High Frequency Data Analysis & Market Micro … Frequency Data Analysis & Market Micro structure T....

High Frequency Data Analysis&

Market Micro structure

T. V. Ramanathan

Department of StatisticsSavitribai Phule Pune University

Pune - 411007 (India).

[email protected]

National Workshop on Financial Data Analytics

C R Rao AIMSCS, Hyderabad

December 27-30, 2014

T. V. Ramanathan High Frequency Data Analysis

Introduction

Observations taken at finer time intervals

Ultimate - Transaction-by-Transaction data - NYSE TAQ(Trades & quotes data base - time duration - second)

HFD is important for studying issues related to tradingprocess and market micro structure

1 To compare the efficiency of different trading systems in pricediscovery

2 To study the dynamics of bid-and-ask quotes of a particularstock

3 To study the order dynamics or to investigate the question ofwho provides the market liquidity.

Cho, Russel, Tiao and Tsay (2003) - Analyzed HFD fromTaiwanese stock exchange.


Some important aspects that can be looked in to:

1 Non synchronous trading

2 Bid-ask spread

3 Transaction Data

4 Price movements - Modeling

5 Duration - Modeling


Non synchronous trading

Different stocks have different trading frequencies, (even forsame stock has different trading frequencies) - For dailyreturns, non synchornous trading can introduce the following

1 lag-1 cross correlation between stock returns2 lag-1 serial correlation in a portfolio return3 negative serial correlations of the return series of a single stock.

Illustrate with stock A (more frequently traded than) andstock B. News impact at the close of the day.

see Campbell, Lo, and MacKinlay (1997) and the referencestherein.



Let rt be the continuously compounded return of a security atthe time index t, (iid assumption for simplicity) withE (rt) = µ and Var(rt) = σ2.

Let π be the probability that the security is not traded at aparticular time period (assumed to be same for all timeperiods)

Let r0t be the observed return (= 0 when there is no trade at

time t). Then we have



r0t =

0 with prob. πrt with prob. (1− π)2

rt + rt−1 with prob. (1− π)2πk∑

i=0rt−i with prob. (1− π)2πk ,

.. ..

.. ..

.. ..

, k = 0, 1, 2, ...



It can be proved that

E (r0t ) = µ, Var(r0

t ) = σ+ 2πµ2

1− π, Cov(r0

t , r0t−j) = −µ2πj , j > 1

Thus, when µ 6= 0, the non synchronous trading inducesnegative autocorrelations in an observed security return series.

The above discussion can be generalized to the return seriesof a portfolio that consists of N securities; see Campbell et al.(1997, Chapter 3).


Bid-Ask Spread

In some stock exchanges (e.g., NYSE), market makers play animportant role in facilitating trades. They provide marketliquidity by standing ready to buy or sell whenever the publicwishes to buy or sell.

By market liquidity we mean the ability to buy or sellsignificant quantities of a security quickly, anonymously, andwith little price impact.

Market makers are granted monopoly rights by the exchangeto post different prices for purchases and sales of a security.

Buy at the bid price Pb and sell at a higher ask price Pa.

The difference Pa − Pb is called as the bid-ask spread, whichis the primary source of compensation for market makers.


Bid-Ask Spread

Bid-ask spread introduces negative lag-1 serial correlation inan asset return.

Model introduced by Roll (1984): Pt = P∗t + It

S2

Pt - Observed market price of an asset, S = Pa − Pb, thebid-ask spread, Pt∗ - fundamental value of the asset in africtionless market, It , sequence of independent randomvariables taking values +1 and -1 with equal probabilities 1/2.

It can be interpreted as an order-type indicator, with 1signifying buyer-initiated transaction and -1 seller-initiatedtransaction.

Whenever there is no change in P∗t , the observed price change

is

∆Pt = (It − It−1)S

2


Bid-Ask Spread

It can be easily seen that

E (∆Pt) = 0, Var(∆Pt) = S2/2,

Cov(∆Pt ,∆Pt−1) = −S2/4, Cov(∆Pt ,∆Pt−j) = 0 for j > 1

The autocorrelation function of ∆Pt is

ρj(∆Pt) =

{−0.5 if j = 1

0 if j > 1

Known as bid-ask bounce.


Bid-Ask Spread

Intuitive interpretation:

Assume P∗t = (Pa + Pb)/2. Then Pt will be Pa or Pb. If

previous obs. value is Pa (higher value), then the currentobserved value will be either 0 or Pb (lower value). Thus, ∆Pt

is either 0 or −S . In the case of Pb, it will be 0 or S , and thusthe lag-1 negative correlation in ∆Pt becomes apparent.


Bid-Ask Spread

More realistic formulation: Assume P∗t as a random walk,

P∗t − P∗

t−1 = εt ,

Then,

Var(∆Pt) = σ2 + S2/2, Cov(∆Pt ,∆Pt−j) = 0 for j > 1

ρj(∆Pt) =−S2/4

S2/2 + σ2< 0

To know more about components of bid-ask spread, referCampbell et al. (1997).


Transaction Data

Transaction data: ti - time measured in seconds frommidnight, at which the i-th transaction of an asset,transaction price, the transaction volume, the prevailing bidand ask quotes etc. constitute the transactions data.

Unequally Spaced Time Intervals

Discrete-Valued Prices

Existence of a Daily Periodic or Diurnal Pattern

Multiple Transactions within a Single Second.


Models for Price Changes

Ordered Probit Model: (Hauseman, Lo and MacKinlay (1992))

y∗t = xiβ + εi

y∗t - unobservable price change of the asset under study,y∗t = P∗

ti− P∗

ti−1, P∗

t is the virtual price of the asset at timet, xi is the p-dimensional vector of explanatory variablesavailable at time ti−1, E (εi |xi ) = 0, Var(εi |xi ) = σ2

i andCov(εi , εj) = 0 for i 6= j .

Let the observed price change yi assumes k possible valuess1, s2, ..., sk

The ordered probit model postulates the relationship betweenyi and y∗ as

yi = sj , if αj−1 < y∗i ≤ αj , j = 1, 2, ..., k ;−∞ < α1, ..., αk <∞


Some Basics of Duration Data

Three types of durations: trade, price and volume - proxies fortrading intensity, trading volatility and liquidity.

Trade durations (intensity): Time interval betweenconsecutive trades.

Price durations (volatility): Minimum duration that is requiredto observe a price change not less than a given amount.

Volume durations (liquidity): The time spells such that thetotal traded volume is not smaller than (lets say) 25000shares.

For trade durations zero duration are common.

Intra-day seasonality is observed - To be removed - (Cubicspline function suggested by Engle and Russell (1998))


Duration Models - ACD

Why duration models? Time series cannot accommodateirregularly spaced data.

High frequency data - Transaction data - Irregularly spaced

Engle and Russell (1998) developed autoregressive conditionalduration (ACD) model

{t0, t1, ..., tn, ...} - sequence of arrival times of events0 = t0 ≤ t1 ≤ t2 ≤ ... ≤ tn ≤ .....Duration: xi = ti − ti−1

xi = ψi εi , ψi = E (xi |Fi−1) where εi are iid, E (εi ) = 1.

ψi = ω +

p∑j=1

αjxi−j +

q∑j=1

βjψi−j

For a survey on ACD: Pacurar (2008)


ACD - Some Properties

ARMA(max(p, q), q) formulation possible.p∑

i=1αi +

q∑i=1

βi < 1, condition for cov. stationarity

Stationarity and invertibility conditions need the roots of1− α(L)− β(L) and 1− β(L) respectively to lie outside theunit circle.

E (xi |Fi−1) = ψi , E (xi ) =ω

1−p∑

i=1αi −

q∑i=1

βi

Var(xi |Fi−1) = ψ2i Var(εi )

Var(xi ) = [E (xi )]2

1− 2(p∑

i=1αi )(

q∑i=1

βi )− (q∑

i=1βi )

2

1− 2(p∑

i=1αi )2 − 2(

p∑i=1

αi )(q∑

i=1βi )− (

p∑i=1

αi )2

.


SCD Models

Let {τi} be the occurrence time of a certain event anddi = τi − τi−1, i = 1, 2, ..., n be the durations.

In the case of a SCD model, these observed durations di aremodeled as the product of a latent variable Ψi and a positiverandom variable εi .

That is,

di = Ψi εi , Ψi = eψi , ψi = α + βψi−1 + ui ,

with εi |Fi−1 independent and identically distributed (i.i.d.)random variables having a positive support, ui i.i.d. withsupport on the real line R and εi is independent of uj for alli and j , where Fi−1 denotes the information set available atthe end of duration di−1.

It is assumed that the initial value ψ0 is drawn from thestationary distribution of ψ.


Ongoing work : DST-SERB Project

Developing a method of estimation for tv-ACD model andestablishing the probabilistic properties of the estimators.

Proposing a method to remove the intra-day seasonality inhigh frequency data using wavelets

Generalized class of ACD model incorporating structuralbreaks which accommodates other successful ACD models aswell

Proposing non/semi parametric SCD models and estimatingthe parameters using different approaches and comparing allthe available estimation methods

Modeling durations using point process theory

Developing duration models using new statistical tools whichcapture the dependence in the data very well like copulas andwavelets


Ongoing work : DST-SERB Project

Employ the Bayesian non-parametric techniques of Ghosh andRamamoorthi (2002)for modeling and estimation in almost allof our research works. The advances in Bayesiannon-parametric inference methods have not received a fullcritical and comparative analysis of their scope and limitationsin financial modelling;

Ghosh et. al. (2011) studied Bayesian inference fornon-parametric state-space model. These techniques will beuseful for us while estimating non-parametric SCD models.

Adams (2009) studied the Bayesian inference for pointprocesses which we will be using while working on intensitymodelling of durations.


References

1 Adams (2009)

2 Bauwens, L. and Veredas, D. (2004) The stochasticconditional duration model: a latent factor model for theanalysis of fnancial durations, Journal of Econometrics, 119,381-41

3 Ghosh, J. K. and R. V. Ramamoorthi. (2002). BayesianNonparametrics. Springer, 2002.

4 Campbell, Lo, and MacKinlay (1997)

5 Cho, Russel, Tiao and Tsay (2003)

6 Engle, R.F. and Russell, J.R. (1998). Autoregressiveconditional duration: a new model for irregularly spacedtransaction data. Econometrica, 66, 1127-1162.


References

1 Ghosh, A., Mukhopadhyay, S., Roy, S., Bhattacharya, S.(2011). Bayesian Inference in Non-parametric DynamicState-Space Models. arXiv preprint arXiv:1108.3262

2 Hauseman, Lo and MacKinlay (1992)

3 Pacurar, M. (2008). Autoregressive conditional durationmodels in finance: A survey in the theoretical and empiricalliterature. Journal of Economic Surveys 22, 711-751.


THANK YOU


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