key: cord-0631669-29qy41by
authors: Nzokem, A. H.; Montshiwa, V. T.
title: Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach
date: 2022-05-02
journal: nan
DOI: nan
sha: 61931c68821af7146691c0280c76ae29cc453a08
doc_id: 631669
cord_uid: 29qy41by

The paper investigates the rich class of Generalized Tempered Stable distribution, an alternative to Normal distribution and the $alpha$-Stable distribution for modelling asset return and many physical and economic systems. Firstly, we explore some important properties of the Generalized Tempered Stable (GTS) distribution. The theoretical tools developed are used to perform empirical analysis. The GTS distribution is fitted using three indexes: S&P 500, SPY ETF and Bitcoin BTC. The Fractional Fourier Transform (FRFT) technique evaluates the probability density function and its derivatives in the maximum likelihood procedure. Based on the three sample data, The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the GTS distribution fits both sides of the underlying distribution for SPY EFT index and Bitcoin BTC returns. Regarding the S&P 500 index, the Tempered Stable distribution fits the right side of the underlying distribution, while the compound Poisson distribution fits the left side.

Modelling the return assets with the normal distribution is the underlying assumption in many financial tools, such as the Black-Scholes-Merton option pricing model and the risk metric Variancecovariance technique to Value at Risk (VAR). However, substantial empirical evidence rejects the normal distribution for various asset classes and financial markets. The normal distribution's symmetric and rapidly decreasing tail properties cannot describe the skewed and fat-tailed properties of the high-frequency asset return distribution. The α-stable distribution has been proposed [1, 2] as an alternative to the normal distribution for modelling asset return and many types of physical and economic systems. Besides the Theoretical and empirical arguments for modelling with a stable distribution, the third and most important argument is that the Central Limit theorem can be generalized by the stable distribution. Regardless of the variance nature (finite or infinite), an appropriately standardized large sum of independent identical distribution (i.i.d) random variable converges to an α-stable random variable [3] . Although the stable distribution allows for varying degrees of tail heaviness and skewness; it has two major drawbacks [2] : firstly, the lack of closed formulas for densities and distribution functions, except for the Normal distribution(α = 2), the Cauchy distribution (α = 1) and the Lévy distribution (α = 1 2 ); secondly, most of the moments of the stable distribution are infinite. An infinite variance of the asset return leads to infinite price for derivative instruments such as options. The Tempered Stable (TS) distribution TS(β ,α,λ ) was developed to overcome the shortcomings from those two distributions specifically in modelling high-frequency asset returns. The tails of the TS distribution for asset returns are heavier than the normal distribution but thinner than the stable distribution [4] . A more general form of the Tempered Stable distribution, called Generalized Tempered Stable (GTS) distribution GTS(β + , β − , α + ,α − , λ + , λ − ) will be considerated in the paper. The Lévy measure (V (dx)) of the Generalized Tempered Stable distribution is provided by (1.1).

The Generalized Tempered Stable Distribution can be used to control the level of skewness, tail heaviness and symmetry of the distribution. In addition, It has finite moments; just like the stable distributions, the closed-form formulas exist only for characteristic function and not for the density or the distribution function. The paper aims to evaluate the assumption that the asset return distribution follows a Generalized Tempered Stable distribution GTS(β + ,β − ,α + ,α − ,λ + ,λ − ) and to compare their fitting performance to the Normal distribution , which is the standard in practice. The statistical inference is based on the Maximum Likelihood (ML) method. The Fractional Fourier Transform (FRFT) technique will be implemented to provide a good approximation of the density function and its derivatives in the optimization process. The data comes from two sources: the Standard & Poors 500 Composite Stock Price Index (S&P 500) and the SPDRS & P 500 ETF (SPY ETF); and the period spans from January 4, 2010, to February 02, 2022. The paper is structured as follows: the following section presents the theoretical framework of the Generalized Tempered Stable (GTS) distribution. The characteristic exponent closed-form formula is developed base on the Lévy measure of the GTS distribution. The third section reviews the Maximum likelihood method implemented in the optimization process. The fourth section presents the Generalized Tempered Stable (GTS) distribution fitting results and the theoretical moments. And in the fifth section, the GTS distribution and the sample data distribution are compared through the Kolmogorov-Smirnov (KS) goodness-of-fit test.

The Lévy measure of the generalized tempered stable (GTS) distribution (V (dx)) is defined in (2.3) as a product of a tempering function (q(x)) and a Lévy measure of the α-stable distribution (V stable (dx)).

where 0 ≤ β + ≤ 1, 0 ≤ β − ≤ 1, α + ≥ 0, α − ≥ 0, λ + ≥ 0 and λ − ≥ 0. α + and α − are the scale parameter, also called the process intensity [5] and has a similar role as the variance parameter in the Brownian motion process. These parameters play an essential role in the Levy process. λ + and λ + control the rate of decay on the positive and negative tails, respectively. When λ + > λ − , the distribution is skewed to the left; the left tail is thicker. When λ + < λ − , the distribution is skewed to the right. And when λ + = λ − , the distribution is symmetric [3] . β + and β − are the indexes of stability, also called tail indexes, tail exponents or characteristic exponents, which determine the rate at which the tails of the distribution taper off [6] . See [3, 7] For more details on tempering function and Lévy measure of tempered stable distribution. The Generalized Tempered Stable (GTS) distribution can be denoted by

From (2.4), it results that when β + < 0, TS(β + , α + ,λ + ) is of finite activity and can be written as a Compound Poisson process on the right side (X + ). we have similar pattern when β − < 0. However, when 0 ≤ β + ≤ 1, X + is an infinite activities process with an infinite number of jumps in any given time interval. We have similar pattern when 0 ≤ β − ≤ 1. In addition to the infinite activities process, we have

By adding the location parameter of the distribution, the GTS distribution becomes GTS(µ, β + , β − , α + ,α − , λ + , λ − ) and we have:

2.1 Tempered Stable Distribution and Lévy Process.

The characteristic exponent can be written

Proof.

(2.3) is a Lévy measure and (2.4) is satisfied. We can applied the Lévy-Khintchine representation for non-negative processes theorem.

Similarly, we have :

The expression in (2.8) becomes:

, GTS becomes Bilateral Gamma distribution with the characteristic exponent:

When (β − , β + ) → (0, 0) and α − = α + = α, GTS becomes Variance-Gamma (VG) distribution (µ, λ − − λ + , 1, α, 1 λ − λ + ) and the characteristic exponent:

Proof.

Similarly, (2.16) works for β − → 0. From (2.7), the characteristic exponent becomes (2.14).

In addition, if α − = α + = α, from (2.14), the characteristic exponent becomes (2.15), which is a Variance-Gamma distribution with parameter (µ, λ − − λ + , 1, α, 1 λ − λ + ). See [8] for more details on Variance-Gamma (VG) model and VG parameter notation.

The cumulants κ k of the Generalized Tempered Stable distribution is defined as follows.

we consider the characteristic exponent Ψ(ξ ) in (2.9).

Hence, the k-th order cumulant κ k is given by comparing the coefficients of both polynomial functions in (2.19 ) in iξ .

Theorem 2.5 (Asymptotic distribution of Generalized Tempered Stable distribution process) Let Y = Y t be a Lévy process on R generated by GTS(µ, β + , β − , α + ,α − , λ + , λ − ). Then Y t converges in distribution to a Lévy process driving by a Normal distribution with mean κ 1 and variance κ 2 Y t d ∼ N(tκ 1 ,tκ 2 ) as t → +∞

It was shown in [9] that Ψ(ξ ) generates the cumulants (κ j ) j∈N in (??) such that

is the characteristic function of the stochastic process Y t −tκ 1 √ tκ 2 and we have the following expression

We have:

We have convergence in distribution to a normal distribution.

3.1 S&P 500 and SPY ETF Data.

The Standard & Poor's 500 Composite Stock Price Index, also known as the S&P 500, is a stock index that tracks the share prices of 500 of the largest public companies in the United States. It is often treated as a proxy for describing the overall health of the stock market or even the U.S. economy. The SPDR S&P 500 ETF (SPY), also known as the SPY ETF, is an Exchange-Traded Fund (ETF)that tracks the performance of the S&P 500. Like S&P, SPY provides the diversification of a mutual fund, the flexibility of a stock and lower trading fees. The S&P 500 and SPY ETF data were extracted from Yahoo finance. The daily prices were adjusted for splits and dividends. The period spans from January 4, 2010 to February 02, 2022.

The daily price dynamics are provided in Fig 4 and Fig 1c for respectively S&P 500 and SPY ETF. Both indexes have an increasing trend, but the scale level are different , S&P500 is priced in thousand US Dollar and SPY ETF in hundred Us Dollar The increasing trend is temporally disrupted in the first quarter of 2020 by the coronavirus pandemic. 

One observation will be lost in the computation process. The results of the daily return are shown in Fig 4. The volatility of both daily returns is higher in the First quarter of 2020 amid the coronavirus pandemic and massive disruptions in the global economy. SPY ETF aims to mirror the performance of the S&P 500. Fig 4a and Fig 4c show that the daily return patterns of S&P 500 and SPY ETF are similar, which is consistent with the goal of SPY ETF. Some daily return observations were identified as outliers and removed from the data set to avoid a negative impact on statistics. By assuming the independent, identically distribution of each return in the sample, the method of moments was used to compute statistics based on data without outliers. The data were summarised and provided statistical information in Table 1 . From a probability density function f (x,V ) with parameter V = (µ, β + , β − , α + , α − , λ + , λ − ) of size p = 7 and the sample data x of size m, we definite the Likelihood Function and its derivatives as follows:

In order to perform the Maximum of the likelihood function (3.2), the quantities dl(x,V ) dV j and d 2 l(x,V ) dV k dV j , which are the first and second order derivative of the density function respectively, must be computed. The quantities d 2 l(x,V ) dV k dV j in (3.3) are critical in computing the Hessian Matrix and the Fisher Information Matrix. Given the parameters V = (µ, β + , β − , α + , α − , λ + , λ − ) and the sample data set X, we have the following quantities (3.4) from the previous development and computations

I (x,V ) should be a negative semi-definite matrix. that is :

we have a local solution when I (x,V ) = 0 in (3.4) and I (x,V ) in (3.5) is met as well . The solutions of (3.4) is provided by the Newton-Raphson Iteration Algorithm (3.6) .

For more detail on Maximum likelihood and Newton-Raphson Iteration procedure, see [10] .

The Fractional Fourier Transform (FRFT) technique computes the likelihood function and its derivatives in the optimization process for a given asset return data. See [11, 12] for a short review of the FRFT technique and the choice of FRFT parameters. The results of the GTS Parameter Estimation are summarised in Table 2 . It appears that the scale parameter β , α, and λ are all positive as expected; except β − for S&P 500 return data, which is negative and suggests that the left side return variable (X − ) is a compound Poisson distribution. More details regarding the Parameter estimation are provided in Appendix A.1. For S&P 500 return data, the maximization procedure convergences after 25 iterations as shown in Table 5 . When comparing empirical and GTS distribution statistics in Table 1 &3, It appears that The estimation of the Maximum likelihood over-estimates the Variance, Kurtosis and skewness statistics. 

See Appendix A.1 in [12] for (5.3) proof. The two-sided Kolmogorov-Smirnov goodness-of-fit statistic (D m ) is defined as follows.

where m is the sample size, F m (x) denotes the empirical cumulative distribution of {y 1 , y 2 . . . y m }. The distribution of the Kolmogorov's goodness-of-fit measure D m has been studied extensively in the litterature [13] . It was shown that the D m distribution is independent of the theoretical distribution (F(x)) under the null hypothesis (H0). More recently, the numerical computation of the exact distribution of D m was developed in [14] along with the R package (KSgeneral). For m = 3048, the probability density function of D m was computed under the null hypothesis (H 0 ). As shown in Fig 

For each Index and model, KS-Statistics (d n ) and P values were computed and summarized in Table 4 . For futher details on the computation results of KS-Statistics (d n ), see table in Appendix E.5 in [12] . Table 4 . In fact, the P value is respectively 7.48% for the S&P500 and 8.95% for the SPY ETF. These P value are higher than the threshold 5%, and we can not reject the null hypothesis that the Generalised Tempered Stable distribution fits the empirical distribution. The histogram of the daily return for each asset was compared graphically to the GTS and CLM density probability. As shown in Fig 4, CLM in red performs poorly.

The paper investigates the rich class of Generalized Tempered Stable distribution, an alternative to Normal distribution and the α-Stable distribution for modelling asset return and many physical and economic systems. We show some important properties of the GTS distribution related to the Lévy process; and how the GTS distributions cover the Bilateral Gamma distribution and the Variance-Gamma distribution. we fit three index returns to the Generalized Tempered Stable distribution GTS(β + ,β − ,α + ,α − ,λ + ,λ − ) and compare their fitting performance to the Normal distribution. The FRFT-based technique evaluates the probability density function and its derivatives in the maximum likelihood procedure. The GTS distribution fits the S&P500 return, the SPY ETF return and the Bitcoin BTC return. We have the sign of expected parameters. The estimation suggests that the left side (negative side) of the S&P 500 return can be modelled by a compound Poisson and the right side by a Stable tempered distribution. The GTS distribution better fits than the Classical Lognormal Model (CLM). The Kolmogorov-Smirnov (KS) goodness-of-fit test shows that the maximum likelihood method with FRFT produces a good estimation for the GTS distribution parameter for S&P 500 and SPY ETF, which fits the empirical distribution of the sample data. The peakedness, the skewness and the tail-heaviness of the histogram are captured to a certain degree by the GTS distribution. The good quality of the GTS distribution fitness is also shown by the low KS-Statistics and the high P values. 

Lévy processes and infinitely divisible distributions

Univariate stable distributions

Financial models with Lévy processes and volatility clustering

Do financial returns have finite or infinite variance? a paradox and an explanation

Non-Gaussian Merton-Black-Scholes Theory

Stable distributions

Tempered stable distributions and processes

Pricing european options under stochastic volatility models: Case of five-parameter gammavariance process

The advanced theory of statistics. The advanced theory of statistics

Gamma variance model: Fractional fourier transform (FRFT)

Fitting infinitely divisible distribution: Case of gamma-variance model

The kolmogorov-smirnov test for goodness of fit

Computing the kolmogorov-smirnov distribution when the underlying cdf is purely discrete, mixed, or continuous

GTS(β + ,β − ,α + ,α − ,λ + ,λ − ) Parameter Estimations for SPY ETF data

We would like to thank an anonymous referee for valuable comments and suggestions which lead to the improvement of this version.