Measuring Toxic Flow for Trading & Risk Management

A common theme of microstructure modeling is that trade flow is often predictive of market direction.  One concept in particular that has gained traction is flow toxicity, i.e. flow where resting orders tend to be filled more quickly than expected, while aggressive orders rarely get filled at all, due to the participation of informed traders trading against uninformed traders.  The fundamental insight from microstructure research is that the order arrival process is informative of subsequent price moves in general and toxic flow in particular.  This is turn has led researchers to try to measure the probability of informed trading  (PIN).  One recent attempt to model flow toxicity, the Volume-Synchronized Probability of Informed Trading (VPIN)metric, seeks to estimate PIN based on volume imbalance and trade intensity.  A major advantage of this approach is that it does not require the estimation of unobservable parameters and, additionally, updating VPIN in trade time rather than clock time improves its predictive power.  VPIN has potential applications both in high frequency trading strategies, but also in risk management, since highly toxic flow is likely to lead to the withdrawal of liquidity providers, setting up the conditions for a flash-crash” type of market breakdown.

The procedure for estimating VPIN is as follows.  We begin by grouping sequential trades into equal volume buckets of size V.  If the last trade needed to complete a bucket was for a size greater than needed, the excess size is given to the next bucket.  Then we classify trades within each bucket into two volume groups:  Buys (V(t)B) and Sells (V(t)S), with V = V(t)B + V(t)S
The Volume-Synchronized Probability of Informed Trading is then derived as:

Typically one might choose to estimate VPIN using a moving average over n buckets, with n being in the range of 50 to 100.

Another related statistic of interest is the single-period signed VPIN. This will take a value of between -1 and =1, depending on the proportion of buying to selling during a single period t.

Fig 1. Single-Period Signed VPIN for the ES Futures Contract

It turns out that quote revisions condition strongly on the signed VPIN. For example, in tests of the ES futures contract, we found that the change in the midprice from one volume bucket the next  was highly correlated to the prior bucket’s signed VPIN, with a coefficient of 0.5.  In other words, market participants offering liquidity will adjust their quotes in a way that directly reflects the direction and intensity of toxic flow, which is perhaps hardly surprising.

Of greater interest is the finding that there is a small but statistically significant dependency of price changes, as measured by first buy (sell) trade price to last sell (buy) trade price, on the prior period’s signed VPIN.  The correlation is positive, meaning that strongly toxic flow in one direction has a tendency  to push prices in the same direction during the subsequent period. Moreover, the single period signed VPIN turns out to be somewhat predictable, since its autocorrelations are statistically significant at two or more lags.  A simple linear auto-regression ARMMA(2,1) model produces an R-square of around 7%, which is small, but statistically significant.

A more useful model, however , can be constructed by introducing the idea of Markov states and allowing the regression model to assume different parameter values (and error variances) in each state.  In the Markov-state framework, the system transitions from one state to another with conditional probabilities that are estimated in the model.

An example of such a model  for the signed VPIN in ES is shown below. Note that the model R-square is over 27%, around 4x larger than for a standard linear ARMA model.

We can describe the regime-switching model in the following terms.  In the regime 1 state  the model has two significant autoregressive terms and one significant moving average term (ARMA(2,1)).  The AR1 term is large and positive, suggesting that trends in VPIN tend to be reinforced from one period to the next. In other words, this is a momentum state. In the regime 2 state the AR2 term is not significant and the AR1 term is large and negative, suggesting that changes in VPIN in one period tend to be reversed in the following period, i.e. this is a mean-reversion state.

The state transition probabilities indicate that the system is in mean-reversion mode for the majority of the time, approximately around 2 periods out of 3.  During these periods, excessive flow in one direction during one period tends to be corrected in the
ensuring period.  But in the less frequently occurring state 1, excess flow in one direction tends to produce even more flow in the same direction in the following period.  This first state, then, may be regarded as the regime characterized by toxic flow.

Markov State Regime-Switching Model

Markov Transition Probabilities

P(.|1)       P(.|2)

P(1|.)        0.54916      0.27782

P(2|.)       0.45084      0.7221

Regime 1:

AR1           1.35502    0.02657   50.998        0

AR2         -0.33687    0.02354   -14.311        0

MA1          0.83662    0.01679   49.828        0

Error Variance^(1/2)           0.36294     0.0058

Regime 2:

AR1      -0.68268    0.08479    -8.051        0

AR2       0.00548    0.01854    0.296    0.767

MA1     -0.70513    0.08436    -8.359        0

Error Variance^(1/2)           0.42281     0.0016

Log Likelihood = -33390.6

Schwarz Criterion = -33445.7

Hannan-Quinn Criterion = -33414.6

Akaike Criterion = -33400.6

Sum of Squares = 8955.38

R-Squared =  0.2753

R-Bar-Squared =  0.2752

Residual SD =  0.3847

Residual Skewness = -0.0194

Residual Kurtosis =  2.5332

Jarque-Bera Test = 553.472     {0}

Box-Pierce (residuals):         Q(9) = 13.9395 {0.124}

Box-Pierce (squared residuals): Q(12) = 743.161     {0}

 

A Simple Trading Strategy

One way to try to monetize the predictability of the VPIN model is to use the forecasts to take directional positions in the ES
contract.  In this simple simulation we assume that we enter a long (short) position at the first buy (sell) price if the forecast VPIN exceeds some threshold value 0.1  (-0.1).  The simulation assumes that we exit the position at the end of the current volume bucket, at the last sell (buy) trade price in the bucket.

This simple strategy made 1024 trades over a 5-day period from 8/8 to 8/14, 90% of which were profitable, for a total of $7,675 – i.e. around ½ tick per trade.

The simulation is, of course, unrealistically simplistic, but it does give an indication of the prospects for  more realistic version of the strategy in which, for example, we might rest an order on one side of the book, depending on our VPIN forecast.

Figure 2 – Cumulative Trade PL

References

Easley, D., Lopez de Prado, M., O’Hara, M., Flow Toxicity and Volatility in a High frequency World, Johnson School Research paper Series # 09-2011, 2011

Easley, D. and M. O‟Hara (1987), “Price, Trade Size, and Information in Securities Markets”, Journal of Financial Economics, 19.

Easley, D. and M. O‟Hara (1992a), “Adverse Selection and Large Trade Volume: The Implications for Market Efficiency”,
Journal of Financial and Quantitative Analysis, 27(2), June, 185-208.

Easley, D. and M. O‟Hara (1992b), “Time and the process of security price adjustment”, Journal of Finance, 47, 576-605.

 

Forecasting Financial Markets – Part 1: Time Series Analysis

The presentation in this post covers a number of important topics in forecasting, including:

  • Stationary processes and random walks
  • Unit roots and autocorrelation
  • ARMA models
  • Seasonality
  • Model testing
  • Forecasting
  • Dickey-Fuller and Phillips-Perron tests for unit roots

Also included are a number of detailed worked examples, including:

  1. ARMA Modeling
  2. Box Jenkins methodology
  3. Modeling the US Wholesale Price Index
  4. Pesaran & Timmermann study of excess equity returns
  5. Purchasing Power Parity

     

     

    Master’s in High Frequency Finance

    I have been discussing with some potential academic partners the concept for a new graduate program in High Frequency Finance.  The idea is to take the concept of the Computational Finance program developed in the 1990s and update it to meet the needs of students in the 2010s.

    The program will offer a thorough grounding in the modeling concepts, trading strategies and risk management procedures currently in use by leading investment banks, proprietary trading firms and hedge funds in US and international financial markets.  Students will also learn the necessary programming and systems design skills to enable them to make an effective contribution as quantitative analysts, traders, risk managers and developers.

    I would be interested in feedback and suggestions as to the proposed content of the program.

    A Practical Application of Regime Switching Models to Pairs Trading

    In the previous post I outlined some of the available techniques used for modeling market states.  The following is an illustration of how these techniques can be applied in practice.    You can download this post in pdf format here.

    The chart below shows the daily compounded returns for a single pair in an ETF statistical arbitrage strategy, back-tested over a 1-year period from April 2010 to March 2011.

    The idea is to examine the characteristics of the returns process and assess its predictability.

    The initial impression given by the analytics plots of daily returns, shown in Fig 2 below, is that the process may be somewhat predictable, given what appears to be a significant 1-order lag in the autocorrelation spectrum.  We also see evidence of the
    customary non-Gaussian “fat-tailed” distribution in the error process.

    An initial attempt to fit a standard Auto-Regressive Moving Average ARMA(1,0,1) model  yields disappointing results, with an unadjusted  model R-squared of only 7% (see model output in Appendix 1)

    However, by fitting a 2-state Markov model we are able to explain as much as 65% in the variation in the returns process (see Appendix II).
    The model estimates Markov Transition Probabilities as follows.

    P(.|1)       P(.|2)

    P(1|.)       0.93920      0.69781

    P(2|.)     0.060802      0.30219

    In other words, the process spends most of the time in State 1, switching to State 2 around once a month, as illustrated in Fig 3 below.


    In the first state, the  pairs model produces an expected daily return of around 65bp, with a standard deviation of similar magnitude.  In this state, the process also exhibits very significant auto-regressive and moving average features.

    Regime 1:

    Intercept                   0.00648     0.0009       7.2          0

    AR1                            0.92569    0.01897   48.797        0

    MA1                         -0.96264    0.02111   -45.601        0

    Error Variance^(1/2)           0.00666     0.0007

    In the second state, the pairs model  produces lower average returns, and with much greater variability, while the autoregressive and moving average terms are poorly determined.

    Regime 2:

    Intercept                    0.03554    0.04778    0.744    0.459

    AR1                            0.79349    0.06418   12.364        0

    MA1                         -0.76904    0.51601     -1.49   0.139

    Error Variance^(1/2)           0.01819     0.0031

    CONCLUSION
    The analysis in Appendix II suggests that the residual process is stable and Gaussian.  In other words, the two-state Markov model is able to account for the non-Normality of the returns process and extract the salient autoregressive and moving average features in a way that makes economic sense.

    How is this information useful?  Potentially in two ways:

    (i)     If the market state can be forecast successfully, we can use that information to increase our capital allocation during periods when the process is predicted to be in State 1, and reduce the allocation at times when it is in State 2.

    (ii)    By examining the timing of the Markov states and considering different features of the market during the contrasting periods, we might be able to identify additional explanatory factors that could be used to further enhance the trading model.

    Regime-Switching & Market State Modeling

    The Excel workbook referred to in this post can be downloaded here.

    Market state models are amongst the most useful analytical techniques that can be helpful in developing alpha-signal generators.  That term covers a great deal of ground, with ideas drawn from statistics, econometrics, physics and bioinformatics.  The purpose of this short note is to provide an introduction to some of the key ideas and suggest ways in which they might usefully applied in the context of researching and developing trading systems.

    Although they come from different origins, the concepts presented here share common foundational principles: 

    1. Markets operate in different states that may be characterized by various measures (volatility, correlation, microstructure, etc);
    2. Alpha signals can be generated more effectively by developing models that are adapted to take account of different market regimes;
    3. Alpha signals may be combined together effectively by taking account of the various states that a market may be in.

    Market state models have shown great promise is a variety of applications within the field of applied econometrics in finance, not only for price and market direction forecasting, but also basis trading, index arbitrage, statistical arbitrage, portfolio construction, capital allocation and risk management.

     REGIME SWITCHING MODELS

    These are econometric models which seek to use statistical techniques to characterize market states in terms of different estimates of the parameters of some underlying linear model.  This is accompanied by a transition matrix which estimates the probability of moving from one state to another.

     To illustrate this approach I have constructed a simple example, given in the accompanying Excel workbook.  In this model the market operates as follows:

      Where

    Yt is a variable of interest (e.g. the return in an asset over the next period t) 

    et is an error process with constant variance s2 

    S is the market state, with two regimes (S=1 or S=2) 

    a0 is the drift in the asset process 

    a1 is an autoregressive term, by which the return in the current period is dependent on the prior period return 

    b1 is a moving average term, which smoothes the error process 

     This is one of the simplest possible structures, which in more general form can include multiple states, and independent regressions Xi as explanatory variables (such as book pressure, order flow, etc):

     

    The form of the error process et may also be dependent on the market state.  It may simply be that, as in this example, the standard deviation of the error process changes from state to state.  But the changes can also be much more complex:  for instance, the error process may be non-Gaussian, or it may follow a formulation from the GARCH framework.

    In this example the state parameters are as follows:

      Reg1 Reg 2
    s 0.01 0.02
    a0 0.005 -0.015
    a1 0.40 0.70
    b1 0.10 0.20

     

    What this means is that, in the first state the market tends to trend upwards with relatively low volatility.  In the second state, not only is market volatility much higher, but also the trend is 3x as large in the negative direction.

    I have specified the following state transition matrix:

      Reg1 Reg2
    Reg1 0.85 0.15
    Reg2 0.90 0.10

     

    This is interpreted as follows:  if the market is in State 1, it will tend to remain in that state 85% of the time, transitioning to State 2 15% of the time.  Once in State 2, the market tends to revert to State 1 very quickly, with 90% probability.  So the system is in State 1 most of the time, trending slowly upwards with low volatility and occasionally flipping into an aggressively downward trending phase with much higher volatility.

    The Generate sheet in the Excel workbook shows how observations are generated from this process, from which we select a single instance of 3,000 observations, shown in sheet named Sample.

    The sample looks like this:

    
     
     
     

     As anticipated, the market is in State 1 most of the time, occasionally flipping into State 2 for brief periods.

     

     

     It is well-known that in financial markets we are typically dealing with highly non-Gaussian distributions.  Non-Normality can arise for a number of reasons, including changes in regimes, as illustrated here.  It is worth noting that, even though in this example the process in either market state follows a Gaussian distribution, the combined process is distinctly non-Gaussian in form, having (extremely) fat tails, as shown by the QQ-plot below.

     

     

    If we attempt to fit a standard ARMA model to the process, the outcome is very disappointing in terms of the model’s poor explanatory power (R2 0.5%) and lack of fit in the squared-residuals:

     

     

    ARIMA(1,0,1)

             Estimate  Std. Err.   t Ratio  p-Value

    Intercept                      0.00037    0.00032     1.164    0.244

    AR1                            0.57261     0.1697     3.374    0.001

    MA1                           -0.63292    0.16163    -3.916        0

    Error Variance^(1/2)           0.02015     0.0004    ——   ——

                           Log Likelihood = 7451.96

                        Schwarz Criterion = 7435.95

                   Hannan-Quinn Criterion = 7443.64

                         Akaike Criterion = 7447.96

                           Sum of Squares =  1.2172

                                R-Squared =  0.0054

                            R-Bar-Squared =  0.0044

                              Residual SD =  0.0202

                        Residual Skewness = -2.1345

                        Residual Kurtosis =  5.7279

                         Jarque-Bera Test = 3206.15     {0}

    Box-Pierce (residuals):         Q(48) = 59.9785 {0.115}

    Box-Pierce (squared residuals): Q(50) = 78.2253 {0.007}

                  Durbin Watson Statistic = 2.01392

                        KPSS test of I(0) =  0.2001    {<1} *

                     Lo’s RS test of I(0) =  1.2259  {<0.5} *

    Nyblom-Hansen Stability Test:  NH(4)  =  0.5275    {<1}

    MA form is 1 + a_1 L +…+ a_q L^q.

    Covariance matrix from robust formula.

    * KPSS, RS bandwidth = 0.

    Parzen HAC kernel with Newey-West plug-in bandwidth.

     

     

    However, if we keep the same simple form of ARMA(1,1) model, but allow for the possibility of a two-state Markov process, the picture alters dramatically:  now the model is able to account for 98% of the variation in the process, as shown below.

     

    Notice that we have succeeded in estimating the correct underlying transition probabilities, and how the ARMA model parameters change from regime to regime much as they should (small positive drift in one regime, large negative drift in the second, etc).

     

    Markov Transition Probabilities

                        P(.|1)       P(.|2)

    P(1|.)            0.080265      0.14613

    P(2|.)             0.91973      0.85387

     

                                  Estimate  Std. Err.   t Ratio  p-Value

    Logistic, t(1,1)              -2.43875     0.1821    ——   ——

    Logistic, t(1,2)              -1.76531     0.0558    ——   ——

    Non-switching parameters shown as Regime 1.

     

    Regime 1:

    Intercept                     -0.05615    0.00315   -17.826        0

    AR1                            0.70864    0.16008     4.427        0

    MA1                           -0.67382    0.16787    -4.014        0

    Error Variance^(1/2)           0.00244     0.0001    ——   ——

     

    Regime 2:

    Intercept                      0.00838     2e-005   419.246        0

    AR1                            0.26716    0.08347     3.201    0.001

    MA1                           -0.26592    0.08339    -3.189    0.001

     

                           Log Likelihood = 12593.3

                        Schwarz Criterion = 12557.2

                   Hannan-Quinn Criterion = 12574.5

                         Akaike Criterion = 12584.3

                           Sum of Squares =  0.0178

                                R-Squared =  0.9854

                            R-Bar-Squared =  0.9854

                              Residual SD =  0.002

                        Residual Skewness = -0.0483

                        Residual Kurtosis = 13.8765

                         Jarque-Bera Test = 14778.5     {0}

    Box-Pierce (residuals):         Q(48) = 379.511     {0}

    Box-Pierce (squared residuals): Q(50) = 36.8248 {0.917}

                  Durbin Watson Statistic = 1.50589

                        KPSS test of I(0) =  0.2332    {<1} *

                     Lo’s RS test of I(0) =  2.1352 {<0.005} *

    Nyblom-Hansen Stability Test:  NH(9)  =  0.8396    {<1}

    MA form is 1 + a_1 L +…+ a_q L^q.

    Covariance matrix from robust formula.

    * KPSS, RS bandwidth = 0.

    Parzen HAC kernel with Newey-West plug-in bandwidth.

    There are a variety of types of regime switching mechanisms we can use in state models:

     

    Hamiltonian – the simplest, where the process mean and variance vary from state to state

    Markovian – the approach used here, with state transition matrix

    Explained Switching – where the process changes state as a result of the influence of some underlying variable (such as interest rate volatility, for example)

    Smooth Transition – comparable to explained Markov switching, but without and explicitly probabilistic interpretation.

     

     

    This example is both rather simplistic and pathological at the same time:  the states are well-separated , by design, whereas for real processes they tend to be much harder to distinguish.  A difficulty of this methodology is that the models can be very difficult to estimate.  The likelihood function tends to be very flat and there are a great many local maxima that give similar fit, but with widely varying model forms and parameter estimates.  That said, this is a very rich class of models with a great many potential applications.

     

     

    Volatility Forecasting in Emerging Markets

    The great majority of empirical studies have focused on asset markets in the US and other developed economies.   The purpose of this research is to determine to what extent the findings of other researchers in relation to the characteristics of asset volatility in developed economies applies also to emerging markets.  The important characteristics observed in asset volatility that we wish to identify and examine in emerging markets include clustering, (the tendency for periodic regimes of high or low volatility) long memory, asymmetry, and correlation with the underlying returns process.  The extent to which such behaviors are present in emerging markets will serve to confirm or refute the conjecture that they are universal and not just the product of some factors specific to the intensely scrutinized, and widely traded developed markets.

    The ten emerging markets we consider comprise equity markets in Australia, Hong Kong, Indonesia, Malaysia, New Zealand, Philippines, Singapore, South Korea, Sri Lanka and Taiwan focusing on the major market indices for those markets.   After analyzing the characteristics of index volatility for these indices, the research goes on to develop single- and two-factor REGARCH models in the form by Alizadeh, Brandt and Diebold (2002).

    Cluster Analysis of Volatility
    Processes for Ten Emerging Market Indices

    The research confirms the presence of a number of typical characteristics of volatility processes for emerging markets that have previously been identified in empirical research conducted in developed markets.  These characteristics include volatility clustering, long memory, and asymmetry.   There appears to be strong evidence of a region-wide regime shift in volatility processes during the Asian crises in 1997, and a less prevalent regime shift in September 2001. We find evidence from multivariate analysis that the sample separates into two distinct groups:  a lower volatility group comprising the Australian and New Zealand indices and a higher volatility group comprising the majority of the other indices.

    Models developed within the single- and two-factor REGARCH framework of Alizadeh, Brandt and Diebold (2002) provide a good fit for many of the volatility series and in many cases have performance characteristics that compare favorably with other classes of models with high R-squares, low MAPE and direction prediction accuracy of 70% or more.   On the debit side, many of the models demonstrate considerable variation in explanatory power over time, often associated with regime shifts or major market events, and this is typically accompanied by some model parameter drift and/or instability.

    Single equation ARFIMA-GARCH models appear to be a robust and reliable framework for modeling asset volatility processes, as they are capable of capturing both the short- and long-memory effects in the volatility processes, as well as GARCH effects in the kurtosis process.   The available procedures for estimating the degree of fractional integration in the volatility processes produce estimates that appear to vary widely for processes which include both short- and long- memory effects, but the overall conclusion is that long memory effects are at least as important as they are for volatility processes in developed markets.  Simple extensions to the single-equation models, which include regressor lags of related volatility series, add significant explanatory power to the models and suggest the existence of Granger-causality relationships between processes.

    Extending the modeling procedures into the realm of models which incorporate systems of equations provides evidence of two-way Granger causality between certain of the volatility processes and suggests that are fractionally cointegrated, a finding shared with parallel studies of volatility processes in developed markets.

    Download paper here.

    Resources for Quantitative Analysts

    Two of the smartest econometricians I know are Prof. Stephen Taylor of Lancaster University, and Prof. James Davidson of Exeter University.

    I recall spending many profitable hours in the 1980′s with Stephen’s book Modelling Financial Time Series, which I am pleased to see has now been reprinted in a second edition.  For a long time this was the best available book on the topic and it remains a classic. It has been surpassed by very few books, one being Stephen’s later work Asset Price Dynamics, Volatility and Prediction.  This is a superb exposition, one that will repay close study.    

    James Davidson is one of the smartest minds in econometrics. Not only is his research of the highest caliber, he has somehow managed (in his spare time!) to develop one of the most advanced econometrics packages available.  Based on Jurgen Doornik’s Ox programming system, the Time Series Modelling package covers almost every conceivable model type, including regression models, ARIMA, ARFIMA and other single equation models, systems of equations, panel data models, GARCH and other heteroscedastic models and regime switching models, accompanied by very comprehensive statistical testing capabilities.  Furthermore, TSM is very well documented and despite being arguably the most advanced system of its kind it is inexpensive relative to alternatives.  James’s research output is voluminous and often highly complex.  His book, Econometric Theory, is an excellent guide to the state of the art, but not for the novice (or the faint hearted!).

    Those looking for a kinder, gentler introduction to econometrics would do well to acquire a copy of Prof. Chris Brooks’s Introductory Econometrics for Finance. This covers most of the key ideas, from regression, through ARMA, GARCH, panel data models, cointegration, regime switching and volatility modeling.  Not only is the coverage comprehensive, Chris’s explanation of the concepts is delightfully clear and illustrated with interesting case studies which he analyzes using the EViews econometrics package.    Although not as advanced as TSM, EViews has everything that most quantitative analysts are likely to require in a modeling system and is very well suited to Chris’s teaching style.  Chris’s research output is enormous and covers a great many topics of interest to financial market analysts, in the same lucid style.