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<!DOCTYPE html>
<html>
<head>
<title>A Hidden Markov Model Approach to Evaluating VPIN</title>
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<meta name="description" content="A Hidden Markov Model Approach to Evaluating VPIN">
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<body style="opacity: 0">
<slides class="layout-widescreen">
<!-- LOGO SLIDE -->
<slide class="title-slide segue nobackground">
<hgroup class="auto-fadein">
<h1>A Hidden Markov Model Approach to Evaluating VPIN</h1>
<h2>Market microstructure analysis via machine learning</h2>
<p>Nicholas Head<br/></p>
</hgroup>
<article></article>
</slide>
<!-- SLIDES -->
<slide class="" id="slide-1" style="background:;">
<article data-timings="">
<!-- Limit image width and height -->
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img {
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<script type="text/javascript" src="http://ajax.aspnetcdn.com/ajax/jQuery/jquery-1.7.min.js"></script>
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});
</script>
<h2>The Lemon Problem</h2>
<p><img src="assets/img/lemons.jpg" alt="'Lemon'"></p>
<ol class = "build incremental">
<li>Introduction to Adverse Selection, PIN, VPIN</li>
<li>Sequential Trade Model and Simulation of Data</li>
<li>Demonstration of HMM decoding of states with cluster analysis</li>
<li>Regressing VPIN against hidden states</li>
<li>Empirical study results</li>
</ol>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="class" id="id" style="background:;">
<hgroup>
<h2>The Lemon Problem</h2>
</hgroup>
<article data-timings="">
<p><img src="assets/img/lemons.jpg" alt="'Lemon'"></p>
<ol class = "build incremental">
<li>'Passive' market makers providing liquidity to 'aggressive' informed traders </li>
<li>Adverse Selection from Toxic Order Flow</li>
<li>Order flow is signed to determine direction of trading activity</li>
<li>Quote Rule, Lee-Ready Rule help infer toxicity...</li>
</ol>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-3" style="background:;">
<hgroup>
<h2>Flash Crash - May 6th 2010</h2>
</hgroup>
<article data-timings="">
<p><img src="assets/img/Flash_Crash.jpg" alt="'FlashCrash'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-4" style="background:;">
<hgroup>
<h2>Sequential Trading Model and Classic PIN</h2>
</hgroup>
<article data-timings="">
<div style='float:left;width:50%'>
<p><img src="assets/img/PIN_Model_1996.jpg" alt="'PIN'"></p>
</div>
<div style='float:right;width:50%'>
<ul class = "build incremental">
<li><p>\(\alpha\): Probability of an 'Information Event'</p></li>
<li><p>\(\delta\): Probability of it being bad news</p></li>
<li><p>\(\mu\): Arrival rate of informed traders</p></li>
<li><p>\(\epsilon\): Arrival rate of uninformed traders</p></li>
<li><p>Assume arrival rates follow Poisson mixture distributions, then parameters can be estimated via MLE hence enabling Probability of Information-based Trading (PIN) to be computed:
\[\text{PIN} = \frac{\alpha \mu}{\alpha \mu + 2 \epsilon}\]</p></li>
</ul>
<p>(Easley & O’Hara - 1996)</p>
</div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-5" style="background:;">
<hgroup>
<h2>High Frequency PIN</h2>
</hgroup>
<article data-timings="">
<ol class = "build incremental">
<li><p>Trades grouped into equal sized volume buckets \(\tau = 1 \dots n\) each of size \(V\)</p></li>
<li><p>Perform bulk classification of buckets e.g. Z% classified as buy, and 1-Z% classified as sell:
\[V_{\tau}^B = \sum_{i = t(\tau - 1) + 1}^{t(\tau)} V_i Z \left(\frac{S_i - S_{i-1}}{\sigma_{\Delta S}}\right)\]
\[V_{\tau}^S = V - V_{\tau}^B\]</p></li>
<li><p>The total expected arrival rate is:
\[\underbrace{\alpha (1 - \delta) (\epsilon + \mu + \epsilon) }_\text{Volume from good news} + \underbrace{\alpha \delta (\mu + \epsilon + \epsilon) }_\text{Volume from bad news} + \underbrace{(1 - \alpha) (\epsilon + \epsilon) }_\text{Volume from no news} = \alpha \mu + 2 \epsilon\]</p></li>
<li><p>Volume Synchronised Probability of Information-based Trading (VPIN) can hence be derived:
\[VPIN = \frac{\alpha \mu}{\alpha \mu + 2 \epsilon} = \frac{\alpha \mu}{V} \approx \frac{\sum_{\tau = 1}^n |V_{\tau}^S - V_{\tau}^B|}{n V}\]
Volume-Synchronized Probability of Informddddde</p></li>
</ol>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-6" style="background:;">
<hgroup>
<h2>Simulation Study</h2>
</hgroup>
<article data-timings="">
<p>\[\alpha =0.28, \delta = 0.33, \mu = 55, \epsilon=22.3\]</p>
<div style='float:left;width:50%'>
<p><img src="assets/img/PIN_Model_1996.jpg" alt="'PIN'"></p>
</div>
<div style='float:right;width:50%'>
<pre><code class="r">while (j <= n) {
if (runif(1) < alpha) {
if (runif(1) < delta) {
Vbuy[j] = rpois(1, epsilon)
Vsell[j] = rpois(1, mu + epsilon)
} else {
Vbuy[j] = rpois(1, mu + epsilon)
Vsell[j] = rpois(1, epsilon)
}
} else {
Vbuy[j] = rpois(1, epsilon)
Vsell[j] = Vbuy[j]
}
j=j+1
}
</code></pre>
</div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-7" style="background:;">
<hgroup>
<h2>Exploratory Data Analysis #1</h2>
</hgroup>
<article data-timings="">
<p>\[\alpha =0.28, \delta = 0.33, \mu = 55, \epsilon=22.3\]</p>
<p><img src="assets/img/sim-time-series.jpg" alt="'Timeseries'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-8" style="background:;">
<hgroup>
<h2>Exploratory Data Analysis #2</h2>
</hgroup>
<article data-timings="">
<p>\[\alpha =0.28, \delta = 0.33, \mu = 55, \epsilon=22.3\]</p>
<p><img src="assets/img/sim-poisson-mixture.jpg" alt="'Poisson'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-9" style="background:;">
<hgroup>
<h2>Fitting a Bivariate Poisson Hidden Markov Model</h2>
</hgroup>
<article data-timings="">
<p>Let \(\{X_t \equiv (B_t, S_t): t = 1,2, \dots T \}\) and \(\{C_t \equiv (C_{b;t}, C_{s;t}) : t = 1,2, \dots T\}\)</p>
<ul class = "build incremental">
<li><p>Conditional probability of observing \(b_t\) buy orders and \(s_t\) sell orders
\[\begin{align}
\begin{aligned}\label{eq:bivarPois}
p_{i,j}(X) &= Pr(X_t = x | C_{b;t} = i, C_{s;t} = j) \\
&= e^{- \lambda_{b;i}} \frac{(\lambda_{b;i})^{b_t}}{b_t !} e^{- \lambda_{s;j}} \frac{(\lambda_{s;j})^{s_t}}{s_t !} \\
\end{aligned}
\end{align}\]</p></li>
<li><p>In matrix form:
\[\mathbf{P}(x) = \begin{pmatrix}
p_1(b_t) p_1(s_t) & & 0 \\
& \ddots & \\
0 & & p_m(b_t) p_n(s_t)
\end{pmatrix}\]</p></li>
<li><p>Unconditional hidden state distribution:
\[u_{i,j}(t) = Pr(C_{b;t} = i, C_{s;t} = j), t = 1,\dots,T\]</p></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-10" style="background:;">
<hgroup>
<h2>Fitting a Bivariate Poisson Hidden Markov Model</h2>
</hgroup>
<article data-timings="">
<p>Let \(\{X_t \equiv (B_t, S_t): t = 1,2, \dots T \}\) and \(\{C_t \equiv (C_{b;t}, C_{s;t}) : t = 1,2, \dots T\}\)</p>
<ul class = "build incremental">
<li>Marginal distribution of \(X\), \(Pr(X_t = x)\):
\[\begin{align}
\begin{aligned}\nonumber
&= \sum_{i=1}^m \sum_{j=1}^n Pr(C_{b;t} = i, C_{s;t} = j) Pr(X_t = x | C_{b;t} = i, C_{s;t} = j) \\
&= \sum_{i=1}^m \sum_{j=1}^n u_{i,j}(t) p_{i,j}(x) \\
&= \begin{pmatrix}
u_{1,1}(t),\dots,u_{1,n}(t), \dots u_{m,1}(t), \dots u_{m,n}(t)
\end{pmatrix}
\begin{pmatrix}
p_1(b_t) p_1(s_t) & & 0 \\
& \ddots & \\
0 & & p_m(b_t) p_n(s_t)
\end{pmatrix}
\begin{pmatrix}
1 \\
\vdots \\
1
\end{pmatrix} \\
&= \mathbf{u}(t) \mathbf{P}(x) \mathbf{1'}
\end{aligned}
\end{align}\]</li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-11" style="background:;">
<hgroup>
<h2>Fitting a Bivariate Poisson Hidden Markov Model</h2>
</hgroup>
<article data-timings="">
<p>Let \(\{X_t \equiv (B_t, S_t): t = 1,2, \dots T \}\) and \(\{C_t \equiv (C_{b;t}, C_{s;t}) : t = 1,2, \dots T\}\)</p>
<ul class = "build incremental">
<li><p>Let
\[\gamma_{i,j;k,l}(t) = Pr(C_{b;t+1} = k, C_{s;t+1} = l | C_{b;t} = i, C_{s;t} = j)\]</p></li>
<li><p>Transition probability matrix:
\[\boldsymbol{\Gamma}(1) =
\begin{pmatrix}
\gamma_{1,1;1,1} \qquad \gamma_{1,1;1,2} & \cdots & \gamma_{1,1;m,n-1} \qquad \gamma_{1,1;m,n}\\
\gamma_{1,2;1,1} \qquad \gamma_{1,2;1,2} & & \gamma_{1,2;m,n-1} \qquad \gamma_{1,2;m,n}\\
\vdots & \ddots & \vdots \\
\gamma_{m,n-1;1,1} \qquad \gamma_{m,n-1;1,2} & & \gamma_{m,n-1;m,n-1} \qquad \gamma_{m,n-1;m,n}\\
\gamma_{m,n;1,1} \qquad \gamma_{m,n;1,2} & \cdots & \gamma_{m,n;m,n-1} \qquad \gamma_{m,n;m,n}
\end{pmatrix}\]</p></li>
<li><p>Hence the likelihood (where \(\boldsymbol{\delta}\) is the stationary hidden state distribution, assuming it exists)
\[L_T = \boldsymbol{\delta} \mathbf{P}(x_1) \boldsymbol{\Gamma} \mathbf{P}(x_2) \dots \boldsymbol{\Gamma} \mathbf{P}(x_T) \mathbf{1'}\]</p></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-12" style="background:;">
<hgroup>
<h2>Exploratory Data Analysis #3</h2>
</hgroup>
<article data-timings="">
<p>\[\alpha =0.28, \delta = 0.33, \mu = 55, \epsilon=22.3\]</p>
<p><img src="assets/img/sim-clusters.jpg" alt="'Poisson'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-13" style="background:;">
<hgroup>
<h2>Decoding the Fitted Hidden Markov Model</h2>
</hgroup>
<article data-timings="">
<ul class = "build incremental">
<li><p>Simulation parameters: \[\lambda_b =
\begin{pmatrix}
\epsilon_b \\
\epsilon_b + \mu_b \\
\epsilon_b
\end{pmatrix} =
\begin{pmatrix}
22.3 \\
77.3 \\
22.3
\end{pmatrix} \qquad
\lambda_s =
\begin{pmatrix}
\epsilon_s \\
\epsilon_s \\
\epsilon_s + \mu_s
\end{pmatrix}=
\begin{pmatrix}
22.3 \\
22.3 \\
77.3
\end{pmatrix}\]</p></li>
<li><p>Fitted estimates: \[\hat{\lambda}_b =
\begin{pmatrix}
\epsilon_b \\
\epsilon_b + \mu_b \\
\epsilon_b
\end{pmatrix} =
\begin{pmatrix}
22.21 \\
79.83 \\
22.12
\end{pmatrix} \qquad
\hat{\lambda}_s =
\begin{pmatrix}
\epsilon_s \\
\epsilon_s \\
\epsilon_s + \mu_s
\end{pmatrix}=
\begin{pmatrix}
21.95 \\
21.95 \\
73.38
\end{pmatrix}\]</p></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-14" style="background:;">
<hgroup>
<h2>Decoding the Fitted Hidden Markov Model</h2>
</hgroup>
<article data-timings="">
<ul class = "build incremental">
<li><p>Conditional distribution of the hidden states given the observations
\[\begin{align}
\begin{aligned}\label{eq:localDecoding}
Pr(C_t = i | \mathbf{X}^{(T)} = \mathbf{x}^{(T)}) &= \frac{Pr(\mathbf{X}^{(T)} = \mathbf{x}^{(T)}, C_t = i)}{Pr(\mathbf{X}^{(T)} = \mathbf{x}^{(T)}}\\
&= \frac{\alpha_t(i) \beta_{t}(i)}{L_T}
\end{aligned}
\end{align}\]</p></li>
<li><p><img src="assets/img/sim-decoded-states.jpg" alt="'States'"></p></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-15" style="background:;">
<hgroup>
<h2>Translating the Decoding States to Volume Bars</h2>
</hgroup>
<article data-timings="">
<p><img src="assets/img/sim-vpin-states-translated.png" alt="'States'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-16" style="background:;">
<hgroup>
<h2>Translating the Decoding States to Volume Bars</h2>
</hgroup>
<article data-timings="">
<p><img src="assets/img/sim-vpin-states.jpg" alt="'States'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-17" style="background:;">
<hgroup>
<h2>Empirical Data Analysis</h2>
</hgroup>
<article data-timings="">
<p><img src="assets/img/empirical-timeseries.png" alt="'States'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-18" style="background:;">
<hgroup>
<h2>Empirical Data Analysis</h2>
</hgroup>
<article data-timings="">
<p><img src="assets/img/empirical-kernels.jpg" alt="'States'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-19" style="background:;">
<hgroup>
<h2>Empirical Data Analysis</h2>
</hgroup>
<article data-timings="">
<p><img src="assets/img/empirical-clusters.jpg" alt="'States'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-20" style="background:;">
<hgroup>
<h2>Empirical Data Analysis</h2>
</hgroup>
<article data-timings="">
<p><img src="assets/img/empirical-vpin-states.jpg" alt="'States'"></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-21" style="background:;">
<hgroup>
<h2>Conclusions</h2>
</hgroup>
<article data-timings="">
<ol class = "build incremental">
<li><p>Mismatch between theoretical model and empirical data</p></li>
<li><p>Possible issues with translation process from time to volume bars </p></li>
<li><p>Data issues: E-Mini S&P 500 futures contracts vs SPY ETF</p></li>
<li><p>Numerical underflow with computational implementation</p></li>
<li><p><img src="http://i2.kym-cdn.com/photos/images/original/000/879/348/504.gif" alt="'States'"></p></li>
</ol>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-22" style="background:;">
<hgroup>
<h2>Questions?</h2>
</hgroup>
<article data-timings="">
</article>
<!-- Presenter Notes -->
</slide>
<slide class="backdrop"></slide>
</slides>
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data-slide=5 title='High Frequency PIN'>
5
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<li>
<a href="#" target="_self" rel='tooltip'
data-slide=6 title='Simulation Study'>
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<li>
<a href="#" target="_self" rel='tooltip'
data-slide=7 title='Exploratory Data Analysis #1'>
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</li>
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<a href="#" target="_self" rel='tooltip'
data-slide=8 title='Exploratory Data Analysis #2'>
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</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=9 title='Fitting a Bivariate Poisson Hidden Markov Model'>
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</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=10 title='Fitting a Bivariate Poisson Hidden Markov Model'>
10
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=11 title='Fitting a Bivariate Poisson Hidden Markov Model'>
11
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=12 title='Exploratory Data Analysis #3'>
12
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=13 title='Decoding the Fitted Hidden Markov Model'>
13
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=14 title='Decoding the Fitted Hidden Markov Model'>
14
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=15 title='Translating the Decoding States to Volume Bars'>
15
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=16 title='Translating the Decoding States to Volume Bars'>
16
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=17 title='Empirical Data Analysis'>
17
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=18 title='Empirical Data Analysis'>
18
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=19 title='Empirical Data Analysis'>
19
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=20 title='Empirical Data Analysis'>
20
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=21 title='Conclusions'>
21
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</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=22 title='Questions?'>
22
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</li>
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