In the past few days, some unusual behaviour has been occurring in after-hours trading of Earthlink shares.

Are there any humans in this market? Hello?

* * * * * * * * * * *

A geological time series may be a representation of midsummer temperature, captured at thousand-year intervals. We don't know what the temperature does in between each of our observations, but it would be reasonable to assume that it varies, probably in quasiperiodic fashion. Worse, our control over the timing of our samples is nowhere near as nice as we like to pretend. Ask a geologist if his samples really are separated by thousand-year intervals and he will smile and have a distant look in his eye. In reality, the samples are at uncertain intervals, and the time series is fitted to some sort of time scale, and the geological parameters of interest have been interpolated (usually in a linear fashion).

Pricing series are different. Each of our observations is one sale. There is no doubt what the price is between sales. By convention the price between sales is that price of the last sale. So there is no need to interpolate data.

Let's look at a simple example. Brown-Forman Corp. (BF.A) had some interesting gyrations on July 12, 2011, as detailed on the Nanex strange days page.

We observe 46 trades time-stamped 09:30:01. Notice the stock trades from $68 down to $23 during this second. The trades are not quite evenly spaced, but I have created the time-delay pseudo phase space plot by assuming they are, and plotting the price of one of the trades with this time stamp against the fifth trade prior (with the same time stamp). Hence we have 42 paired trades to put on a scatter plot. By convention we draw a trajectory through them in sequence. Here is what the resulting pseudo phase space plot looks like.

A masterpiece of flash impressionism! Look at the elegant lines. It looks ready to take flight, free at last from human meddling with the stock price! The initial trades are near the upper right, the final trades took place at the left lower tip.

Now we can add some trading density to the graph. We know the location of each of the paired trades. We choose select the volume--either that of the original trade or that of the lagged trade--it doesn't matter which, but be consistent! I have chosen the lagged volume and contoured using various bin sizes. In these graphs, the bins are 2x2 squares, centred in the midpoint of the four squares.

Here is the same plot with smaller bins.

Smaller binning gives a better image of what's going on. Here we see the greatest trading density was actually in the $50 range. There are five disjoint basins (six disjoint areas, maybe). Other than that I don't know how to interpret this. I'm not sure whether there is any point in trying to tease out any more information from it.

Let us look at trades for ASIA on July 14, 2011.

The stock began trading near $16 and within 1 s had retreated to $14.

Trading density plot.

Here I've used an absolute trading density (i.e. number of shares traded). The most shares traded in one bin was in excess of 50,000 (labelled on diagram). Instead of contouring, I shaded the bins in accordance with the legend. The labelled dot is the first state at 9:30:01.

This exercise is really about displaying the data in a different form in the hopes that we can make some kind of interpretation of it. It is always possible that no interpretation is possible. This has made me dizzy. I am posting these (and will post a few more shortly) in the hopes that someone sees something of note.

Or perhaps this is the correct interpretation.

Are there any humans in this market? Hello?

*After hours pricing on ELNK, August 2, 2011. Lots of action*.

*Image from Nanex.*

*Details of the above image. Same source.*

*And here's the trading action. Not much considering all the bidding activity.*

I think what we are seeing is the elimination of humans from the market. Two algos, using their own stat-arb approaches have a differing opinion about ELNK. One thinks it is a buy at any price below, say, $8--the other thinks it a sell at any price better than, say, $7.95. It is normal for such differences of opinion to exist--indeed, they have to exist for the market to exist. When two humans meet in the market, with just such a difference in opinion, they would soon come to an agreement, the price being dependent on which participant gives away his opinion first.

The algos each try to maximize its own gain. And they do this by showing only a small offering at the best price that doesn't attract any attention. As soon as some interest is shown in their bid, it is cancelled and moved to a much more favourable price. It would be as if one of the human traders had opined "I might be interested in selling some ELNK at $7.95", and then when anyone expresses an interest, suddenly changes his mind, and say, "actually, I meant $8.15." Then the other trader says, "well, if you came down to $8.10, I might be interested," and just as the first trader goes to agree, the other suddenly says, "actually I mean $7.75." After this goes back and forth for awhile until the inevitable fistfight breaks out.

No trading would occur. This approach provides no liquidity.

It is a contest, like the game where you try to step on your opponents foot. One favoured tactic is to dangle your foot in front, luring your opponent into an attack, pulling it out of the way as he does so, and then quickly counterattacking your opponent's extended foot. Every so often one of the opponents manages to touch the other and a trade goes through. Otherwise, the bids and offers just go up and down furiously.

In the last "Deconstructing algos" article we looked at two-dimensional reconstructed phase space portraits of busted trade data for CNTY; original data acquired from the Nanex site here.

As described earlier, one approach to creating a geometric representation of a phase space from a time series is to generate a time-delay plot, in which the values of our time series are plotted against lagged values of the same series. We use a constant lag for reasons described here.

Now, in the CNTY data (and in the data series in today's articles) the time control isn't as fine as we would like. In particular, even though the trades are presented in order, the time stamps only extend to the second. We may have 250 transactions in order in that second, but we don't actually know the length of time between any of them. How do we come up with a constant lag?

We can't. What I did in the last episode was assume that all trades were evenly spaced. In reality, this was unlikely. The result is that my phase space portraits were distorted somewhat from reality. How much distortion depends on how far from evenly spaced the samples are. In practice, with lots of points, the distortion isn't really going to be bad unless you have more than 80% of the trades compressed into an interval comprising less than 20% of the time investigated. This seems unlikely, but it would be nice to be able to check. Intuitively, it seems likely that the many trades at similar values occur close together in time.

Pricing series are different. Each of our observations is one sale. There is no doubt what the price is between sales. By convention the price between sales is that price of the last sale. So there is no need to interpolate data.

Let's look at a simple example. Brown-Forman Corp. (BF.A) had some interesting gyrations on July 12, 2011, as detailed on the Nanex strange days page.

We observe 46 trades time-stamped 09:30:01. Notice the stock trades from $68 down to $23 during this second. The trades are not quite evenly spaced, but I have created the time-delay pseudo phase space plot by assuming they are, and plotting the price of one of the trades with this time stamp against the fifth trade prior (with the same time stamp). Hence we have 42 paired trades to put on a scatter plot. By convention we draw a trajectory through them in sequence. Here is what the resulting pseudo phase space plot looks like.

A masterpiece of flash impressionism! Look at the elegant lines. It looks ready to take flight, free at last from human meddling with the stock price! The initial trades are near the upper right, the final trades took place at the left lower tip.

Now we can add some trading density to the graph. We know the location of each of the paired trades. We choose select the volume--either that of the original trade or that of the lagged trade--it doesn't matter which, but be consistent! I have chosen the lagged volume and contoured using various bin sizes. In these graphs, the bins are 2x2 squares, centred in the midpoint of the four squares.

The above plot used fairly large bins. Each bin has a $20 trading range. I had to use such large bins because there weren't very many trades. The contours are at 10% intervals, meaning that all bins (2x2 boxes) centred within the first shaded contour contain at least 10% of all trades during the one second interval represented in the plot. Most trades occur in the $60-$70 range. The trading density thins out at the lower price intervals.

Smaller binning gives a better image of what's going on. Here we see the greatest trading density was actually in the $50 range. There are five disjoint basins (six disjoint areas, maybe). Other than that I don't know how to interpret this. I'm not sure whether there is any point in trying to tease out any more information from it.

Let us look at trades for ASIA on July 14, 2011.

The stock began trading near $16 and within 1 s had retreated to $14.

Trading density plot.

Here I've used an absolute trading density (i.e. number of shares traded). The most shares traded in one bin was in excess of 50,000 (labelled on diagram). Instead of contouring, I shaded the bins in accordance with the legend. The labelled dot is the first state at 9:30:01.

This exercise is really about displaying the data in a different form in the hopes that we can make some kind of interpretation of it. It is always possible that no interpretation is possible. This has made me dizzy. I am posting these (and will post a few more shortly) in the hopes that someone sees something of note.

Or perhaps this is the correct interpretation.

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