The role of orthogonal polynomials in adjusting hyperpolic secant and logistic distributions to analyse financial asset returns

Abstract

In this paper, we will tackle the issue of accounting for skewness and potentially severe excess kurtosis of the empirical distribution of a random variable of interest by adjusting a parent leptokurtic distribution, using orthogonal polynomials. We will show that the polynomial shape adapter that allows the transformation from a given parent to a target distribution is a linear combination of the orthogonal polynomials associated to the former with coefficients depending on the difference between the moments of these two distributions. A recent work (Zoia, Commun Stat Theory Methods 39(1):52–64, 2010) has shown how to adjust the normal density by using Hermite polynomials but this application is suitable only for series with moderate kurtosis (lower than 5). This is why we provide two other parent distributions, the logistic and the hyperbolic secant which, once polynomially adjusted, can be used to reshape series with higher degrees of kurtosis. We will apply these results for modelling heavy-tailed and skewed distributions of financial asset returns by using both the conditional and unconditional approaches. We empirically demonstrate the advantages of using the polynomially adapted distributions in place of popular alternatives.

Appendix

The probability density function of the hyperbolic secant is

$$\begin{aligned} f(x;a,b)=a\ \text {sech}(bx), \end{aligned}$$

where $a$ and $b$ are parameters. By choosing these parameters so that the first two central moments of the variable $x$ are zero and one, respectively, we get the density function of the normalized hyperbolic secant, that is

$$\begin{aligned} f(x)=\frac{1}{2}\text {sech}\left( \frac{\pi x}{2} \right) . \end{aligned}$$

(49)

Bearing in mind that the characteristic function of $\text {sech}(x)$ is $\pi \text {sech} \left( \displaystyle \frac{\pi t}{2} \right)$ and the time scaling property of the Fourier integrals (see Papoulis and Maradudin 1963) we conclude that the characteristic function of (49) is $\text {sech}(it)$ which, likewise the Gaussian case, has the same functional form as the density to within a scalar factor. Expanding $\text {sech}(it)$ about $t=0$, yields

$$\begin{aligned} \text {sech}(it)=1+\frac{(it)^2}{2!}+\frac{(it)^4}{4!}5+\cdots +\frac{(it)^{r}}{r!}E_{r}+.... \end{aligned}$$

The coefficients of the expansion at stake, which are the Eulero numbers $E_{r}$, provide the moments of this density. The first even Eulero numbers are

$$\begin{aligned} E_{2}=-m_{2}=-1,\ E_{4}=m_{4}=5,\ E_{6}=-m_{6}= -61,\ E_{8}=m_{8}=1385 . \end{aligned}$$

By taking the square of (49) we get the logistic density, that is

$$\begin{aligned} f(y)&= \frac{1}{4}\text {sech}^2 \left( \frac{\pi y}{2} \right) \nonumber \\&= \exp (-y)[1+\exp (-y)]^{-2}, \end{aligned}$$

(50)

Normalizing, we eventually get the density of a zero-mean and unit-variance variable, that is

$$\begin{aligned} f(x)&= \frac{\pi }{4\sqrt{3}}\text {sech}^2\left( \frac{ \pi x}{2\sqrt{3}}\right) \nonumber \\&= \frac{\pi }{\sqrt{3}}\exp \left( -\frac{\pi x}{\sqrt{3}}\right) \left[ 1+\exp \left( -\frac{\pi x}{\sqrt{3}}\right) \right] ^{-2} . \end{aligned}$$

(51)

Upon noting that the characteristic function of the random variable $y$ with density (50) is

$$\begin{aligned} E[\exp (ity)]=it\pi \ \text {cosech}(it\pi ), \end{aligned}$$

and bearing in mind the time scaling property of the Fourier integrals, we conclude that the characteristic function corresponding to (51) is given by

$$\begin{aligned} E[\exp (itx)]=\frac{it\pi ^2}{3}\text {cosech} \left( \frac{it\pi ^2}{3} \right) . \end{aligned}$$

(52)

Expansion of the characteristic function (52) about $t=0$ eventually yields to find out the moments. Referring to Feller (1974), the moments can be expressed in closed form as follows

$$\begin{aligned} E(x^r)=2\Gamma (r+1)(1-2^{r-1})\xi (r)\left( \frac{3}{\pi ^2}\right) ^{\frac{r}{2}} \end{aligned}$$

where $\Gamma (\cdot )$ is the gamma function and $\xi (r)=\sum _{j=1}^{\infty }j^{-r}$ is the Riemann zeta function. Upon noting that $\Gamma (r+1)=r!$ and that

$$\begin{aligned} \xi (2)=\frac{\pi ^2}{6},\; \xi (4)=\frac{\pi ^4}{90},\; \xi (6)=\frac{\pi ^6}{945},\; \xi (8)=\frac{\pi ^8}{9450}, \end{aligned}$$

simple computations provide the moments of the said distribution.