Market Affine Aggregated ARCH process with EMA

This process is build similarly to the LM-Aff-Agg-ARCH, but the historical volatilities are computed on a set of predefined time horizons instead of a geometric series. The historical returns are measured on a set of time steps l_k, and the historical volatilities at a corresponding set of time horizons τ_k for the EMAs. The historical volatilities are defined by:

r [l δt](t) = x(t) --x√-(t---lkδt) (1) k lk 2 2 2 σk(t) = μk σk(t - δt) + (1 - μk)r[lkδt] (t) k = 1,⋅⋅⋅,n, (2)

with l_k and τ_k taken as process parameters, and with the decay coefficient of the EMA given by μ_k = exp(-δt ∕ τ_k). The effective variance is a convex combination of the mean variance and of the historical variances with weights given by χ_k, as for the multicomponent ARCH processes. This process has been introduced in Zumbach and Lynch [2001] and Lynch and Zumbach [2003].

For the simulation, 4 components are used, corresponding to the intra-day, daily, weekly and monthly horizons. The lags for computing the returns r[l_kδt] are 1, 8, 17 and 205 steps (on the grid given by δt). The characteristic time spans τ_k of the EMAs are 0.02, 0.7, 3.5 and 14 days. The remaining parameters are:

σ∞ = 0.11
w∞ = 0.084
χ₁ = 0.56
χ₂ = 0.24
χ₃ = 0.06
χ₄ = 0.14

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

References

Paul Lynch and Gilles Zumbach. Market heterogeneities and the causal structure of the volatility. Quantitative Finance, 3:320–331, 2003.

Gilles Zumbach and Paul Lynch. Heterogeneous volatility cascade in financial markets. Physica A, 298(3-4):521–529, September 2001.