Quantitative thoughts

When dealing with transaction cost analysis, a stock’s volume is assumed to be stable or foreseeable.  However, there is different picture, then we are dealing with an illiquid stock.

It is relatively easy to forecast the volume of a liquid stock, because trading volume has high autocorrelation – the volumes at t and t+1 are correlated. For example, let’s take a look at BPN Paribas stock  autocorrelation figure:

X axis shows the lag between the days and Y axis shows percentage of the correlation. For BNP Paribas stock we have 60 % correlation between t and t+1 and 30 % between t and t+2.

Now, let’s look what is autocorrelation  of an illiquid stock:

The figure above shows, that autocorrelation for this emerging market stock is zero. That means, that we can’t forecast tomorrow’s volume, based on today’s volume.

Imagine, that a portfolio manager has to liquidate 100 000 of illiquid stock where the median of daily volume is 4000 and participation rate has to be maximum 20 %. How many days it will take to liquidate the position?

Because the daily volume of the stock is very volatile, we need more randomness in our forecast. To do that, we can use bootstrap – let’s take the last 250 data-points of the volume and generate 10 000 or 100 000 new time series. Once the have a bunch of new time series, let’s check how many days it would take to liquidate 100 000 stock position in each. Finally, we collect the numbers of the days needed for liquidation  and form a new vector. The result is following:

The histogram above shows, that it will take maximum 80 days to liquidate 100 000 of illiquid stock with 95% confidence.

#volume - vector of the stock's volume
#shareNumber - number of the shares to liquidate
#loop - number of the time-series to be created
#participationRate - what is going to be participation rate 
simNumberOfDays<-function(volume,shareNumber,loop=10000,participationRate=0.05)
{
  rez=matrix(nrow=loop)
  for(i in 1:loop)
  {
    x=nasa[sample(NROW(volume))]
    y=cumsum(x*participationRate)
    rez[i,]=which(y>shareNumber)[1]
  }
  return(rez)
}
rez=simNumberOfDays(illiquidStock,100000,10000,0.2)

I will be giving a presentation on “Optimal transaction cost” in Vilnius on  16  August. While preparing the presentation and looking for an optimal execution solution, a natural question arises: does the size of the trade affect stock market price? I’m sure, you would say 100 % yes. Well, you would be right, but what is the scale of such effect? Is it possible to profit from execution of the big block trades?

Such test is not trivial and to conduct it, you need high frequency data, which is messy in most of the cases. For testing purpose I chose BNP Paribas stock from February 2011 to May 2011. Initially, I had more than 460 k. trades and more than 320k. quotes. However, the data was filtered by buyers initiated trades. To find buyers initiated trades, I used Lee-Ready Rule – short description can be found here on page 2. I found about Lee – Ready rule while reading Maxdama last post and a damn good summary (check page 42).

The first chart below shows the average return  one trade later (within seconds in most of the cases), when big or small trade was done. X axis represents difference between the trade and following trade, Y axis represents the trade size and the dot size represents number of trades within that cluster of volume. As you can see, small trades add 0.0004% to the price, while big ones (more than 980 of shares) increase the price on average 0.0007%

The next figure shows average return one minute later. This time the different between small trades and big one are almost3 times!

While we can see, that stock market prices are affected by big blocks, there’s no easy way to profit from it. You have to take into account bid/ask spread, plus you are becoming liquidity demander when liquidity is dry. On other end, this test shows the cost for each volume cluster and this cost can be used when choosing an optimal strategy for portfolio/stock liquidation.

Transaction cost analysis (TCA) is the framework to achieve best execution in trading context. TCA can be split into three groups: pre-trade analysis, intraday analysis, and post-trade measurement.

Pre-trade analysis allows us to get insight about the future volatility of the price, forecast intra-day and daily volumes, market impact. It evaluates all strategies and advises the strategy that is most consistent with manager preferences for the risk.

Intraday analysis is real time analysis, where the system ensure that the strategy performs in line with the forecast.

Post-trade analysis measures the implementation of the investment decision to ensure, that pre-trade models are accurate and best execution is delivered.

If you want to learn more about the transaction cost analysis, I highly recommend Optimal Trading Strategies: Quantitative Approaches for Managing Market Impact and Trading Risk by Robert Kissell. The book not only covers all type of mentioned analysis, but also considers the practical aspects of the implementation of trading strategy and algorithms. The book includes very interesting part about Principal Bid Transactions or blind bids.

To build a forecast model we need some input parameters. Pre-trade analysis can be split into two parts – fundamental part and statistically based part. It is much ease to derive fundamental facts such as market capitalization, company profile, the sectors in which company operates, than statistically based parameters. For the latter I’m going to use R and the raw stock price data to derive it. The former can easily be found on Internet, Bloomber or Reuter.

The first input parameter that I’m are going to derive is average daily trading volume of the traded security. It is very easy to obtain it – only daily data is necessary. I used 20 days rolling window to get average volume. Here’s an example of IBM security:

The graph shows, how the 20 days average volume evolved since last September – the minimum was 3.5 millions and the maximum was 6 millions traded a day. But the average can be misleading – it takes into account last 20 days and then it derives one number. In most of the case, we are going to deal with short time prediction such as 1 day or 1 hour, so approximation of 20 days does not make a lot sense.
One of possible solutions would be using confidence intervals estimate daily volume. 95% confidence interval is wide used in finance – I will use the same interval answering the following question: based on this interval, what volume will be traded next day.

The density diagram shows distribution of daily volume of IBM security. As you can see most of the days the volume was above 3 millions. However, 5% of the days the volume was below 3 millions. Based on this diagram, we can predict, that tomorrow’s volume will be higher than 3 millions.

Before we finish with the daily volume, we have to test for weekly seasonality. If there is weekday seasonality, then the volume forecast has to be adjusted.

The chart above clearly indicates, that the traded volume on Monday is below (~5%) the day average. There is increase in the volume on Friday, but the significance is under question. Let’s check density diagram to get rid of any doubt about the volatility on Monday and Friday.

The density diagram above shows, that Monday’s peak and the body are shifted to the left. On the other end we can see, that Friday’s body is aligned with others days, only it has a fat tail. Based on this diagram we can eliminate Friday’s volume adjustment and apply Monday’s adjustment only.

Once we finished with daily data we can move to intra-day. Then liquidating (or acquiring) a position in a security, the position has to be sliced into smaller parts to avoid influence of the market and to hide the intentions. For this reason, intra-day volume patterns have to be known. With that in mind, let’s look at how the volume is distributed in relation to the trading hour.

The graph above shows hourly volume pattern, where the volume is grouped by hour. The black dots indicate the median of the hour. Please keep in mind, that the trading starts at 9:30 and the first trading hour has only 30 minutes (if you want align the first hour to the others, then you need to multiply the volume of the first hour by two). As we can see, the first and the last are most traded and the volume drops in the middle of the day.

The next useful thing then liquidating a big position is average trade size. We need to know, how behaves average Joe and what he trades.

The chart above shows all trades grouped by hour – the black dots indicate median of the trades. The following table is supplementary to the chart – here you find median of the hour.

Once again, average trade size is much higher in the first and last hours and it drops ~2.5 times during the trading session.

The final chart shows the volatility grouped by hour. There is a lot jittery, then the market opens, it becomes calmer during the lunch time and slightly increases then the market closes. These are the numbers for each hour:

In this article we analyzed a list of potential input parameters for pre-trade analysis and further forecast. The list is not static and can be extended with supplementary parameters, such as bid-ask spread distribution and etc. The next step is to aggregate these parameters and build the models to forecast volatility and volume.