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Trade Scheduling in Equity Markets:Theory and Practice

Michael Simmonds

Liquid Markets Analytics

Contents

Nomura (Company and Analytics Teams)

Trade Scheduling Framework

Transaction Cost Estimation

Liquidity Prediction

Risk Estimation

Trade Scheduling Optimisation

Applications

Source:

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1

14th Sept 2009: Opened discussion with Lehman administrators

22nd Sept 2009: Announced acquisition of Asia-Pacific, including Japan and Australia

23rd Sept 2009: Announced acquisition of Europe and Middle Eastern equities and investment banking operations

7th Oct 2009: Hired selected former Lehman Brothers fixed income staff

14th Oct 2009: Completed acquisition of three Lehman companies in India

Europe & Middle East

Acquisition of equities and investment banking operations

Approx 2,500 people Hired ex-Lehman fixed income

staff: interest rate, credit and currency linked operations

Approx 250 people

India

Acquired three subsidiaries: LB service India IT, Global Servicing; LB Financial Services (India) Research services; LB Structured Finance Services Capital Markets Support and Analytics

Approx 2,900 people

Japan

Acquired Japan franchise Approx 1,100 people

Asia (ex Japan)

Acquired Asia Pacific franchise Approx 1,500 people

Nomura moved quickly and decisively

Lehman Acquisition

2

Geography of Nomura

4

Europe & Middle East Asia-Pacific Americas

1,060 employees in 3 countries with presence in:

North America:

– New York

– San Francisco

– Toronto

South America:

– Sao Paolo

20,500 employees in 13 countries with presence in:

Asia ex-Japan:

– Bangkok

– Beijing

– Hanoi

– Hong Kong

– Jakarta

– Kuala Lumpur

– Manila

Japan:

– 171 branches countrywide

– Tokyo headquarters

– Melbourne

– Mumbai

– Seoul

– Shanghai

– Singapore

– Sydney

– Taipei

Note: (1) Subject to regulatory approval.All headcount figures are approximate.

4,500 employees in 18 countries with presence in:

Europe:

– Amsterdam

– Budapest

– Dublin

– Frankfurt

– Geneva

– London

– Luxembourg

– Madrid

Middle East:

– Bahrain

– Dubai

– Saudi Arabia

– Qatar(1)

– Milan

– Moscow

– Paris

– Rome

– Stockholm

– Vienna

– Warsaw

– Zurich

3

5

London Stock Exchange

80

70

60

50

40

30

20

10

Dec-08 Jan-09 Feb-09 Mar-09 Apr-09 May-09 Jun-09 Jul-09

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

Rank 66 25 11 11 11 8 3 1

Market Share 0.11% 0.69% 2.71% 3.19% 3.15% 4.14% 6.24% 7.45%

Dec-08 Jan-09 Feb-09 Mar-09 Apr-09 May-09 Jun-09 Jul-09

200

150

100

50

Dec-08 Jan-09 Feb-09 Mar-09 Apr-09 May-09 Jun-09 Jul-09

0.00%

3.00%

6.00%

9.00%

Rank 182 61 6 6 4 3 3 1

Market Share 0.00% 0.15% 4.26% 4.92% 5.55% 6.35% 7.34% 10.66%

Dec-08 Jan-09 Feb-09 Mar-09 Apr-09 May-09 Jun-09 Jul-09

Eurex Derivatives Exchange

Note: London Stock Exchange statistics are whole trading volumes, weighted by value tradedEurex statistics are for Listed Equity Index Volume whole traded volumes, weighted by value traded

4

Analytics Team

Based in London, New York, Tokyo, Hong Kong and Mumbai

London office quants are approximately 70% have PhDs

Highest degrees typically in Mathematics, Physics, Engineering, Computer Science and

Economics

Location of highest degree concentrated in UK/US/France

Focus areas (in Equities) include algorithmic trading, market microstructure modelling, risk

estimation, structured product creation and volatility modelling

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UK

France

USA

Japan

Canada India

5

The Troika of Quantitative Investment

Primary focus of the Quant community Factor models to exploit behavioural biases in security valuation Represent systematisation of the stock selection process

Focus on loss preservation and efficient capital allocation

Estimated using fundamental/statistical factor models

Generally purview of third-party vendors but recently an area of internal focus

Measures shortfall due to the implementation process

Depends critically on the execution style and strategy (front-loaded, passive, back-loaded, etc)

Usually receives the least focus by Quant Portfolio Managers

Risk

Return

Cost

6

Trade Implementation as a Scientific Process

Market impact modeling (Transaction Cost Modeling)

Model estimation principles similar to multi-factor modeling in alpha research Markets have memory so static impact models are not adequate Example: Nomura METRIC model

Liquidity, volume profile and volatility prediction

PCA decomposition of volume into systematic and idiosyncratic components Estimating volatility using non-stationary and non-synchronous tick data Example: Nomura Volume Prediction and Volatility Prediction Models

Optimal trade scheduling

Non-linear optimisation techniques similar to multi-period portfolio construction Example: Nomura PortfolioIS Algorithm

7

Trade Scheduling Algorithms are typically formulated as optimisation problems

Price evolution model: Random walk, Short-term momentum, Mean-reversion Market impact model: Instantaneous, with Memory Performance criteria – deviation from a target benchmark Trade as quickly as possible to reduce opportunity cost without causing market impact

Construction of Trade Scheduling Algorithms

Order parameters

Trade Schedule:

Number of shares to trade in each bin

Price evolution model

Market impact model

Performance criterion

Tra

de

Sch

ed

ulin

g A

lgo

rith

m

Current market conditions

8

Execution Algorithms Systematise Implementation

Execution algorithms implement a systematic trade implementation process

process vast amount of real-time market data make simultaneous trading decisions at different time scales

Execution algorithms can be decomposed into three modules

Trade scheduling algorithm slices the original institutional size order into a sequence of smaller trades (minutely horizon decisions)

Order placement algorithm decides type and timing of trades to send to the market (secondly horizon decisions)

Market access algorithm decides which destination to route each order (millisecond horizon decisions)

Trade Scheduling

Order Placement

Market Access

trade motivation order parameters liquidity profiles

limit order model short-term alpha signals

dynamic venue execution quality analysis

9

10

METRIC

Model Estimated TRade Impact Cost (METRIC)

Focused on Execution Costs

Cost models have limited constraints (other than

matching the data), but some no-arbitrage

constraints can be applied

Data set is large (~1M trades used in a calibration)

and noisy with ~40% of orders rejected using

reasonable criteria

Calibration methodology is critical, as is correct time

frame selection (matching sample size versus slow

timescale effects) to maintain stable parameters of

multiple data sets

execution costs

feestaxescommissions

fixed costs

trading costs

instantaneous impacttransient impactpermanent impact

opportunity costs

11

METRIC: Observations

The dependence of execution cost on many descriptive variables is quite intuitive and is easily

verified:

Large orders are relatively expensive to trade.

Stocks with high volume tend to be cheaper to trade

Stocks with higher bid-ask spreads tend to be more expensive to trade

Volatile stocks tend to be more expensive to trade than stocks that stay in tight trading ranges

Similar stocks in different countries and on different exchanges within a country may be more

or less expensive, depending on exchange structure and data reporting conventions.

12

METRIC: Structure

Decompose cost into three parts:

Instantaneous impact: A measure of our micro execution skills which only affects child orders individually and then dissipates immediately.

Transient impact: Caused by temporary imbalances between supply and demand caused by our trades which lead to temporary price movement from equilibrium. Transient impact induced price will reverse after our trade and decay to 0 at the end.

Permanent impact: Impact due to changes in the equilibrium price caused by our trading, which accumulates and remains for the life of the trade. Permanent impact induced price will not mean reverse and stay at the end price level after trading. Therefore, we can capture permanent impact if and only if we wait long enough.

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METRIC

0.0

3.5

7.0

10.5

14.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2Time

Impact

Transient Impact Temporary Impact Permanent Impact

Total impact over the trade period

Trade PeriodPost-Trade

Period

2TvTvSMETRIC

Where S is the average bid-ask spread, is the volatility, v is the trade rate and T is the trade duration

14

METRIC: Model QualityOut of Sample Performance Performance versus Trade Rate

Performance versus Period VolatilityPerformance versus Bid-Ask Spread

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Risk Modeling Variance of cost model is closely correlated with period (time scaled) volatility

Stock price moves are heavily correlated, though stock-wise correlation is not found in transaction cost estimates

Principles are fundamentally based on a linear mappings (given a return vector R with expected return µ one assumes that for a set of factors with returns F then there exist L such that R - µ = L.F + where E( ) = 0 and the matrix L is the factor loading matrix. If E( F ) = 0 and cov( L. ) =0 then

cov( R - µ ) = cov(L.F + ) = L.cov( F ).LT + cov( ) = L.cov( F ).LT + where = cov( ) Therefore the risk matrix, , is defined by = L.cov( F ).LT + and is constant for rotations of factors (i.e. if a

new family of factors F’=Q.F and one defines L’=L.QT such that Q.QT = I then ’ = ) Weighting schema, time scale and factor selection are critical to producing good quality risk estimates

0

10

20

30

40

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2%

Period Volatility

Impact Standard Deviation (bps)

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Liquidity Prediction The METRIC and Nomura algorithms are very sensitive to the intra-day liquidity profiles used Major project to improve liquidity prediction versus using historic profiles

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Liquidity Prediction Focus on volume, but same methodology is applied to volatility and spread Profile shows a characteristic and persistent U shape Suggest:

Stock Profile = “Market Profile” + Stock Specific Deviation Given a list of stocks i=1, …., N and intraday time bins t=1, … , 35 can define a matrix of profiles

for any given day Xi,t and hence a correlation matrix can be defined

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Liquidity Prediction

First examine the eigenvalues: first mode is largest and explains more than 40% of the variance, magnitude of first three eigenvalues are much larger than the others

Eigenvalues of the correlation matrix of X First eigenmode

19

Liquidity Prediction: Stock Specific

The following is observed for the profile after discounting the market profile for each stock:

Null hypothesis of stochastic non-stationarity is rejected using Augmented Dickey Fuller Test (ADF)

Box-Jenkins (noting ACF and partial ACFs decay exponentially) suggests that ARMA(1,1) is optimal; describing next bin in terms of the current one and the deviation of the previous bin:

Important to note that : mean reversion effect

t

p

iiti

p

iitit bYaY

11

0,0 11 ba

20

Liquidity Prediction: Model Quality

Define quality measure so that for time bin t on day j for stock i via:

So then P defines the improvement of our methodology versus the static predictions where

Similar results for volatility but

minimal improvement versus

historic for spread profiles

jti

D

d

djit

jit XD

X

1

,,

1

)var(

)var(1

i

iiP

Universe Min 1st Qu.

Median

Mean

2nd Qu. Max

FTSE 100 -0.23 0.20 0.27 0.27 0.33 0.58

FTSE MidCap -0.20 0.14 0.18 0.18 0.24 0.50

FTSE Small Cap -2.81 0.06 0.13 0.08 0.18 0.45

DAX 30 -0.44 0.22 0.26 0.25 0.33 0.46

Cac 40 -0.76 0.21 0.27 0.22 0.31 0.44

Tokyo (300 stocks) 0.01 0.15 0.24 0.25 0.35 0.56

Korea (200 stocks) 0.07 0.24 0.30 0.29 0.34 0.48

Hong Kong -0.06 0.25 0.31 0.31 0.41 0.58

NYSE -0.14 0.24 0.34 0.34 0.45 0.78

Nasdaq -0.10 0.17 0.29 0.29 0.39 0.78 21

Liquidity Prediction: Enhancements

The model predicts the next bin, but can be extended to produce an expected profile for the remainder of the day at any point through the day

However can improve upon this by adjusting according to the volume traded so far

22

Trade Scheduling

Can combine risk, liquidity prediction and cost models to run mean-variance minimisation of the objective function (for a given set of positions X(t)):

Computed trade schedule is kept constant throughout trading interval (e.g., VWAP, TWAP) (i.e. pick appropriate discretisation)

TtttttMETRIC )(ˆ)()(ˆ))(;( X..σXX

23

Trade Scheduling

Static Trade Scheduling Algorithms optimisation to compute trade schedule is performed initially

computed trade schedule is kept constant throughout trading interval (e.g., VWAP, TWAP)

Dynamic Trade Scheduling Algorithms trade schedule is re-optimized at the beginning of each bin

optimisation criterion is fixed but depends on market conditions (e.g., Participation, Dynamic VWAP)

Adaptive Trade Scheduling Algorithms trade schedule is re-optimized at the beginning of each bin

optimisation criterion changes in response to market condition (e.g., Aggressive/Passive In The Money)

24

Conclusions

Cash markets require a variety of different modeling techniques to trade

effectively

Calibration methodology is critical to maintain stable and explanatory models

Trade data and intraday data are both critical to effective trade scheduling

Most clients of top-tier brokers have insufficient data (and possibly quantitative

resources) to manage this process themselves

Second tier brokers will struggle to keep pace with developments and may be

forced to “white label” algorithms

25

References

Almgren, Chriss, “Optimal Execution of Portfolio Transactions” (2001)

Almgren, Lorenz, “Adaptive Arrival Price” (2006)

Bialkowski, Darolles and Le Fol, “Decomposing Volume for VWAP Strategies”

(2005)

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