Complex Numbers and the Stable Characteristic Function.

Where z is complex and r,  θ, u, and v are Real, z can be represented in polar and Cartesian forms:

z=r (e^(i θ))=u+i v

Expanding the polar form.

z=r cos(θ)+i r sin(θ)=u+i v

Therefore,

u=r cos(θ);  v=r sin(θ)

Absolute values

(Abs[z])=r=(Sqrt[(u^2)+(v^2)])

Argument

arg(z)=θ=(tan^(-1))(u,v)

Conjugate

(Conjugate[z])=r (e^(-i θ))=u-i v

The stable characteristic function, 1 - parameterization

ϕ(t)=exp(i t δ-(γ^α) ((Abs[t])^α) (1-i β sgn(t) tan(((π α)/2))))

This can be rewritten in polar complex form

ϕ(t)=(e^(-(γ^α) ((Abs[t])^α))) (e^(i β sgn(t) (γ^α) tan(((π α)/2)) ((Abs[t])^α)+i t δ))

Thus

(Abs[ϕ(t)])=r=(e^(-(γ^α) ((Abs[t])^α)))

arg(ϕ(t))=θ=β sgn(t) (γ^α) tan(((π α)/2)) ((Abs[t])^α)+t δ

The Quick approximation of stable γ, using empirical characteristic function, ecf, where (Subscript[X, k]) is each random variable uses the absolute value of the characteristic function.

ecf(t)=(1/n) (∑_(k=1)^n)(e^(i t (Subscript[X, k])))

If (Subscript[X, k]) is from a stable distribution then

(Abs[(((∑_(k=1)^n)(e^(i t (Subscript[X, k]))))/n)])=(e^(-(γ^α) ((Abs[t])^α)))

Letting t = 1,

(Abs[(((∑_(k=1)^n)(e^(i (Subscript[X, k]))))/n)])==(e^(-(γ^α)))

Since stable distributions are scalable, if we divide each (Subscript[X, k]) by γ the scaled stable characteristic function will have γ = 1 and

(Abs[(((∑_(k=1)^n)(e^(i (Subscript[X, k])/γ)))/n)])==(e^(-1))

In Mathematica this can be rewritten and solved numerically:

Abs[Mean[Exp[ I sample/γ]]] == 1/E

Thus γ can be approximated numerically from the empirical characteristic function independently of α for any stable distribution including the normal distribution.  This is how we approximate market volatility, using a small enough sample over which γ hasn't changed much.

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