In order to test our model against real financial data, we need to find a measurable quantity that can be computed without specifying the ω distribution. For this reason, a direct comparison of the logarithmic prices x(t) with financial datasets cannot be of some help. On the contrary the variance of any time window is independent on the details of ω distribution. It is, therefore, intuitive to consider ρ(T), the T-days quadratic volatility scaled with T, defined as
In the context of our model, the above quantity is
Notice that for α = 0 this quantity would be a constant, while it increases with T when α is positive (repulsive moving averages), and it decreases with T when α is negative (attractive moving averages). Notice also that the one-step (T = 1) quadratic volatility is ρ(1) = σ2(1 − e 2 + α2)/(1 − e 2), which is always larger than σ2.
In other terms, the one-step volatility is not simply σ, the component due to the random variables, but it is systematically larger due to the influence of the moving average. In the long range (T --> ∞) ρ(T) tends to σ2(1 − β)2/(1 − ε )2, which is larger (smaller) then σ2 for positive (negative) α.
The expression (6) of ρ(T) can be fitted with real data varying the free parameters α, β and σ. In fig. 1 ρ(T) is plotted for three datasets: Nasdaq index (fig. 1a), Mibtel index (fig. 1b) and 1998 USD-DM high frequency exchange rate (figs. 1c and 1d).
In the first two cases the agreement between data and expression (6) predicted by our model is excellent up to a critical value of T, which depends on the size of the dataset. For T larger than the critical value insucient statistics makes the experimental values no longer significant.
In particular, we have obtained the following results:
a) Nasdaq index: α = (9.51 ± 0.01) · 10−2, β = (2.49 ± 0.01) · 10−1 and σ = (1.187 ± 0.001) · 10−2 in the range 1 ≤ T ≤ 12;
b) Mibtel index: α = (5.54 ± 0.01) · 10−2, β = (3.15 ± 0.01) · 10−1 and σ = (1.370 ± 0.001) · 10−2 in the range 1 ≤ T ≤ 10;
The two daily indices not only behave qualitatively the same, but also exhibit similar parameters. The main result is that they have a positive α, which means that the moving average repulses the spot price.
Moreover, in both cases the typical memory length is about 2 time steps, which corresponds to a couple of trading days, and σ, which is nearly the daily market volatility, turns out to be about 1%. Let us stress that this is only the random component of daily volatility, this last being larger because of the deterministic component contribution.
The 1998 USD-DM exchange rate case is more complex, since it exhibits a sort of transition phase. In fact ρ(T) turns out to be the union of two di erent curves of type (6) (see fig. 1c):
c) 1998 USD-DM exchange rate: α = (−1.14 ± 0.01) · 10−1, β = (1.57 ± 0.01) · 10−1 and σ = (4.584 ± 0.001) · 10−4 in the range 1 ≤ T ≤ 8;
α = (−9.92 ± 0.01) · 10−1, β = (9.00 ± 0.01) · 10−1 and σ = (8.481 ± 0.001) · 10−4 in the range 8 ≤ T ≤ 150.
The agreement between data and the two fits is remarkable. The transition point corresponds to T = 8 (about 2.5 minutes), as shown in fig.1d where the range 1 ≤ T ≤ 20 is magnified. This peculiar behaviour can be explained taking into account that the price behaves like a quantum variable on time scales from seconds to one or two minutes, according to a sort of indeterminacy principle [16].
In fact, in [16] it is shown that Tρ(T) does not vanish in the T --> 0 limit, which means that two almost contemporary exchange rate bids not necessary coincide. It is than reasonable to argue that the transition at T = 8 in fig. 1c and fig. 1d corresponds to the transition between quantum behaviour and classical behaviour for the price.
In any case the USD-DM high frequency exchange rate has a negative α, which means that the moving average attracts the spot price. Moreover it exhibits a more consistent influence of the moving average with respect to the indices, since |α| is about one order of magnitude larger, and the typical memory turns out to be about 7-8 time steps (2.5 minutes).
FIG. 1. T-days quadratic volatility scaled with T, ρ(T) (circles), as a function of T, for: a) Nasdaq index; b) Mibtel index; c) 1998 USD-DM high frequency exchange rate; d) magnification of the previous case in the range 1 ≤ T ≤ 20. The continuous line represents the best fit of expression (6), which gives:
a) Nasdaq index: α= (9.51 ± 0.01) · 10−2, β = (2.49 ± 0.01) · 10−1 and σ = (1.187 ± 0.001) · 10−2;
b) Mibtel index: α = (5.54 ± 0.01) · 10−2, β = (3.15 ± 0.01) · 10−1 and σ = (1.370 ± 0.001) · 10−2;
c-d) 1998 USD-DM exchange rate: α= (−1.14 ± 0.01) · 10−1, β= (1.57 ± 0.01) · 10−1 and σ = (4.584±0.001) · 10−4 in the range 1 ≤ T ≤ 8; β= (−9.92±0.01) ·10−1 , α= (9.00±0.01) ·10−1 and σ = (8.481 ± 0.001) · 10−4 in the range 8 ≤ T ≤ 150.
By Prof. R. Baviera, Prof. M. Pasquini, Prof. J. Raboanary and Prof. M. Serva
Next: Efficiency
Summary: Index