The one component discrete exponential stochastic volatility model is defined by:

∘ --- δt r (t + δt) = ---σeff(t) ϵr(t) (1a) 1y σeff(t) = σ ∞ exp (h(t)) (1b) h(t) = h (t - δt) + dh(t) ∘ --- (1c) δt δt dh(t) = - -- h(t - δt) + γ -- ϵσ(t) (1d) τ τ

The parameters of the process are σ∞, τ, γ and the PDF for ϵ_r and ϵ_σ.

The various terms are:

δt: the time step for the discrete process.
The term ’1y’ denote the one year time interval, and the factor brings the annualized volatility σ∞ to the scale δt.
ϵ_r: random variable, with zero mean and unit variance.
σ∞: Proportional to the annualized volatility (up to correction in exp(< h² >)).
τ: The characteristic time of return for h(t) toward zero.
γ: The strength of the volatility noise.
ϵ_σ: random variable, with zero mean and unit variance.

The parameters for the simulations are:

The simulation time corresponds to 200 years with a time increment δt = 3 minutes.

Simulation with p(ϵ_σ) = Gaussian gives with similar results.