The valid regions of Gram–Charlier densities with high-order cumulants

Abstract

Based on derivatives of a Gaussian density, the Gram–Charlier series is an infinite expansion. Its truncated series is often used in many fields to approximate probability density functions. Although the expansions are useful, there are constrained regions on the value of the cumulants (or moments) that admit a valid (nonnegative) probability density function. When the truncation order is low (just at fourth-order), the truncated Gram–Charlier density may be difficult to approximate an implied probability distribution as closely as possible, especially for distributions that are not sufficiently close to a normal distribution. One might increase the order after which the series is truncated until a perfect fit is achieved. However, the series expansion is usually truncated in the existing literature until the fourth-order term because it becomes difficult to find valid regions. This paper shows how the valid region of higher cumulants can be numerically implemented by the semi-definite algorithm, which ensures that a series truncated at a cumulant of arbitrary even order represents a valid probability density. We provide examples of two valid regions of the sixth and eighth Gram–Charlier densities (i.e., truncated at the sixth and eighth terms). Our analysis proves the fact that valid regions can be broadened with the higher-order expansions. Furthermore, the impact of higher cumulants on the valid regions has been shown.

Introduction

The primary historical purpose of the standard Gram–Charlier (GC hereafter) expansion and Edgeworth series is to extend the Gaussian distribution when new information, such as moments or cumulants, is given which does not match that of the Gaussian. The GC and Edgeworth expansions were established at the end of the 19th century and the beginning of the 20th century. Since then, these expansions have been commonly used in many fields from mathematics, physics, finance, astronomy to oceanography. For applications in finance, this approach comes from the pioneering work of Sargan [1], [2] and Phillips [3], [4], who developed the theory of Edgeworth and GC expansions to derive the approximate distribution function of econometric estimators. Jarrow and Rudd [5] show how an arbitrary distribution can approximate a given stochastic process of an underlying security in terms of a series expansion involving second and higher moments. Jondeau and Rockinger [6] use the constrained expansion to the estimation of risk-neutral densities. Corrado [7] gives a hidden martingale restriction which is developed for option pricing models based on the GC expansions of the normal density function. For application in astronomy, Blinnikov and Moessner [8] draw attention to the form of this expansion and present an algorithm to illustrate the problems arising when fitting spectral line profiles of galaxies, supernovae, or other stars. For application in oceanography, [9] use the GC series to analyze the capabilities and limitations of the simulation of the probability density of rough sea surface elevations. For applications in mathematics, [10] introduces a new approximation that involves a four-dimensional dynamical system that uses the skewness and kurtosis of the queueing distribution via the GC expansion.

Although convenient, it should be mentioned that the application of the series expansion of probabilities has the shortcoming that once truncated, the GC density is not well-defined; i.e., it is not strictly nonnegative for all values of its parameters in the parametric space. Several authors have disregarded this issue, and some have come up with parameters that do not yield a proper probability density function. Without keeping the nonnegative of a truncating GC density, there might be inconsistencies among option prices. For example, if the function used as density is negative over one interval $(a,b)$, then a digital option that pays off only when the log return is in that interval will have a negative price. As a consequence, a proper probability density function is the only safe choice. For this purpose, different alternatives have been proposed in the many comparative studies depending on the end-use of a model, namely: (i) accurate selection of initial values for the maximum likelihood (ML) algorithms [11]; (ii) density function transformations; and (iii) parametric constraints.

The first method to estimate GC densities by ML also presents important drawbacks. The first one is the shape of the ML function surface that presents flat areas and jumps. These features may lead the maximization algorithms to local optima or even to non-convergence. Compared with ML algorithms, [12] propose the use of the method of moments (MM) to obtain estimates of the GC densities parameters and show this method is extremely straightforward to implement because these parameters are linear functions of the density moments. However, both ML and MM algorithms still do not solve the “nonnegative problem” unless a very large expansion is implemented.

The second method is based on the density function transformations. Sargan [1] brought GC and Edgeworth series into semi-nonparametric (SNP hereafter) econometrics. Many related studies have developed both univariate and multivariate distributions based on GC densities, attempting to ensure the nonnegativity by exploiting the orthogonal properties of the Hermite polynomials. There exist fruitful methods to ensure nonnegative, such as [13], [14], who introduced straightforward solutions to this problem that define the density in terms of the squared weighted sum of Hermite polynomials. The positive Edgeworth–Sargan distribution, the simplest case, is used in some studies to forecast heavy-tailed densities, such as [15]. For applications in asset pricing, [16] derive the statistical properties of the SNP densities whose advantage is always positive and flexible. In that paper, financial derivatives valuation by SNP densities has been shown. For application in risk modeling, [17] introduce the SNP approach for modeling Bitcoin risk relatively to other parametric distributions and volatility models. Ñíguez et al. [18] show that flexible probability distribution functions, in addition to being able to capture stylized facts of financial returns, can be used to identify pure higher-order effects of investors’ optimizing behavior. Cohen [19] presents an inspiring approach by defining the probability density functions as the square of other complex functions, which are analogous to quantum mechanical wave functions. The advantage of this approach is that the nonnegative restriction is automatically satisfied.

Parametric constraints to avoid the density approximation taking negative values derive from how to characterize the set of moments or cumulants since polynomial factors are determined by the moments or cumulants of the distribution. The problem was first highlighted by Shenton [20], who guarantees the nonnegative by evaluating the efficiency of moment estimation for the GC densities. Since conditions on coefficients of a quartic polynomial to ensure the absence of real roots were known by the dialytic method of Sylvester, [21] extend the work of Shenton to propose a set of polynomial inequalities in the skewness and kurtosis. Then, the kurtosis-skewness pairs on the boundary of the positive definite and unimodal regions can be shown by solving these inequalities numerically. Draper and Tierney [22] re-examine the results given by Barton and Dennis and correct a minor error in their plot of the unimodal region for the Edgeworth series. In [6], the nonnegative problem has been tackled through parametric restrictions. Later, a fantastic analytical expression on the polynomial equations to determine the positive definite and unimodal regions for the GC densities is given by Kwon [23], who first establishes a direct relationship between the skewness and kurtosis. In Kwon’s paper, polynomial equations are solved explicitly to express the standardized skewness on the two boundaries analytically as functions of the standardized kurtosis. This enables the skewness corresponding to any kurtosis on the two boundaries to be computed exactly and analytically, and avoids the reliance on tabulated values that require further interpolation. However, these solutions are not always the best option since imposing nonnegative regions is not easy to define beyond the simpler cases. The applications of these GC densities in finance, mainly for asset or option pricing, have not usually considered expansions beyond the fourth-order term (e.g., for expansions defined in terms of a couple of moments, usually skewness and kurtosis). “Unfortunately, [6] only determined those restrictions for $n=4$, because it becomes exceedingly difficult to find them for higher $n$”, as [16] point out. Our goal in this paper is to fill the vacuum.

In this paper, the higher moments or cumulants regions to determine the positive definite are solved by semi-definite programming (SDP hereafter). Theoretically, the domain of any order moments and cumulants to ensure the truncated GC density nonnegative could be estimated by SDP. According to SDP results, adding the higher terms to the GC density can significantly improve the possible values of cumulants, which allows a closer fit of the cumulants or moments observed in the market. The second contribution in this paper is that we gauge and analyze the impact of higher cumulants on skewness and kurtosis. We use SDP to calculate all possible values of skewness and kurtosis when both higher-order cumulants are fixed at all possible values. Nevertheless, we also present a shortcoming in GC densities when one cumulant reaches an extreme value (minimum or maximum). When one of the cumulants is fixed at a possible extreme value, SDP constrains other cumulants in a quite narrow space. Finally, we focus on a theoretical extension of the truncated GC series for a better application in this paper and leave the application to asset pricing and risk measuring in my following research considering the length of this paper.

The remainder of this paper is organized as follows. In Section 2, we provide some necessary mathematical foundations of the GC densities. In Section 3, we review the positive region and skewness of the truncated GC expansion in terms of the first two moments. The impact of higher cumulants on the positive definite region is discussed in Section 4, together with the convex optimization, SDP, outlined in Appendix A. Three specific cases are described in Section 4 to show the range of higher cumulants. In Section 5, we emphasize the analysis of the impact of higher cumulants on the positive definite region. Concluding remarks are offered in Section 6.