Future ? Index

The "official" intraday index is based on the constituents' last trade prices. But it is possible (and as we will see beneficial) to use the constituents' mid or microprices. Both alternatives have the advantage of receiving more frequent updates. The microprice has the additional advantage of not being quantized by the finite tick size.

Fig. 1 shows that visually they track each other and the index future almost perfectly in case of the DAX40.

I have used the future as reference since "the future leads the cash" - this will also show in our data.

NB: The cash market solves a very complex multi-dimensional optimization problem with many hard and soft constraints. Suppose the future goes up by 2 bps. How does each constituent need to react to bring cash in line with the future? There are hard tick size constraints, so not each stock can simply go up by the same amount. And even if, not all stocks have the same beta. Then there are the various correlations to consider. And the weights. Truly amazing how the market solves this.

We now want to measure how the different flavors of the index react to a move in the future. We have two asynchronous timeseries, each at irregular timestamps. We use the Hayashi-Yoshida covariance

where ri and rj are the price changes of the DAX future and the index, respectively (no need to distinguish between returns, log returns, and raw price changes over such short time scales). Fig. 2 below shows the result averaged over a month of data. At t=0, a change of the microprice of the FDAX future occurs. The right half of the chart shows subsequent changes in the index. The left half, for negative lags, can be interpreted as the future reacting to past index changes.

A few points to note:

• The covariances for positive lags are much stronger than for negative lags. Hence, "future leads cash."
• The index based on the constituents' microprices reacts quickest - many milliseconds ahead of the "official" index based on the last trade prices.
• The index based on the microprices reacts strongest although I would phrase this as it incorporating changes in the future's microprice most efficiently. Due to tick size constraints and the cost of crossing the spread, minor changes do not feed into trades in the constituents.
• The dip at -100 μs is a bit curious. Not sure how to explain this.
• The round-trip latency is in the tens of microseconds. So the plateaus inside ±30 μs represent reactions to prior changes. - Or are due to participants sending orders to the future and the cash simultaneously.

References

Nicolas Huth and Frédéric Abergel, High frequency lead/lag relationships — Empirical facts, Journal of Empirical Finance 26:41-58, 2014.

T. Hayashi and N. Yoshida. On covariance estimation of non-synchronously observed diffusion processes, Bernoulli, 11(2):359–379, 2005.