We will once again draw attention to the fact that the solution presented
in Equation 4.15 stays valid only when the state variance is equal to the ob-
servation variance. In other situations, we need to obtain an estimate of the
state and observation variance at each time step. We will therefore conclude
this section with a brief discussion on the estimation of the state and obser-
vation variances.
In general random walk terms, the state variance can be estimated as the
variance of the innovations. Modeling the stock price series as a random
walk, the innovations correspond to the period returns of the price series.
The state variance in this case is therefore the variance of the period returns.
This calculation is rather commonplace in financial circles and is often re-
ferred to as historic volatility. The observation variance is, however, a tricky
issue. Let us assume that in addition to the closing price in a time interval,
we also observe the high and low stock prices within the interval. The vari-
ance of the error in the observation must be the volatility of the stock price
in the time period. This volatility is characterized by high and low values.
Several methods to estimate the volatility based on the high–low prices exist,
and the references are provided in the reference section. One may use any
one of the methods described in the papers in the appendix to estimate the
observation variance. Once the state and observation variances are known,
we are ready to apply the Kalman-filtering approach.
Application: Example with the Standard & Poor Index
Is it conceivable that the observation and innovation variance will be the
same? Why not? After all, they are both representative of the volatility of the
same underlying random walk. Let us therefore see how we fare in practice.
We apply the random walk-filtering process with all its assumptions on the
Standard & Poor (S&P) index. We use the closing prices of the S&P index
depository receipt (spidr) with ticker SPY in our example. As discussed in
the section Filtering the Random Walk, we use Equation 4.13 to determine
the state of the process at a given time. The weights in Equation 4.15 are
determined using Fibonacci numbers. Note that if we decide to use T+1
observations in the state estimation process, the weight of the last data
point according to Equation 4.14 is given by. This is the fraction
of the oldest data point used in the estimate. It turns out that this value ap-
proaches zero rather quickly, and its contribution to the value of the state
becomes insignificant as Tincreases. In order to demonstrate the point, we
constructed a plot of the reciprocals of the Fibonacci series in Figure 4.3.
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64 BACKGROUND MATERIAL