Skewed Normal Variance-Mean Models for Asset Pricing and the Method of Moments

Abstract

Financial returns (log-increments) data, Y_t, t=1,2,..., are treated as a stationary process, with the common distribution at each time point being not necessarily symmetric.

We consider as possible models for the common distribution four instances of the General Normal Variance-Mean Model (GNVM), which is described by $Y|V\sim N(a(b+V),c^{2}V+d^2)$ where V is a non-negative random variable and a, b, c and d are constants. When V is Gamma distributed and d=0, Y has the skewed Variance-Gamma distribution (VG). When V follows a Half Normal distribution and c=0, Y has the well-known Skew Normal (SN) distribution. We also consider two cases where V is Exponentially distributed. Bounds for skewness and kurtosis in each case are found in terms of the moments of the V. These are useful in determining whether the Method of Moments for a given model is feasible. The problem of overdetermination of parameters via estimating equations is examined. 5 data sets of actual returns data, chosen because of their earlier occurrence in the literature, are analysed using each of the 4 models.