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Market Affine Aggregated ARCH process with MA

This process is build similarly to the Mkt-Aff-Agg-ARCH with EMA, but the historical volatilities are computed with a more ”rectangular” moving average instead of the simple EMA. The moving average is defined by a set of iterated exponential moving averages, as defined in Zumbach and Müller [2001].

For a time series z, a recursive definition of an iterated EMA[τ,n] of order n is

(1)

with EMA[τ, 1; z] = EMA[τ; z] the usual exponential moving average. The MA operator is defined so as to have a more rectangular kernel

For the aggregated version of this model, the historical returns r[l_kδt] are measured on a set of time steps l_k. The historical volatilities are computed with MAs for a corresponding set of time horizons τ_k

(3)

with l_k and τ_k taken as process parameters. 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, 40, 80 and 672 steps (on the grid given by δt). The characteristic time spans τ_k of the EMAs are 0.06, 0.7, 3.5 and 14 days. The remaining parameters are:

• σ∞ = 0.11
• w∞ = 0.1285
• χ₁ = 0.49
• χ₂ = 0.24
• χ₃ = 0.09
• χ₄ = 0.18

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.

References