LM-Aff-Agg-ARCH with stochastic volatility mean

The general motivation for mixing a stochastic volatility mean with a GARCH structure for the volatility feed-back is presented for the GARCH(1,1) process with a stochastic volatility mean.

The present process uses a long memory aggregated ARCH for the volatility feed-back component. The equations for the ARCH volatility components σ_k are described in the page for this process and in microscopic ARCH process for the parameters χ_k. The combined volatility equations are as follow:

∘ ---- δt δt h(t) = - ----h(t - δt) + γSV ---- ϵSV (t) τSV τSV σSV (t) = σ∞ exp (h(t)) ∑ σ2 (t) = w σ2 (t) + (1 - w ) χ σ2(t). eff ∞ SV ∞ k k k

The parameters have been chosen so as to have statistics close to the empirical ones, with a power law decay for the coefficient χ_k. They are

σ∞ = 0.11.
w∞ = 0.1.
τ₁ = exp(-3) day (i.e. τ₁ = 1h12).
ρ =
ρ^n-1 = 512 (i.e. n = 19), leading to τ_n = 25.5 day.
λ = 0.15 (power law decay for χ_k with exponent λ).
τ_SV = 50 day (mean reversion time for the stochastic volatility dynamics).
γ_SV = 0.1 (the vol-of-vol for the stochastic volatility).

The innovations have a Student distribution with 3.3 degree of freedom, while the innovations ϵ_SV for the stochastic volatility have a normal distribution. The simulation time corresponds to 200 years with a time increment δt = 3 minutes.