pected profit is then easily calculated by multiplying the number of trades
with the profit per trade. This calculation can be done for different thresh-
old values, and the value that yields the maximum profit is chosen as the
threshold.
The preceding approach necessitates that we know somehow the rate of
crossing of a particular level for an ARMA series. Luckily for us, the prob-
ability or rate of zero crossing or level crossing for an ARMA process may
be calculated using Rice’s formula. Armed with this information, we can
now say that we are ready to handle spreads modeled as ARMA processes.
Case 3: Hidden Markov ARMA Models
Recall that the ARMA series is a linear combination of past white noise re-
alizations. Traditionally, the white noise series used in the construction of
ARMA series are assumed to be Gaussian. But from our earlier discussion
involving securities that enjoy a strict parity relationship, we expected the
spread to be a mixture Gaussian white noise. It is therefore not much of a
stretch to speculate that the underlying white noise series in the ARMA case
to also be a mixture Gaussian white noise series and exhibit GARCH-like
properties.
The most generalized model to cover these cases could be to say that the
underlying white noise series is generated by drawing a sample from a
Gaussian distribution. The Gaussian distribution is at a given time instant,
however, chosen by rolling the dice. Better still, we say that the exact distri-
bution to use is decided by a Markov process. A Markov processis a process
where the set of outcomes of the dice rolling is dependent on the current dis-
tribution. After the distribution is decided, we then use it to draw the white
noise realization. Note that the white noise generation in this case is a two-
step process. The first step decides the distribution to sample from, and the
second step actually draws a sample from the distribution. Models of this
kind have been used in speech processing and are termed hidden Markov
models. The parameters of such models may be evaluated using the popular
Baum-Welch algorithm.
Our model construction process is not done as yet. Once the white noise
process is generated using the mechanism just described, we construct an
ARMA process by taking linear combinations of the past white noise real-
izations at each time step. (Whew!) Note that the modeling process described
in this section is actually a synthesis of the models described in the two ear-
lier sections.
So, how do we decide the threshold values in this case? As a matter of
fact, at this level of complication there are no known methods as of now to
evaluate the zero-crossing or level-crossing rates other than by way of sim-
ulation. One could extract the model parameters from the existing sample
and generate time series data using these parameters. The profit potential of
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