EGARCH(1,1)

The EGARCH process, standing for exponential GARCH, has been introduced by Nelson [1991]. The equations are

r(t + δt) = σ∘eff(t)ϵr(t) δt σeff(t) = --- eh(t) 1y ( |r(t)| ) h (t) = (1 - β1)ln(σ∞ ) + α1 ------------ E [|ϵ|] + β1 h(t - δt) σeff(t - δt)

with the three parameters ln(σ∞),α₁ and β₁, and with the constraint β₁ < 1. The parameter ln(σ∞) fixes the mean annualized volatility. The constant E[|ϵ|] depends on the probability distribution for the resisual ϵ_r.

The parameters for the simulations are

ln(σ∞) = 1.0 (annualized)
α₁ = 0.015
β₁ = 0.999.

The innovations have a Student distribution with 3.3 degrees of freedom. The simulation time corresponds to 200 years with a time increment δt = 3 minutes.

References

Daniel B. Nelson. Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59:347–370, 1 1991.