Long-Memory Affine Microscopic ARCH process

In order to build processes with long memory, a set of (historical) volatilities are computed over increasing time horizons. Each historical volatility is measured by an exponential moving average (EMA):

τk = lkτ1 k = 1,⋅⋅⋅,n μk = μ1∕lk = exp (- δt∕τk) (1) 2 1 2 2 σ k(t) = μkσk(t - δt) + (1 - μk )r (t).

The time structure l_k of the process is a geometric series l_k = ρ^k-1. The time horizons τ_k correspond to the characteristic times of the EMA at which the historical volatility is measured. The usual parameters μ_k entering in the EMA are related to the EMA time scales τ_k with the second equation. The last equation computes an historical volatility using an EMA (exponential moving average) of the returns at the shortest time horizon δt.

The effective volatility is defined by:

n σ2 (t) = σ2 + (1 - w ) ∑ χ (σ2(t) - σ2 ) eff ∞ ∞ k k ∞ n k=1 ∑ 2 2 = wk σk(t) + w ∞ σ∞ (2) k=1 wk = (1 - w ∞ )χk

The coefficients χ_k are such that ∑ χ_k = 1 and ∑ w_k + w∞ = 1. They are defined with a simple function, so as to obtained the desired decay for the volatility lagged correlation. For a power law decay:

( ) λ - (k-1)λ 1- χk = cρ = c lk (3) n ∑ -(k-1)λ 1 ∕c = ρ . k=1

A similar definition is used for a logarithmic decay. This process is described in details in Zumbach [2004] and Zumbach [2012].

The simulation uses a power law decay for the coefficients, with parameters:

σ∞ = 0.11
w∞ = 0.068
τ₁ = exp(-4) day = 26 min
ρ =
ρ^n-1 = 1024 (i.e. n = 21)
λ = 0

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

Gilles Zumbach. Volatility processes and volatility forecast with long memory. Quantitative Finance, 4:70–86, 2004.

Gilles Zumbach. Discrete Time Series, Processes, and Applications in Finance. Springer Finance, 2012.