multiplicatively
constrained bivariate lognormal distribution with
applications to futures hedging
Donald Lien
a , ∗
, N. Balakrishnan
b
a
Department of Economics, College of Business, University of Texas, San Antonio, TX 78249 0633, USA
b
Department of Mathematics and Statistics, McMaster University, Hamilton, Ont., Canada L8S 4K1
Received 8 August 2003; accepted 7 October 2004
Available online 13 November 2004
Abstract
In this paper, we derive explicit expressions for marginal and product moments of a bivariate
lognormal distribution when a multiplicative constraint is present. We show that the coefficients of
variation always decrease regardless of the multiplicative constraint imposed. We also evaluate the
effects of the constraint on the variances and covariance, and present conditions under which the
correlation coefficient increases under the presence of such a multiplicative constraint. We finally
apply these results to futures hedging analysis and some other financial applications.
© 2004 Elsevier B.V.All rights reserved.
MSC: 62G30; 62P05
Keywords: Bivariate lognormal distribution; Multiplicative constraint; Moments; Coefficient of variation;
Correlation coefficient; Financial applications
1. Introduction
Let (X, Y )
T
be a bivariate random vector such that (ln X, ln Y)
T
has a bivariate nor-
mal, N
2
(
X
,
Y
,
X
,
Y
, ), distribution. Equivalently, let X = exp(
X
+
X
U) and Y =
∗
Corresponding author. Tel.: +1 210 458 4315; fax: +1 210 458 5837.
E-mail address: don.lien@utsa.edu (D. Lien).
0378-3758/$ - see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jspi.2004.10.004