Mean and standard deviation of price series with Kalman

Question

I like to calculate the mean and standard deviation of a price series, using the Kalman filter. I am somehow stuck with the deviation, or have some problem in understanding, which my research could not solve.

mean(t) =  mean(t-1) + K(t) * ( price(t) - mean(t-1) )

with Kalman gain K(t) = R(t-1) / (R(t-1) + Ve), state variance R(t) = (1 - K(t)) * R(t-1) and measurement error Ve practically as some pre-defined parameter, similarly to the lookback period in a simple mean.

I've read a few times that the variance R should give kind of variance (and thus standard deviation) of the price series. But with a K < 1, R with every iteration just gets smaller and is no way the deviation of the price series. This only would make sense for a constant value to measure, where with every measurement iteration we get more certainty. Is my concept of the Kalman filter too simplistic? Can anybody give me a hint please.

Answer 1

I would suggest check out the Wikipedia page first and use more stylized notations.

In your update equation mean(t) = mean(t-1) + K(t) * ( price(t) - mean(t-1) ) you are basically saying that your state process is mean(t) and price(t) is a measurement of mean(t). This doesn't sound legit

On the other hand, you could have a mean reverting process $$\text{price}(t) = \text{price}(t-1) + \alpha (\text{mean}(t-1) - \text{price}(t-1))$$

Although it looks similar, it's fundamentally different from the update equation in Kalman filter.

Then the state vector for this process could be $X_t = \begin{bmatrix}\text{price}(t) \\ \text{mean}(t) \end{bmatrix}$ and state transition equation could be $$\begin{bmatrix}\text{price}(t) \\ \text{mean}(t) \end{bmatrix} = \begin{bmatrix} 1-\alpha & \alpha \\ 0 & 1 \end{bmatrix} \begin{bmatrix}\text{price}(t) \\ \text{mean}(t) \end{bmatrix} + \begin{bmatrix}\epsilon_1(t) \\ \epsilon_2(t) \end{bmatrix}$$

Denote $F_t = \begin{bmatrix} 1-\alpha & \alpha \\ 0 & 1 \end{bmatrix}$ then the above equation is simply $$X_t = F_tX_t + \epsilon_t$$

The measurement equation could be

$$Z_t = H_tX_t + \nu_t$$

where $Z_t$ is the acutal price series and $H_t = \begin{bmatrix} 1 & 0 \end{bmatrix}$

The you can use the Kalman filter two-step recursion to estimate the mean(t)

Answer 2

Rt in your notation is "filtered" variance R(t|t). The prediction of variance R(t+1|t) adds another term which is not guaranteed to be decreasing overtime.

I think another critical assumption is Ve in your equation. How do you define Ve? For price series Ve as a proxy for volatility makes sense to be time-varying, and probably exhibit some auto-correlation.

Answer 3

You need to run over the times series once for initial variance, and a second time with updated Kalman params for the state variance (your Rt). This is two step recursion