Skew Generalized Secant Hyperbolic Distributions: Unconditional and Conditional Fit to Asset Returns

A generalization of the hyperbolic secant distribution which allows both for skewness and for leptokurtosis was given by Morris (1982). Recently, Vaughan (2002) proposed another flexible generalization of the hyperbolic secant distribution which has a lot of nice properties but is not able to allow for skewness. For this reason, Fischer and Vaughan (2002) additionally introduced a skewness parameter by means of splitting the scale parameter and showed that most of the nice properties are preserved. We briefly review both classes of distributions and apply them to financial return data. By means of the Nikkei225 data, it will be shown that this class of distributions – the socalled skew generalized secant hyperbolic distribution – provides an excellent fit in the context of unconditional and conditional return models. JEL classification: C22; G12


Introduction
The hyperbolic secant distribution -which was first studied by Baten (1934) and Talacko (1956) -seems to be an appropriate candidate as a starting point for financial return models: Firstly, it exhibits more leptokurtosis than the normal and even more than the logistic distribution. Secondly, the cumulative distribution function admits a closed form implying that, for example, risk neutral probabilities of option prices can be calculated fast and accurate. Thirdly, this distribution is reproductive (i.e. the class is preserved under convolution), infinitely divisible with existing moment-generating function and has finite moments. Since 1956, two generalization have been proposed which both incorporate most of these properties, too and, in addition, allow for a more flexible form concerning skewness and leptokurtosis. The first generalization was proposed by Morris (1982) in the context of natural exponential families (NEF) with quadratic variance function (i.e. the variance is a quadratic function of the mean). In this class, consisting of six members, one distribution -the so-called NEF-GHS distribution -is generated by the hyperbolic secant distribution. The NEF-GHS distribution allows for skewness and arbitrarily high excess kurtosis. Morris (1982) showed that this class is again reproductive, infinitely divisible with existing moment-generating function and existing moments. However, the corresponding cumulative distribution function doesn't admit a closed form.
Recently, Vaughan (2002) suggested a family of symmetric distributions -the socalled generalized secant hyperbolic (GSH) distribution -with kurtosis ranging from 1.8 to infinity. This family includes both the hyperbolic secant and the logistic distribution and closely approximates the Student t-distribution with corresponding kurtosis. In addition, the moment-generating function and all moments exist, and the cumulative distribution is given in closed form. Unfortunately, this family does not allow for skewness. For this purpose, Fischer and Vaughan (2002) introduced a skewness parameter by means of splitting the scale parameter according to Fernandez, Osiewalski and Steel (1995). This method preserves the closed form for the density, the cumulative distribution function, the inverse cumulative distribution function. It will be shown that this family -termed as skew generalized secant hyperbolic distribution (SGSH) -provides an excellent fit to the Nikkei225 data. This is verified in the context of unconditional and conditional return models. In particular, we compare the results to other popular return models which have been proposed in the literature in the past: The α-stable distributions (see, for example, Mittnik et al. (1998)), the class of generalized hyperbolic distributions (see, for example, Prause (1999)), the generalized logistic family of McDonald (1991)

(Generalized) Hyperbolic Secant distribution
A symmetric random variable X is said to follow a hyperbolic secant (HS) distribution if its probability density function (with unit variance) is given by or, equivalently, its cumulative distribution function is given by This distribution is more leptokurtic than the normal, even more leptokurtic than the logistic distribution and has a kurtosis coefficient (measured by the fourth standardized moment) of 5. Consequently, it seems to be a reasonable "starting point" as a distribution for leptokurtic data, in particular for financial return data. In order to obtain higher "leptokurtic flexibility", the λ-th convolution of a hyperbolic secant distribution can be considered. This was discussed, for example, by Harkness and Harkness (1968) or Jørgensen (1997). The resulting distribution is commonly known as generalized hyperbolic secant (GHS) distribution. However, GHS offers still no opportunity to take skewness into account.

NEF-GHS Distribution
The NEF-GHS distribution was originally introduced by Morris (1982) in the context of natural exponential families (NEF) with specific quadratic variance functions. Densities of natural exponential families are of the form In the case of the NEF-GHS distribution, ψ(λ, θ) = −λ log(cos(θ)) and ζ(x, λ) equals the probability density function of a generalized hyperbolic secant (GHS) distribution. Hence, the probability density function of the NEF-GHS distribution is given by for λ > 0 and |θ| < π/2. Introducing a scale parameter δ > 0 and a location parameter µ ∈ R, and setting β ≡ tan(θ) ∈ R, equation (2) changes to The NEF-GHS distribution is a flexible class of distribution which allows for skewness and excess kurtosis, which is infinitely divisible with existing moment-generating function and hence, existing moments. However, the cumulative distribution function and the inverse cumulative distribution function is not available in a closed form. These properties facilitate calculating risk measures, constructing multivariate copula-based distributions (see, for example, Fischer (2003)) or valuating option pricing models (see Fischer (2002)).

GSH distribution
Another generalization of the hyperbolic secant distribution -which is able to model both thin and fat tails -was introduced by Vaughan (2002). This distribution family, the so-called standard generalized secant hyperbolic (GSH) distribution with kurtosis parameter t ∈ (−π, ∞), has density The density from (4) is chosen so that X has zero mean and unit variance. The GSH distribution includes the logistic distribution (t = 0) and the hyperbolic secant distribution (t = −π/2) as special cases and the uniform distribution on (− √ 3, √ 3) as limiting case for t → ∞. Vaughan (2002) derives the cumulative distribution function, depending on the parameter t, as for t ∈ (−π, 0), for t > 0. and the inverse cumulative distribution function for t > 0.
The moment-generating function also depends on t and is given by π t sinh(ut/c 2 (t)) csc(uπ/c 2 (t)) für t > 0. Moments of X can be deduced from the last equation. Despite of its nice properties, the GSH distributions is not able to include skewness effects.

SGSH distributions
There are plenty of methods in the literature to make a symmetric distribution skew. As the cumulative distribution function of the GSH distribution is explicitly known, Fischer and Vaughan (2002) decided in favour of splitting the scale parameter, as it was done by Fernandez, Osiewalski and Steel (1995) for the skewed exponential power distribution.
Let γ > 1, I + (x) denote the indicator function for x on R + and I − (x) denote the indicator function for x on R − . Then it can be easily verified that is a density function which is symmetric for γ = 1, skewed to the right for γ > 1 and skewed to the left for 0 < γ < 1. The corresponding distribution is termed as skew generalized secant hyperbolic (SGSH) distribution in the sequel. The effect of the skewness parameter on the density can be seen in figure 1, below. It can also be shown (see, for example, Grottke (2002), p. 21) that the cumulative and the inverse cumulative distribution functions admits a closed form, namely and In addition, the power moments of a SGHS distribution can be deduced from that of a GHS distribution by means of with the partial positive expectation value Note, that for even r the positive half moments E + (X r GSH ) can be deduced from E(X r GSH ) by division with 2: Setting γ = 1 in (5), For odd r, the formula for the half moments E + (X r GSH ) is slightly more complicated. The corresponding results are deduced by Fischer and Vaughan (2002).

The data set
In order to adopt and compare estimation results for a great deal of distributions -in particular the stable distributions (STABLE) -priority is given to the weekly returns of the Nikkei from July 31, 1983 to April 9, 1995, with N = 608 observations. This series was intensively investigated, for example, by Mittnik, Paolella and Rachev (1998) because it exhibits typical stylized facts of financial return data. Figure 2 illustrates the time series of levels and corresponding log-returns.

Unconditional fit to financial return data
Similar to Mittnik, Paolella and Rachev (1998), four criteria are employed to compare the goodness-of-fit of the different candidate distributions. The first is the log-Likelihood value (LL) obtained from the Maximum-Likelihood estimation. The LL-value can be considered as an "overall measure of goodness-of-fit and allows us to judge which candidate is more likely to have generated the data". As distributions with different numbers of parameters k are used, this is taken into account by calculating the Akaike criterion given by The third criterion is the Kolmogorov-Smirnov distance as a measure of the distance between the estimated parametric cumulative distribution function,F , and the empirical sample distribution, F emp . It is usually defined by Finally, the Anderson-Darling statistic is calculated, which weights |F emp (x) −F (x)| by the reciprocal of the standard deviation of F emp , namely Instead of just the maximum discrepancy, the second and third largest value, which is commonly termed as AD 1 and AD 2 , are also taken into consideration. Whereas K emphasizes deviations around the median of the fitted distribution, AD 0 , AD 1 and AD 2 allow discrepancies in the tails of the distribution to be appropriately weighted.  Table 1: Goodness-of-fit for the unconditional case: Nikkei225.
Estimation was performed not only for the two families of generalized hyperbolic secant distributions (NEF-GHS and SGSH with symmetric counterparts GHS and GSH), but also for distribution families which have become popular in finance in the last years: Firstly, the generalized hyperbolic (GH) distributions which were discussed by Prause (1999) and include, for example, the Normal-inverse Gaussian (NIG) distributions (see Barndorff-Nielsen (1997)) as well as the hyperbolic (HYP) distributions (see Eberlein and Keller (1995)) as special cases. Secondly, the exponential generalized beta of the second kind (EGB2) distribution that was introduced by McDonald (1991) as a generalization of the logistic (LOG) distribution and used in various financial applications, see also . Thirdly, a very flexible skew version of the generalized t-distribution (SGT2) proposed by Grottke (2001). Finally, we performed calculations for the gh-transformed normal (gh-NORM) distribution (see Klein and Fischer (2002)).
The estimation results are summarized in table 1, above and are as follows: Firstly, let us focus on the fit of generalized hyperbolic secant families. There seems to be no difference between the GSH distribution of Vaughan (2002) and the GHS distribution of Harkness and Harkness (1968). This is not true if we consider the skewed pendants and compare the NEF-GHS distribution of Morris (1982) with the SGSH distribution of Fischer and Vaughan (2002) which exhibits better goodness-of-fit values with respect to all five criteria. For that reason, we restrict our considerations to the SGHS distribution. Concerning the LL-value, only the SGT2 distribution has a (slightly) higher value. The same is true if we compare the K-values. If we take the number of parameters into account (i.e. focus on the AIC criterion), SGSH even outperforms SGT2. The situation is a little bit different concerning the tail fit.
Here, gh-transformed distributions finished best, followed by SGT2, NIG, STABLE and SGHS (Note, that the last three are close together).

Conditional fit to financial return data
Assuming independent observations -as we did it in the last subsection -is not very realistic. To capture dependency between different log-returns, generalized autoregressive conditionally heteroscedastic (GARCH) models have proposed by Engle (1982) and Bollerslev (1986) as models for financial return data. These models are able to capture the distributional stylized facts (like thick tails or high peakedness), on the one hand, as well as the time series stylized facts (like volatility clustering), on the other hand. The setting for our GARCH framework is similar to Bollerslev (1986) assuming that the log-returns R t of financial data are given by where Θ m (L) is a polynomial in the lag operator L of order m. For reasons of simplicity, assume that Θ m (L) ≡ 1 and µ ≡ 0. The residuals {U t } are assumed to follow a GARCH-D process. That means they follow a distribution 1 D with shape parameter η and time-varying variance h 2 t . In the GARCH(1, 1)-Normal specification from Bollerslev (1986) it is given by Note, that setting β 1 = 0 results in the ARCH model of Engle (1982 Table 2: Goodness-of-fit for GARCH(1,1)-models: Nikkei225.
Again, the SGSH distribution outperforms most of its competitors and even has the lowest K-value. Concerning the LL-value, only SGT2 and GH produced slightly higher values. They same is true for AD 0 . GARCH models have been generalized in many different ways. In order to capture leverage effects, Zakoian (1994) proposed the threshold (T)-GARCH model with standard deviation given by where R + t = max{R t , 0} and R − t = min{R t , 0}. Imposing a Box-Cox-transformation on the conditional standard deviation process and the asymmetric absolute returns leads to the asymmetric power (AP-ARCH) specification of Ding, Engle and Granger (1993), where the variance equation is Equation (10) reduces to (9) for λ = 1, α 1 = α − 1 /(2−α + 1 ) and c = 1−α + 1 (2−α + 1 )/α − 1 . Moreover, equation (8) is achieved for λ = 2 and c = 0. To take also asymmetric effects into account, we end up with an AP-ARCH(1,1)-D specification whose estimation results are summarized in table 3. Note that AP-ARCH(1,1) estimation results for stable distribution were not available. Moreover, * * indicates convergence problems of our algorithm.  Table 3: Goodness-of-fit for AP-ARCH(1,1)-models: Nikkei225.
It can be observed that for most of the distributions the K-values improve (compared to the GARCH(1,1)-fit) whereas the AD-values become worser. SGT2 and SGSH seem to dominate the other distributions. Except of AD 0 , SGSH is close to SGT2. Again, SGSH achieves the smallest K-value.

Conclusions
Two generalizations of the hyperbolic secant distribution have been proposed in the last years which seem to be encouraging as a model for financial return data: The NEF-GHS distribution of Morris (1982) and the SGSH distribution of Fischer and Vaughan (2002). Both incorporate skewness and leptokurtosis. Within this work we applied them to the weekly returns of the Japanese stock index Nikkei225. Firstly, results of the unconditional fit (or of the GARCH(0,0)-D model) were calculated (for a broad class of distributions D which were established as suitable return models in the literature, and for the generalized hyperbolic secant families) as a benchmark. In a second step, volatility cluster were taken into account by means of a GARCH(1,1)-D model. Finally, we also tried to model leverage effects and estimated an AP-