Long-Memory Affine Pure-Aggregated ARCH (LM-Aff-PureAgg-ARCH)

The equations for the LM-Aff-PureAgg-ARCH process are

l∑max σ2eff(t) = w∞ σ2∞ + (1 - w ∞ ) χl r2[l δt](t) (1) l=1 g χl = -α (2) l
and the constant g is fixed by the condition ∑ χl = 1. The model has three parameters σ∞,w∞ and α. The constant w∞ fixes the overall level of correlation (the correlation increases with decreasing w∞), and the constant α fixes the decrease of the lagged correlation with increasing lags. The parameter σ∞ fixes the mean level for the volatility. The upper cut-off lmax is also an implicit parameter in the model.

A few notes on the model

  1. This model is essentially equivalent to the limit τ0 → 0 of the model LM-Aff-Agg-ARCH. The remaining difference is that the LM-Aff-Agg-ARCH process uses a logarithmic set of time horizons.
  2. The computational time is proportional to the cut-off (whereas for the LM-Aff-Agg-ARCH it is proportional to the log of the cut-off). This difference makes the simulation time much (much) longer.
  3. The model has some analytical tractability, see Borland and Bouchaud [2005]. Models with EMAs seem more complicated.

The figures for this model is given for 2 set of parameters. The first set corresponds to a lagged correlation decay in aggreement with the empirical decay.

  1. σ∞ = 0.11
  2. w∞ = 0.10
  3. α = 1.15
  4. lmax = 32768
  5. p(ϵ): Student with 5 degrees of freedom

The simulation time corresponds to 70 years with a time increment δt = 3 minutes.

References

   Lisa Borland and Jean-Philippe Bouchaud. On a multi-timescale statistical feedback model for volatility fluctuations. Preprint, 2005. Available at: arXiv:physics/0507073.