Base information Home Purpose FAQ Notations Formula Figure descriptions Indexes Links to time series Pop-up to time series Two-time series comparison One figure comparison

Market Affine Microscopic ARCH process with MA

This process is build similarly to the Mkt-Aff-Agg-ARCH with MA, but the historical volatilities are computed with a moving average of the microscopic returns instead of the aggregated returns

(1)

with τ_k taken as process parameters.

For the simulation, 4 components are used, corresponding to the intra-day, daily, weekly and monthly horizons. The characteristic time spans τ_k of the EMAs are as for the aggregated process, namely 0.06, 0.7, 3.5 and 14 days. The remaining parameters are:

• σ∞ = 0.11
• w∞ = 0.065
• χ₁ = 0.46
• χ₂ = 0.25
• χ₃ = 0.09
• χ₄ = 0.20

The innovations have a Student distribution with 3.5 degree of freedom. The simulation time corresponds to 200 years with a time increment δt = 3 minutes.