A Kalman filter is a very useful tool in quantitative finance that helps us process noisy measurements to get a better approximation of the true states of the underlying system. A common use of the Kalman filter is in a pairs trading strategy, where the goal is to take advantage of the mean-reversion behavior of two co-integrated assets, such as stocks or ETFs.
Letβs begin with the concept of co-integration between two price series. Assuming that we have two assets, A and B, whose prices are represented by the series PA and PB. If these two series are co-integrated, then a linear combination of them should be stationary, i.e., it should exhibit mean-reverting behavior. Formally, the spread can be defined as:
Spreadtβ=βPA,βtβ
ββ
Ξ²PB,βt
Where Ξ² represents the hedge ratio, which we estimate to find the proper linear combination to make the spread stationary.
The aim of the pairs trading strategy is to exploit the deviations in the spread from its mean. When the spread deviates from its mean, we expect that it will revert back to the mean, and we can take long and short positions in the assets accordingly.
Now, letβs see how we can use the Kalman filter for a pairs trading strategy in three steps:
1. **Use the Kalman filter to estimate the hedge ratio:**
The Kalman filter helps us obtain the hedge ratio Ξ² by taking into account the dynamic nature of the underlying assets. A classic form of a state space model, which can be used to model the regression coefficients using Kalman filter can be presented as follows:
$$\begin{bmatrix}
\alpha_t \\
\beta_t
\end{bmatrix}
=
\begin{bmatrix}
\alpha_{t-1} \\
\beta_{t-1}
\end{bmatrix}
+
\begin{bmatrix}
\varepsilon_{\alpha, t} \\
\varepsilon_{\beta, t}
\end{bmatrix}$$
Spreadtβ=βΞ±tβ
+β
Ξ²tPB,βtβ
+β
Ξ΅Spread,βt
Where Ξ±t is the intercept term, Ξ²t is the slope or hedge ratio, Ρα,βt, and Ρβ,βt denote the process noise values for the intercept and the slope, and Ξ΅Spread,βt is the measurement noise term.
We can use the Kalman filter to estimate the time-varying intercept, slope, and error covariance through iterations as new data points arrive at each time step.
2. **Calculate the spread using the estimated hedge ratio:**
After obtaining the estimates for Ξ±t and Ξ²t, we compute the spread as follows:
$$\hat{Spread}_t = P_{A,t} - \hat{\beta_t} P_{B,t}$$
3. **Generate trading signals based on the spread:**
To generate trading signals, we analyze deviations of spread from its mean using standard deviations or error bands (e.g., 1 or 2 standard deviations). Since our purpose is to take advantage of mean-reverting behavior, we perform the following actions:
- If the spread exceeds a certain threshold (e.g., two standard deviations), we assume that the spread will revert to the mean soon. Hence, we short asset A and long asset B.
- If the spread falls below a certain negative threshold, we assume that the spread will revert to the mean soon. Hence, we long asset A and short asset B.
- If the spread is within a certain range from the mean (e.g., 1 standard deviation), we do not take any position.
This is an outline of how a Kalman filter can be used in a pairs trading strategy. The key advantage is the more robust and adaptive nature of the Kalman filter-based time-varying hedge ratio estimation compared to using a simple linear regression model. In practice, additional considerations such as transaction costs, risk management, and portfolio optimization should also be taken into account to improve the performance of the strategy.