The long memory exponential stochastic volatility model with n components is defined by:

∘ --- r (t + δt) = δt-σ (t) ϵ (t) (1a) 1y eff r ( ) ∑ σeff(t) = σ ∞ exp hk(t) (1b) k hk(t) = hk (t - δt) + dhk(t) k = 1,⋅⋅⋅,n (1c) ∘ --- dhk(t) = - δt (hk(t - δt) - mk ) + √γ- δt ϵσ,k(t) (1d) τk n τk { 0 for k = n mk = (1e) hk+1 for k < n τk = ρk- 1τ0 k = 1,⋅⋅⋅,n (1f)

The parameters of the process are σ∞, τ₁, τ_n, γ and the PDF for ϵ_r and ϵ_σ,k.

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 σeff 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_k: The characteristic time of return for h_k(t) toward the mean m_k.
m_k the mean term for the logarithmic volatility.
In this model, the mean volatility at a given scale k is the volatility h_k+1 at the longer scale k + 1.
γ: The strength of the volatility noise. This parameter fixes the vol-of-vol.
ϵ_σ,k: random variable, with zero mean and unit variance.
For the simulations, the distribution is a Student with 3.0 degree of freedom.

The parameters for the simulations are:

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