Mathematical and Statistical Methods for Actuarial Sciences and Finance

156 L. Grilli, M.A. Russo, and A. Sfrecola

decision and superscript+indicates the updated value of a parameter, the vector of
minority decisionxwhose componentxt∈{+ 1 ,− 1 }is the entry of the time series
at instantt. Let us suppose thateach player is provided with such a neural network
and makes her decision according to the following rules:

σi=sign(x·ωi)

ωi+=ωi−

η

M

xsign

⎛

⎝

∑N

j= 1

sign(x·ωj)

⎞

⎠=ωi+η

M

xS,

whereωis aM-dimensional weight vector from which decisions are made. So each
player uses anM-bit window of the past minority decisions as an input in order to
adjust the weights and try to capture the “regularities” in the time series. As we can
see later, the choice ofMis often crucial in order to determine the best prediction
power. It is possible to compute the number of predictions as a function ofMin
order to obtain the value ofMfor which it is maximum. In this case the parameter
Mindicates how many past price fluctuations are considered by the agent in order to
make a decision. The parameterηis the learning rate.
In [10] it is shown that it is crucial to select the window of past entries to consider
as an input for current decision correctly, that is the choice of parameterM.The
authors show that the number of corrected predictions is maximum if the neural
network uses the sameMas the sequence generator. This suggests that, if anMvalue
exists for which the neural network predictions are maximum then it is possible to
infer that the sequence of minority decisions is generated by a learning algorithm
with exactly the same valueM. If we apply the same arguments to financial markets
time series, the presence of a valueMfor which the number of corrected predictions
is maximum indicates that the time series is generated by a learning algorithm with
that parameterM, that is the length of the time window used by the investor, and this
is key information derived exclusively with this approach.
Moreover, to determine this value we analyse the number of predictions of the
neural network as a function ofM. Figures 1, 2, 4, 5, 6 and 7 show the results of these
simulations. The result is different according to the market considered; in particular
the case of U.S. Treasury Bond seems to be the most interesting. In this market the
maximum is reached forM=32, that is the dimension of the temporal window of
the past minority decisions to consider as an input of the neural network. The case of
S&P500, DJ Eurostoxx 50, Dow Jones, Mibtel and Nikkei 225 is completely different;
the maximum value forMis, in general, very low (M= 3 −5). This can suggest
that in these markets investors look at the very recent past to make decisions and do
not consider what has happened in the remote past. On the other hand, Fixed Income
Market presents a different situation and it seems to be the most predictable since the
number of predicted entries is the highest one (about 60% of corrected predictions).
This can be explained according to some features that make this market different since
it must follow common laws dictated by macroeconomic variables. As a consequence,
the data present a strong positive correlation between bonds [9]. Another reason is that
usually only large investors (like insurance companies or mutual funds) are interested