In addition to that, let us now bring into consideration the empirical ob-
servation of GARCH effects in the volatility of individual stock returns.
GARCH is an acronym for generalized auto-regressive conditional het-
eroskedascity. There is a huge body of literature on it and its ramifications
for options pricing. To describe the phenomenon from a modeling perspec-
tive, let us consider a time series whose value at any point is drawn from a
Gaussian distribution. If the distribution is fixed, we would have Gaussian
white noise. However, we could allow for the variance of the distribution
used at each point in time to vary. In fact, we go even further and prescribe
that the variance of the distribution must follow an ARMA time series. If we
do that, we now have a GARCH process. Although GARCH processes have
been observed and recorded for individual stocks return series and not for
spreads, we assert that it is definitely within the realm of plausibility that
white noise spreads could exhibit the GARCH property.
In any case, to summarize the discussions so far, it seems as though it
would be more realistic to model white noise spreads as values drawn from
normal distributions but with standard deviations that are dependent on
time. The overall distribution of the spread values in this case may be re-
ferred to as a mixture Gaussian distribution.
So, how do we design the trading rules in this case? The solution may
be to resort to multiple threshold levels instead of one to maximize partic-
ipation in the markets at times of both high and low volatility. We would
then need to estimate the component Gaussian distributions and design the
levels for each one of them. An alternate approach would be to do a dy-
namic estimation of the volatility using say Kalman filtering methods and
let the levels vary with time.
Case 2: ARMA Model
Now let us consider the case of a stock that has just experienced a news
event. We can expect the stock to trade in increased volumes with higher
prices leading to higher prices or lower prices leading to lower prices, de-
pending on the nature of the news event. In other words, we could expect
stock prices to exhibit a momentum-like behavior. If this stock were to be
paired up with another stock for pairs trading and the news event was spe-
cific to the particular stock, it would definitely not surprise us to see the mo-
mentum in the stock price series manifest itself as momentum in the spread
series. In other words, we expect to see some correlation between consecu-
tive values of the spread leading to a meaningful autocorrelation function
and therefore an ARMA series for the spread.
So, how do we design our trading rules in this case? The underlying
principles remain the same. Any choice for the threshold level has a profit
per trade associated with it. If we know the rate at which the threshold level
is crossed, we can determine the expected number of trades. The total ex-
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