Inverse Gaussian Distribution

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Inverse Gaussian Distribution

Kazuhisa Matsuda

Department of Economics

The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, NY 10016-4309

Email: [email protected] http://www.maxmatsuda.com/

March 2005 Abstract

This paper presents the basic knowledge of the inverse Gaussian distribution.

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[1] Inverse Gaussian Distribution: JKB (1994) Parameterization

There are many different parameterizations of the inverse Gaussian distribution which can be really confusing to beginners. In this section, basic properties of the inverse Gaussian distribution is presented following Johnson, Kotz, and Balakrishnan (1994)’s parameterization of the equation (15.4a).

The probability density function of the inverse Gaussian distribution is a two parameter family:

( ; , ) ^{3/ 2}exp ₂ ( )² 1 ₀

2 2 ^x

IG z z z

z

λ λ

µ λ µ

π µ

−

>

= ⎧⎨− −

⎩ ⎭

⎫⎬ , (1)

where µ∈ R⁺ and λ∈ R⁺.

By Fourier transforming the IG probability density (1), its characteristic function φ ω_Z( ) is calculated as:

φ ω_Z( ) _Z[IG z( ; , )]( )µ λ ω ^∞e^{i z}^ω IG z dz( )

≡^F ≡

∫

−∞

₂

( ) 0 ^{i z} ( ) exp 2

Z e^ω IG z dz λ λ i

φ ω µ µ

∞ ⎛ ⎞

= = ⎜⎜ −

⎝ ⎠

∫

^λ ⁻ ^ω⎟⎟. (2)

Simplifying the equation (2) yields:

2 2

( ) exp 2 exp 1 2

Z λ λ i λ µ λ i

φ ω λ ω

µ µ ^⎧µ λ µ λ ω ^⎫

⎛ ⎞ ⎪ ⎛ ⎞⎪

= ⎜⎝⎜ − − ⎟⎟⎠= ⎪⎩⎨ ⎜⎜⎝ − − ⎟⎟⎠⎪⎬⎭

2 2 2

( ) exp 1 2

Z λ µ λ µ i

φ ω λ ω

µ λ µ λ

⎧ ⎛ ⎞⎫

⎪ ⎪

= ⎨⎪⎩ ⎜⎜⎝ − − ⎟⎟⎠⎬⎪⎭

2 2

( ) exp 1 1

Z

λ µ i

φ ω µ λ

⎧ ⎛ ⎞⎫

= ⎪⎨ ⎜⎪⎩ ⎜⎝ − − ⎠⎪⎭ ω ⎪

⎟⎬⎟ . (3)

The characteristic exponent (i.e. cumulant generating function) ψ ω_Z( ) of the IG distribution is:

2 2

( ) ln ( ) 1 1

Z Z

λ µ ωi

ψ ω φ ω

µ λ

⎛ ⎞

≡ = ⎜⎜ − −

⎝ ⎠⎟⎟ . (4)

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Using (4), the first four cumulants defined by 1 ( ) ₀ ( )

n Z

n n n

cumulant Z

i ^ω

ψ ω

ω ⁼

≡ ∂

∂ are

calculated as the follows:

cumulant₁= , µ cumulant₂ =µ λ³/ , cumulant₃ =3µ λ⁵/ ²,

7 3

4 15 /

cumulant = µ λ .

Using the above cumulants, the mean, variance, skewness, and excess kurtosis of the IG random variable Z are obtained as (consult Table 4.1 of Matsuda (2004)):

E Z[ ]=µ, (5) Variance Z[ ]=µ λ³/ ,

Skewness Z[ ]=3 µ λ/ , [ ] 15 / Excess Kurtosis Z = µ λ.

The moment generating function M_Z( )ω of the IG distribution can be expressed as:

( ) ^z ( ; , ) 0 ^z ( )

MZ ω ^∞e IG z^ω µ λ dz ^∞e IG z dz^ω

≡

∫

−∞ =

∫

MZ^{( )}ω =^exp

(

ξ ωZ^{( )}

)

, where the Laplace exponent ξ ω_Z( ) is given by:

2 2

( ) 1 1

Z

λ µ ω

ξ ω µ λ

⎛ ⎞

= ⎜⎜⎝ − − ⎠⎟⎟. (6)

Using the moment generating function M_Z( )ω with (6), first four raw moments (i.e.

0 n ( )

Z

n n

r M ω _ω

ω ⁼

≡ ∂

∂ ) of the IG distribution are computed as:

r₁ ≡E Z[ ]= , µ

3

2 2

2 [ ]

r E Z µ µ

≡ = + λ ,

4 5

3 3

3 2

3 3

[ ]

r E Z µ µ µ

λ λ

≡ = + + ,

5 6

4 4

4 2 3

6 15 15

[ ]

r E Z

µ µ µ7

µ λ λ λ

≡ = + + + .

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Note tha form of centered moments of (5) tells us that the IG probability density is always positively skewed and the excess kurtosis is always positive. Figure 1 illustrates the shape of the IG distribution with varying parameters. In Panel A, as

t the

λ increases, its variance, skewness, and excess kurtosis decreases. In Panel B, as µ rises holding λ constant, all moments rise.

0 0.5 1 1.5 2

z 0

0.5 1 1.5 2

ytilibaborPytisneD

λ=16 λ=8 λ=4 λ=2 λ=1 λ=0.5 λ=0.1

A) µ = 1 and varying λ

0 1 2 3 4 5

z 0

0.5 1 1.5

ytilibaborPytisneD

λ=32 µ=16 µ=8 µ=4 µ=2 µ=1 µ=0.5

B) Varying µ and λ = 1

Figure 1 Plot of IG probability density

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Probably, the most important property of the IG distribution is its infinite divisibility. Let Z be an IG random variable with µ∈ R and ⁺ λ∈ R⁺. Then, there exist pieces of

random variable

n . . .

i i d Z Z₁, ₂,...,Z each from the IG distribution with _n and such that:

µ/ n∈ R⁺ λ/ n∈ R⁺

1 2

... _n

Z d Z +Z + +Z ,

which is the definition 3.3 of Matsuda (2005). This indicates that the IG distribution generates a class of increasing Lévy processes (subordinators).

[2] Inverse Gaussian Distribution: Barndorff-Nielsen (1998) Parameterization In this section, basic properties of the inverse Gaussian distribution is presented following Barndorff-Nielsen (1998)’s parameterization.

Reparameterize the IG probability density (1) using µ δ γ= / and λ δ= ²:

( ; , ) ^{3/ 2}exp ₂ ( )² 1 ₀

2 2 ^x

IG z z z

z

λ λ

µ λ µ

π ⁻ µ ^>

⎧ ⎫

= ⎨− − ⎬

⎩ ⎭

2 2 2

3/ 2

2 0

( ; , ) exp 1

2 2( / ) ^x

IG z z z

z

δ δ δ

δ γ π δ γ γ

−

>

⎧ ⎛ ⎞ ⎫

⎪ ⎪

= ⎨− ⎜ − ⎟ ⎬

⎝ ⎠

⎪ ⎪

⎩ ⎭

2 2

3/ 2 2

2 0

( ; , ) exp 2 1

2 2 ^x

IG z z z z

z

δ γ δ δ

δ γ π γ γ

− >

⎧ ⎛ ⎞⎫

⎪ ⎪

= ⎨− ⎜ − + ⎟⎬

⎪ ⎝ ⎠⎪

⎩ ⎭

2 2

3/ 2

( ; , ) exp 1 0

2 2

2 ^x

IG z z z

z

δ γ δ

δ γ δγ

π

−

>

⎧ ⎫

= ⎨− + − ⎬

⎩ ⎭

3/ 2

(

² ¹ ²

)

( ; , ) exp 1 0

2 2 ^x

z z

IG z δ γ δ z δγ δ γ

π

−

>

⎧ + ⎫

= ⎪⎨ −

⎪ ⎪

⎩ ⎭

⎪⎬ , (7)

where δ∈ R⁺ and γ ∈ R⁺.

By Fourier transforming the IG probability density (7), its characteristic function φ ω_Z( ) is calculated as:

φ ω_Z( )≡^F_Z[IG z( ; , )]( )δ γ ω ≡

∫

_−∞^∞e^{i z}^ω IG z dz( )

φ ω_Z( )=

∫

0^∞e^{i z}^ω IG z dz( ) =exp

(

δγ δ− ^γ²⁻2i^ω

)

. (8)

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The characteristic exponent (i.e. cumulant generating function) ψ ω_Z( ) of the IG distribution is:

ψ ω_Z( )≡lnφ ω_Z( )=δγ δ γ− ²−2iω . (9)

Using (9), the first four cumulants defined by 1 ( ) ₀

( )

n Z

n n n

cumulant Z

i ^ω

ψ ω

ω ⁼

≡ ∂

∂ are

calculated as the follows:

cumulant₁ =δ γ/ , cumulant₂ =δ γ/ ³,

cumulant₃=3 /δ γ⁵,

7

4 15 /

cumulant = δ γ .

Using the above cumulants, the mean, variance, skewness, and excess kurtosis of the IG random variable Z are obtained as (consult Table 4.1 of Matsuda (2004)):

E Z[ ]=δ γ/ , (10) Variance Z[ ]=δ γ/ ³,

3 [ ] Skewness Z

= δγ ,

[ ] 15

Excess Kurtosis Z

=δγ .

The moment generating function M_Z( )ω of the IG distribution can be expressed as:

( ) ^z ( ; , ) 0 ^z ( )

MZ ω ^∞e IG z^ω δ γ dz ^∞e IG z dz^ω

≡

∫

−∞ =

∫

MZ^{( )}ω =^exp

(

ξ ωZ^{( )}

)

, where the Laplace exponent ξ ω_Z( ) is given by:

ξ ω_Z( )=δγ δ γ− ²−2ω . (11) Using the moment generating function M_Z( )ω with (11), first four raw moments (i.e.

0 n ( )

Z

n n

r M ω _ω

ω ⁼

≡ ∂

∂ ) of the IG distribution are computed as:

r₁≡E Z[ ]=δ γ/ ,

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₂ ² (1 ₃ ) [ ]

r E Z δ δγ

γ

≡ = + ,

2 2 3

3 5

(3 3 )

[ ]

r E Z δ δγ δ γ γ

+ +

≡ = ,

2 2 3 3

4

4 7

(15 15 6 )

[ ]

r E Z δ δγ δ γ δ γ

γ

+ + +

≡ = .

Note that the form of centered moments of (10) tells us that the IG probability density is always positively skewed and the excess kurtosis is always positive. Figure 2 illustrates the shape of the IG distribution with varying parameters. In Panel A, as γ increases holding δ constant, all the standardized moments decrease. In Panel B, as δ rises holding γ constant, the mean and variance rise while the skewness and excess kurtosis fall.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 z

0 2 4 6 8 10

ytilibaborPytisneD

γ=8 γ=6 γ=4 γ=2 γ=1 γ=0.5 γ=0.1

A) δ = 1 and varying γ

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0 2 4 6 8 10 z

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ytilibaborPytisneD

δ=8 δ=6 δ=4 δ=2 δ=1 δ=0.5 δ=0.1

B) Varying δ and γ = 1

Figure 2 Plot of IG probability density

Probably, the most important property of the IG distribution is its infinite divisibility. Let Z be an IG random variable with µ∈ R and ⁺ λ∈ R⁺. Then, there exist pieces of

random variable

n . . .

i i d Z Z₁, ₂,...,Z each from the IG distribution with _n and such that:

µ/ n∈ R⁺ λ/ n∈ R⁺

1 2

... _n

Z d Z +Z + +Z ,

which is the definition 3.3 of Matsuda (2005). This indicates that the IG distribution generates a class of increasing Lévy processes (subordinators).

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References

Barndorff-Nielsen, O. E., 1998, “Processes of Normal Inverse Gaussian Type,” Finance and Stochastics 2, 41-68.

Johnson, N. L., Kotz, S., and Balakrishnan, N., 1994, Continuous Univariate Distributions, Volume 1, Second Edition, John Wiley & Sons.

Matsuda, K., 2004, “Introduction to Option Pricing with Fourier Transform: Option Pricing with Exponential Lévy Models,” Working Paper, Graduate School and University Center of the City University of New York.

Matsuda, K., 2005, Introduction to the Mathematics of Lévy Processes, Vol. 1 of Ph.D.

thesis expected to be filed on May 2006, Graduate School and University Center of the City University of New York.