2-EPT probability density function: Difference between revisions

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{{AFC submission|t||ts=20120121183210|u=Hcsexton|ns=5}} <!--- Important, do not remove this line before article has been created. --->

{{Probability distribution |

name =2-EPT Density Function|

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pdf_image =|

cdf_image =|

parameters = <math>(\textbf{A}_N,\textbf{b}_N,\textbf{c}_N,\textbf{A}_P,\textbf{b}_P,\textbf{c}_P)</math> <br/> <br/>

<math>\mathbbmathfrak{RRe}(\sigma(\textbf{A}_P))<0</math><br/> <br/>

<math>\mathbbmathfrak{RRe}(\sigma(\textbf{A}_N))>0</math>|

support =<math>x \in (-\infty; +\infty)\!</math>|

pdf = <math>f(x) =

\left\{\begin{matrix}

\textbf{c}_Ne^{\textbf{A}_Nx}\textbf{b}_N & \mboxtext{if }x < 0

\\[8pt]

\textbf{c}_Pe^{\textbf{A}_Px}\textbf{b}_P & \mboxtext{if }x \geq 0

\end{matrix}\right.

</math>|

cdf = <math>F(x) =

\left\{\begin{matrix}

\textbf{c}_N\textbf{A}_N^{-1}e^{\textbf{A}_Nx}\textbf{b}_N & \mboxtext{if }x < 0

\\[8pt]

1 + \textbf{c}_P\textbf{A}_P^{-1}e^{\textbf{A}_Px}\textbf{b}_P & \mboxtext{if }x \geq 0

\end{matrix}\right.

</math>|

Line 35 ⟶ 34:

char =<math> -\textbf{c}_N(Iiu-\textbf{A}_N)^{-1}\textbf{b}_N+\textbf{c}_P(Iiu-\textbf{A}_P)^{-1}\textbf{b}_P</math>|

}}

The '''variance-gamma distribution''', '''generalized Laplace distribution'''<ref name=laplace>{{cite book|title=The Laplace Distribution and Generalizations|author=Kotz, S. et al|page=180|year=2001|publisher=Birkhauser|isbn=0-8176-4166-1}}</ref> or '''Bessel function distribution'''<ref name=laplace/> is a [[continuous probability distribution]] that is defined as the [[normal variance-mean mixture]] where the [[mixture density|mixing density]] is the [[gamma distribution]]. The tails of the distribution decrease more slowly than the [[normal distribution]]. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The distribution was introduced in the financial literature by Madan and Seneta<ref>D.B. Madan and E. Seneta (1990): The variance gamma (V.G.) model for share market returns, ''Journal of Business'', 63, pp. 511–524.</ref>. The variance-gamma distributions form a subclass of the [[generalised hyperbolic distribution]]s.

In [[probability theory]], a '''2-EPT probability density function''' is a class of [[probability density function]]s on the real line. The class contains the density functions of all distributions that have [[Characteristic function (probability theory)|characteristic function]]s that are strictly proper [[rational function]]s (i.e., the degree of the numerator is strictly less than the degree of the denominator).

The class of probability density functions on <math>\mathbb{R}</math> with strictly proper rational characteristic functions are referred to as 2-EPT probability density functions. On <math>[0, \infty)</math> as well as <math>(-\infty, 0)</math> these probability density functions are Exponential-Polynomial-Trigonometric (EPT) functions. An EPT density function on <math>(-\infty, 0)</math> can be represented as <math>f(x)=\textbf{c}_Ne^{\textbf{A}_Nx}\textbf{b}_N</math>. Similarly the EPT density function on <math>[0, -\infty)</math> is expressed as <math>f(x)=\textbf{c}_Pe^{\textbf{A}_Px}\textbf{b}_P</math>. We have that <math>(\textbf{A}_N,\textbf{A}_P)</math> are square matrices, <math>(\textbf{b}_N,\textbf{b}_P)</math> column vectors and <math>(\textbf{c}_N,\textbf{c}_P)</math> row vectors. <math>(\textbf{A}_N,\textbf{b}_N,\textbf{c}_N,\textbf{A}_P,\textbf{b}_P,\textbf{c}_P)</math> is the minimal realization of the 2-EPT function. The general class of probability measures on <math>\mathbb{R}</math> with (proper) rational characteristic functions are densities corresponding to mixtures of the pointmass at zero (``delta distribution") and 2-EPT densities. Unlike phase-type and matrix analytic distributions the 2-EPT probability density functions are defined on the whole real line. It is shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations are illustrated for the two-sided framework in Sexton and Hanzon<ref>C. Sexton and B. Hanzon: State Space Calculations for two-sided EPT Densities with Financial Modelling Applications, ''www.2-ept.com''</ref>. The most involved operation is the convolution of 2-EPT densities using state space techniques. Much of the work centers on the ability to decompose the rational characteristic function into the sum of two rational functions with poles located in the open left and resp. open right half plane. The Variance Gamma density is shown to be a 2-EPT density under a parameter restriction and the Variance Gamma asset price process can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes. Examples of applications provided include option pricing, computing the Greeks and risk management calculations.

==Definition==

The fact that there is a simple expression for the [[moment generating function]] implies that simple expressions for all [[moment (mathematics)|moments]] are available. The class of variance-gamma distributions is closed under [[convolution]] in the following sense. If <math>X_1</math> and <math>X_2</math> are [[statistical independence|independent]] [[random variable]]s that are variance-gamma distributed with the same values of the parameters <math>\alpha</math> and <math>\beta</math>, but possibly different values of the other parameters, <math>\lambda_1</math>, <math>\mu_1</math> and <math>\lambda_2,</math> <math>\mu_2</math>, respectively, then <math>X_1 + X_2</math> is variance-gamma distributed with parameters <math>\alpha, </math> <math>\beta, </math><math>\lambda_1+\lambda_2</math> and <math>\mu_1 + \mu_2.</math>

A 2-EPT probability density function is a [[probability density function]] on <math>\mathbb{R}</math> with a strictly proper rational [[Characteristic function (probability theory)|characteristic function]]. On either <math>[0, +\infty)</math> or <math>(-\infty, 0)</math> these probability density functions are exponential-polynomial-trigonometric (EPT) functions.

See also [[Variance gamma process]].

Any EPT density function on <math>(-\infty, 0)</math> can be represented as

:<math>f(x)=\textbf{c}_Ne^{\textbf{A}_Nx}\textbf{b}_N ,</math>

where ''e'' represents a matrix exponential, <math>(\textbf{A}_N,\textbf{A}_P)</math> are square matrices, <math>(\textbf{b}_N,\textbf{b}_P)</math> are column vectors and <math>(\textbf{c}_N,\textbf{c}_P)</math> are row vectors. Similarly the EPT density function on <math>[0, -\infty)</math> is expressed as

:<math>f(x)=\textbf{c}_Pe^{\textbf{A}_Px}\textbf{b}_P.</math>

The parameterization <math>(\textbf{A}_N,\textbf{b}_N,\textbf{c}_N,\textbf{A}_P,\textbf{b}_P,\textbf{c}_P)</math>

is the minimal realization<ref>Kailath, T. (1980) ''Linear Systems'', Prentice Hall, 1980</ref> of the 2-EPT function.

The general class of probability measures on <math>\mathbb{R}</math> with (proper) rational characteristic functions are densities corresponding to mixtures of the pointmass at zero ("[[delta distribution]]") and 2-EPT densities. Unlike [[Phase-type distribution|phase-type]] and matrix geometric<ref>Neuts, M. "Probability Distributions of Phase Type", Liber Amicorum Prof. Emeritus H. Florin pages 173-206, Department of Mathematics, University of Louvain, Belgium 1975</ref> distributions, the 2-EPT probability density functions are defined on the whole real line. It has been shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations have been illustrated for the two-sided framework in Sexton and Hanzon.<ref>Sexton, C. and Hanzon, B., "State Space Calculations for two-sided EPT Densities with Financial Modelling Applications", ''www.2-ept.com''</ref> The most involved operation is the [[convolution]] of 2-EPT densities using state space techniques. Much of the work centers on the ability to decompose the rational characteristic function into the sum of two rational functions with poles located in either the open left or open right half plane. The [[variance-gamma distribution]] density has been shown to be a 2-EPT density under a parameter restriction.<ref>Madan, D., Carr, P., Chang, E. (1998) "The Variance Gamma Process and Option Pricing", ''European Finance Review'' 2: 79–105</ref>

== Notes ==

<references/>

==External links==

*[http://www.2-ept.com/ 2 - Exponential-Polynomial-Trigonometric (2-EPT) Probability Density Functions] {{Webarchive|url=https://web.archive.org/web/20200708015221/http://www.2-ept.com/ |date=2020-07-08 }} Website for background and Matlab implementations

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Variance-Gamma Distribution}}

[[Category:ContinuousTypes of probability distributions]]

{{statistics-stub}}

[[ru:Распределение variance-gamma]]