Quantitative Trading and Systematic Investing

Letian Wang Blog on Quant Trading and Portfolio Management


Exponential Moving Average (EWA) and Irregular Tick Data Intervals


Exponentially Weighted Moving Average (EWMA or EWA) is a simple and powerful tool in quantitative trading. There are a lot of technical analysis articles about this concept. There are some articles explaining the situation under varying or irregular intervals. I write this note in order to apply it to intra-day trading.

Regular Intervals

EWA is a smoothing technique for both prices and volatilities. Generally, let \(Y\) be EWA, for regular intervals,

\[ \begin{aligned} {Y}_n &= \alpha * X_n + (1-\alpha) *{Y}_{n-1} = Y_{n-1}+\alpha*(X_n-Y_{n-1}) \\ \alpha &= \frac{2}{1+days}=\frac{1}{(1+days)/2}=\frac{1}{G} \end{aligned} \]

The deonominator \(G\) is related to the concept of half-life as described here.

Specifically, consider any time scale, e.g., minutes or seconds instead of days,

\[ \begin{aligned} Y_n &= \alpha X_n+(1-\alpha) Y_{n-1} \\ &= \alpha X_n+\alpha(1-\alpha)X_{n-1}+(1-\alpha)^2Y_{n-2}=... \\ &=\alpha \left[ X_n + (1-\alpha) X_{n-1} + (1-\alpha)^2X_{n-2} + ... \right] \end{aligned} \]

Neglecting first coefficient, assume each step is \(\Delta t\), half life is \(H\) in unit of time or \(H/{\Delta t}\) in steps,

\[ \begin{aligned} (1-\alpha)^{H / \Delta t} &= 0.5 \rightarrow \alpha = 1-e^{ln0.5\frac{\Delta t}{H}} \\ &\approx -ln0.5\frac{\Delta t}{H} \end{aligned} \]

It is obvious to see that in daily case \[ G = \frac{H}{-ln0.5}=1.4427H \]

Irregular Intervals

Now consider an irregular tick arrival. For example, \(G=10s\) and current EWA is \(100\).

Case One: next tick comes in \(1\) second with price \(101\), then new EWA is

\[ \begin{aligned} \alpha &= 1-e^{-1/10}=0.09516\\ Y_1 &= Y_0 + \alpha (X_1-Y_0) = 100.095 \end{aligned} \]

Case Two: If rather the next tick comes in \(400\)ms,

\[ \begin{aligned} \alpha &= 1-e^{-0.4/10}=0.0392\\ Y_1 &= Y_0 + \alpha (X_1-Y_0) = 100.039 \end{aligned} \]

Therefore current EMA \(100\) is more relevant in case two.

Furthermore, in case two, if after 600ms another tick comes in, then \[ \begin{aligned} Y_2 &= \alpha_2 X_2 + (1-\alpha_2)Y_1 \\ &=\alpha_2 X_2 + (1-\alpha_2)\alpha_1 X_1 + (1-\alpha_2)(1-\alpha_1)Y_0 \end{aligned} \]

where the weight of \(Y_0\) or current EWA at \(100\) is \((1-\alpha_2)(1-\alpha_1)=e^{-0.6/10}e^{-0.4/10}=e^{-1/10}\), which is exactly the weight on \(Y_0\) in case one. In other words, current EWA decays at the same rate regardless which path it takes, e.g., either 1sec or 400ms+600ms, or any other route.

In sum, for irregular arrivals, the EWA weight follows expnential distribution. This idea is implemented in the intraday moving average strategy.

DISCLAIMER: This post is for the purpose of research and backtest only. The author doesn't promise any future profits and doesn't take responsibility for any trading losses.