Market Microstructure and Algorithmic Tradingcfrei/PIMS/Almgren4.pdfPIMS Summer School 2016...

Robert Almgren

Market Microstructureand

Algorithmic Trading

PIMS Summer School 2016University of Alberta, Edmonton

Lecture 4: July 6, 2016

Edmonton mini-course, July 2016

Trade trajectory determination

Overall schedule of tradeActual order placement by "micro" algo

Schedule to receive "best" pricerelative to benchmarkminimize variance

Real implementation different than theorybut theory is useful for ideas

2


Real examples

3

Buy 2932 FGBMU6

Produced by Quantitative Brokers from internal and Eurex Market Data

Edmonton mini-course, July 2016 4

2.801

2.802

2.803

2.804

2.805

2.806

2.807

2.808

2.809

2.810

09:03:10 09:03:20 09:03:30 09:03:40 09:03:50 09:04:00 09:04:10 09:04:20 09:04:30

CDT on Tue 05 Jul 2016

BUY 17 NGQ6 BOLT

09:03:19

Done at 09:04:23

Exec 2.806VWAP 2.806

Strike 2.8065

NG

Q6

Passive fillsCumulative execMarket tradesLimit ordersCumulative VWAPMicropriceBid-ask

Exec = 2.806 Cost to strike = -0.38 tick = -$3.82 per lot

09:03:10 09:03:20 09:03:30 09:03:40 09:03:50 09:04:00 09:04:10 09:04:20 09:04:30

0

5

10

15

Exe

cute

d an

d w

orki

ng q

uant

ity

100

200

300

400

500

581

Cum

ulat

ive

mar

ket v

olum

e

09:03:19

Done at 09:04:23

2 @ 2.808

1 @ 2.808

1 @ 2.808

1 @ 2.808

2 @ 2.807

1 @ 2.807

2 @ 2.806

4 @ 2.805

2 @ 2.804

1 @ 2.803

2.9% mkt / Limit = 8%

17 passv 0 aggrPassive fillsWorking quantityCumulative Market VolumeFilled quantityAggressive quantity

2.801

2.802

2.803

2.804

2.805

2.806

2.807

2.808

2.809

2.810

09:03:10 09:03:20 09:03:30 09:03:40 09:03:50 09:04:00 09:04:10 09:04:20 09:04:30

CDT on Tue 05 Jul 2016

BUY 17 NGQ6 BOLT

09:03:19

Done at 09:04:23

Exec 2.806VWAP 2.806

Strike 2.8065

NG

Q6

Passive fillsCumulative execMarket tradesLimit ordersCumulative VWAPMicropriceBid-ask

Exec = 2.806 Cost to strike = -0.38 tick = -$3.82 per lot

09:03:10 09:03:20 09:03:30 09:03:40 09:03:50 09:04:00 09:04:10 09:04:20 09:04:30

0

5

10

15

Exe

cute

d an

d w

orki

ng q

uant

ity

100

200

300

400

500

581

Cum

ulat

ive

mar

ket v

olum

e

09:03:19

Done at 09:04:23

2 @ 2.808

1 @ 2.808

1 @ 2.808

1 @ 2.808

2 @ 2.807

1 @ 2.807

2 @ 2.806

4 @ 2.805

2 @ 2.804

1 @ 2.803

2.9% mkt / Limit = 8%

17 passv 0 aggrPassive fillsWorking quantityCumulative Market VolumeFilled quantityAggressive quantity Slow because

no passive fills

Sudden fillsat end

Produced by Quantitative Brokers from internal and CME Market Data


Outline

1. Benchmarks

2. Market impact models

3. Mean-variance trajectory optimization

4. Option hedging

5


1. Benchmark

“Contract” with clientWhat would constitute an ideal tradeChoice of benchmark determines algoMain benchmarks:

Arrival PriceVWAPTWAP

6


Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

The Implementation Shortfall: Paper Versus RealityPerold, Andre F.Journal of Portfolio Management; Spring 1988; 14, 3; ABI/INFORM Globalpg. 4








IS ≡ "Arrival Price"


Arrival price positives:corresponds to real trade decisioncannot be gamed by executing broker

Arrival price negatives:subject to price motion

impossible to separate alpha from impactstatistically very noisy

8

123−00

123−00+

123−01

123−01+

123−02

123−02+

123−03

08:19:20 08:21:20 08:23:20

CDT time on Wed 06 Jul 2011

2000 lots08:20:34

Strike 123−01¾

●●●●●●●●●●●●●●

●●●●●●●●●●

●

●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●

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●

●●●●● ●●●● ● ● ● ●●●

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●●

●●●● ●● ●●●

●●●●●●

●

●●

●●

●

●●●●

●

●

●●●

●

●

●

●●●●●●●

●

●●●●●●●●

●●●●● ●●●●●●●

●

●●●●

●●●

●

●●

●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●

●

●●●●●●●●●●●●

●●

●●●●●●●●

●●

●●●●●●●●●●●●●●●● ●●●●●●

●

●●●●● ●●●●

●

Arrival Price

Minimize slippage to arrival price benchmarkby completing trade as rapidly as possible



14:00 14:01 14:02 14:03 14:04 14:05 14:06 14:07 14:08 14:09 14:10 14:11

0

20

40

60

80

2k

4k

6k

8k

9k

SELL 80 ZBU1 BOLT

CDT time on Thu 30 Jun 2011

Exec

uted

and

wor

king

qua

ntity

14:01:10

14:10:001 @ 123−00 1 @ 123−00

4 @ 123−00

4 @ 123−00 2 @ 123−01

4 @ 123−01 2 @ 123−01

1 @ 123−01

3 @ 123−01

3 @ 123−01

4 @ 123−01

2 @ 123−01 2 @ 123−01

20 @ 123−02

8 @ 123−03 2 @ 123−03

1 @ 123−03

4 @ 123−03 2 @ 123−03 2 @ 123−03 1 @ 123−03

0.87% mkt

56 passv 24 aggr

Aggressive FillsPassive FillsFilled quantityWorking quantity

Market volume

Working quantity

Filled quantity

Front-loaded (approximately)


Arrival Price Example


VWAP benchmark

Volume-Weighted Average Pricebetween given start time and end time“what the market did”

Positives:easy to evaluate ex post: select siz wavg prc slippage is very consistent

Negatives:Does not correspond to any investment goalCan be gamed by brokerDifficult to forecast volume profile during trading

10

Minimize slippage to VWAP benchmarkby trading exactly along market profile


12:00 12:20 12:40 13:00 13:20 13:40 14:00 14:20 14:40 15:00 15:20

0

500

1000

1500

20k

40k

60k

80k

100k

120k

140k

148k

BUY 1830 ZNU1 STROBE


Exec

uted

and

wor

king

qua

ntity

12:19:07

15:00:00

1 @ 122−28+6 @ 122−30+

11 @ 122−30+

30 @ 123−01 14 @ 123−00+

29 @ 123−00+11 @ 123−00+

5 @ 123−00 6 @ 122−31+

14 @ 122−31+22 @ 123−00+

4 @ 123−00+17 @ 123−00

3 @ 123−00 14 @ 123−00

11 @ 123−00+4 @ 123−00+13 @ 123−01+

17 @ 123−00+1 @ 123−01+

3 @ 123−01 6 @ 123−01

26 @ 123−02 4 @ 123−02

31 @ 123−02 31 @ 123−02

2 @ 123−01+13 @ 123−00+

8 @ 122−31 4 @ 123−00

32 @ 122−31+13 @ 122−30+

1 @ 123−00 20 @ 123−01

8 @ 122−31+8 @ 122−31+

10 @ 123−00+2 @ 123−00

6 @ 122−31+11 @ 122−30+

13 @ 122−31 5 @ 122−31

5 @ 122−30+1 @ 122−30+

1.2% mkt

1611 passv 219 aggr

Aggressive FillsPassive FillsAhead/behind boundariesBase profileFilled quantityWorking quantity

Market volume

Cumulative actual market volume

Executions trackforecast market volume

burst at 14:00


VWAP Example (1)


VWAP Example (2)

12

Blue Line: where we predictedVWAP trajectory, relative to actual

Triangles: where we executed, close to predicted curve

but far from actual

Market printing more than forecast:we fall behind VWAP

Market printing less than forecast:we get ahead of VWAP

122−26

122−27

122−28

122−29

122−30

122−31

123−00

123−01

123−02

123−03

12:00 12:20 12:40 13:00 13:20 13:40 14:00 14:20 14:40 15:00 15:20


BUY 1830 ZNU1 STROBE

12:19:07

15:00:00

Exec 123−00.12VWAP 123−00.05

Strike 122−28¾

Aggressive FillsPassive FillsCumulative execLimit ordersMarket priceCumulative VWAP

Exec = 123−00.12 Cost to VWAP = 0.14 tick = $2.24 per lot

12:00 12:20 12:40 13:00 13:20 13:40 14:00 14:20 14:40 15:00 15:20

−200

−150

−100

−50

0

50

100

−10%

−5%

5%

−0.8

−0.6

−0.4

−0.2

0.2

0.4

Ahea

dBe

hind

Lead

/lag

rela

tive

to re

alis

ed V

WAP

Slip

page

to V

WAP

(tic

ks)

12:19:07

15:00:00

1.2% mkt1611 passv 219 aggr

Take

Post

Aggressive FillsPassive FillsAhead/behind boundariesFilled quantityWorking quantity

Slippage



TWAP benchmark

Time-Weighted Average Priceignoring trade volume

Positives (relative to VWAP):easy to achieveeasy to measure where volume info is unreliable

Negatives (relative to VWAP):not related to market activity

13

Minimize slippage to TWAP benchmarkby trading exactly along straight-line profile


TWAP Example (1)

14

06:00 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00

0

200

400

600

800

1000

1200

20k

40k

60k

80k

100k

120k

140k

150k

SELL 1233 ZTU1 STROBE


Exec

uted

and

wor

king

qua

ntity

07:00:00

15:00:00

1 @ 109−24¾1 @ 109−25 1 @ 109−25

1 @ 109−24¾4 @ 109−24¾

6 @ 109−24¾1 @ 109−24¼17 @ 109−24+

1 @ 109−24¼3 @ 109−24+

1 @ 109−24 1 @ 109−24¼

1 @ 109−24+1 @ 109−24¾

2 @ 109−25 6 @ 109−24¾

1 @ 109−24+6 @ 109−24+

2 @ 109−24+3 @ 109−24¾

1 @ 109−24¾1 @ 109−24+8 @ 109−24¼

1 @ 109−24¼1 @ 109−24¼

6 @ 109−24¼3 @ 109−24¼

1 @ 109−24 17 @ 109−24¼

2 @ 109−24+13 @ 109−24¼

1 @ 109−24+22 @ 109−24+

2 @ 109−24+10 @ 109−24+

17 @ 109−24+1 @ 109−24+

1 @ 109−24¾1 @ 109−24¾

1 @ 109−24¾1 @ 109−24¼

5 @ 109−24¼1 @ 109−24+

8 @ 109−24+1 @ 109−24+

1 @ 109−24¼1 @ 109−24¼

0.82% mkt

849 passv 384 aggr

Aggressive FillsPassive FillsAhead/behind boundariesBase profileFilled quantityWorking quantity

Market volume

Executed quantitytracks straight line

Ignore actual market volume



TWAP example (2)

15

Cumulative fill quantity consistently within ±1% of true TWAP

109−24

109−24¼

109−24+

109−24¾

109−25

109−25¼

109−25+

06:00 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00


16:01

SELL 1233 TUU1 STROBE

07:00:00

15:00:00

Exec 109−24.49TWAP 109−24.51

Strike 109−24.62

Aggressive FillsPassive FillsCumulative execLimit ordersMarket priceCumulative TWAP

Exec = 109−24.49 Cost to TWAP = 0.07 tick = $1.08 per lot

06:00 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00

−30

−20

−10

0

10

20

−2%

−1%

1%

2%

−0.6

−0.4

−0.2

0.2

0.4

Ahea

dBe

hind

Lead

/lag

rela

tive

to T

WAP

Slip

page

to T

WAP

(tic

ks)

07:00:00

15:00:00

16:01

0.82% mkt849 passv 384 aggr

Take

Post

Aggressive FillsPassive FillsAhead/behind boundariesFilled quantityWorking quantity

Slippage

Price slippage within0.1 bid-ask spread



Other benchmarks

Daily close price, or settlementwithout impacting it too much

Option hedgingachieve delta-neutral position, depending on price(work with Tianhui Li, Princeton)

Etc.

16

Benchmark must be agreed in advanceExecution is tailored to benchmarkResults must be evaluated against same benchmark


Execution is uncertain

Market price moves during executionLimit orders may or may not get filledLiquidity appears and disappearsVolume profiles are unpredictable

Result of execution is a “random variable”Execution strategy tailors its properties

e.g. mean and variance

17

Perold's "cost of not trading"


Fundamental Balance of Execution

18

Cost Risk(mean of executed price relative to benchmark)

(variance of executed price relative to benchmark)

Alpha (forecastable price change)Passive fill probabilityMarket impact

Price volatility (most important)

vs


Example #1: Arrival Price

Minimize risk: execute as quickly as possibleButToo fast = lower chance of passive fills when to cross spread to reduce risk?Too fast = market impact pushes price how much is it worth it to push price?Alpha signal says price will get worse how much should you accelerate? increase impactAlpha signal says price will improve how much slower should you trade? increase risk

19


Example #2: TWAP

Ideal profile: evenly spaced in time

Should you cross spread whenever neededor wait for passive fills even if behind?

Alpha signal: how much should you deviate from schedule to exploit signal? (Signal may be very uncertain)

20


Example #3: VWAP

Track volume forecast, fixed or dynamic

Should you worry less about deviating from schedule, since schedule itself is uncertain?

21


2. Price dynamics and market impact

1. Mathematical model for trading and impact

2. Definition of objective function

3. Mathematical optimization

22


Trading trajectory

X = target number of shares to buyT = time limitx(t) = shares remaining at time t x(0) = X, x(T) = 0v(t) = -dx/dt = rate of buyingx(t), v(t) can depend on all information to time t or be fixed in advance (adaptive / non-adaptive)Detailed orders are placed by micro-algorithm

23

t

x(t)

X

00 T


Market model (Almgren/Chriss ‘00)

= stock price

= volatility

24

�(t)

S(t)dS(t) = �(t)dB(t) + g

�v(t)

�dt

g(v) = permanent market impact

Arbitrage exists unless g(v) is linearg(v) = �v

Integrates out so neglect permanent impact


S(t) = S(t) + h�v(t)

�

Temporary market impact

25

Market parameters σ(t) and η(t)- constant, predictable, or random processes- observable in real time

S(t) = realized trade price

Linear case: h(v) = �(t)v


Temporary market impact

26

Time t

Price S(t)

Realised price S(t)

Market price S(t)Pre-tradeprice S(0)

0 T

Other effects:- permanent impact- nonlinear impact- time decay- short-term drift- etc


Cost of trading

27

=� T

0S(t)v(t)dt

= X S0 +� T

0⇥(s)x(s)dB(s) +

� T

0�(t)v(t)2 dt

“Capture”= Dollars

realized onselling X shares

C =� T

0⇥(s)x(s)dB(s) +

� T

0�(t)v(t)2 dt

Random because of price changeand trade trajectory decisions

Cost = capture - initial value X S0


3. Mean variance optimization

Minimize both E = expectation of C V = variance of C

28

minx(s):t�s�T

E(C) + � Var(C)At each time t

E

V

VWAP

Fast

Variance of cost

Expectedcost

Efficient frontier

Risk parameter λ set by client


Solution

29

C = ⇥� T

0x(t)dB(t) + �

� T

0v(t)2 dt

V = ⇥ 2� T

0x(t)2 dt, E = �

� T

0v(t)2 dt

(for a precomputed fixed trajectory, which is optimal solution)

Calculus of variations with quadratic objective: Exponential solution

x(s) = x(t) sinh��(T � s)

⇥

sinh��(T � t)

⇥

v(s) = � x(t) cosh��(T � s)

⇥

sinh��(T � t)

⇥⇥2 = ⇤⌅ 2

�

Time scale


Arrival price solution

30

Time

Shares remaining to execute

x(t)

Orderentry time

Imposedend time

High urgency(immediate)

E large, V small

Low urgency (TWAP)

E small, V large

E

V

VWAP

Fast

Variance of cost

Expectedcost

Efficient frontier

⇥2 = ⇤⌅ 2

�

Time constant

Solution trajectorydoes not change as

price motions are observedduring execution


Extensions

Portfolios (Almgren/Chriss 2000)

Nonlinear market impact (Almgren 2003)

Adaptive trajectories (Lorenz/Almgren 2011)

Stochastic liquidity and volatility (Almgren 2012)

Bayesian update price drift (Almgren/Lorenz 2006)

Option hedging (Almgren/Li 2016)

Many others ...

31


Critique of arrival price solutions

Well calibrated market impact modelKnown & constant risk aversion parameter λReasons to trade slowly: reduce market impact

slow steady trading leaks informationprice drift can lead to trade fasteropportunistic trading

Reasons to trade fast: reduce volatility riskevent riskreduce number of open orders

Market impact is more complicated than thispersistence in time (integral kernels)bursty trading can be cheaper than steady

32


4. Option hedging

33

Black-Scholes with market impactEffect on public markets

Option Hedging with Smooth Market Impact

Robert Almgren*

Quantitative Brokers, New York, NY, [email protected]

Tianhui Michael Li

The Data Incubator, New York, NY, USA

Accepted 15 April 2016Published

We consider intraday hedging of an option position, for a large trader who experi-ences temporary and permanent market impact. We formulate the general modelincluding overnight risk, and solve explicitly in two cases which we believe are rep-resentative. The ¯rst case is an option with approximately constant gamma: theoptimal hedge trades smoothly towards the classical Black–Scholes delta, withtrading intensity proportional to instantaneous mishedge and inversely proportionalto illiquidity. The second case is an arbitrary non-linear option structure but with nopermanent impact: the optimal hedge trades toward a value o®set from the Black–Scholes delta. We estimate the e®ects produced on the public markets if a largecollection of traders all hedge similar positions. We construct a stable hedge strategywith discrete time steps.

Keywords: Option hedging; market impact; optimal trading.

1. Introduction

Dynamic hedging of an option position is one of the most studied problems inquantitative ¯nance. But when the position size is large, the optimal hedgestrategy must take account of the transaction costs that will be incurred byfollowing the Black–Scholes solution. This large position may be the positionof a single large trader, or may be the aggregate position of a collection oftraders, for example the entire sell-side community, who all hold similarpositions, and all of whom hedge while their counterparties do not. In

*Corresponding author.

June 13, 2016 5:43:10pm WSPC/298-MML 1650002 ISSN: 2382-62662nd Reading

Market Microstructure and LiquidityVol. 2, No. 1 (2016) 1650002 (26 pages)°c World Scienti¯c Publishing CompanyDOI: 10.1142/S2382626616500027

1650002-1


Equity price swings on July 19 2012 (one day prior to options expiration)

34

...

http://www.bloomberg.com/news/2012-07-20/hourly-price-swings-whipsaw-investors-in-ibm-coke-mcdonald-s.html

Marko Kolanovic, global head of derivatives and quantitative strategy at JPMorgan & Chase Co

http://www.bloomberg.com/news/2012-07-20/hourly-price-swings-whipsaw-investors-in-ibm-coke-mcdonald-s.html


CA CHEUVREUX QUANTITATIVE RESEARCH

QUANT NOTE

28th of August, 2012

www.cheuvreux.com

What does the saw-tooth pattern on US markets on 19 July 2012 tell us about the price formation process The saw-tooth patterns observed on four US securities on 19 July provide us with an opportunity to comment on common beliefs regarding the market impact of large trades; its usual smoothness and amplitude, the subsequent “reversal” phase, and the generic nature of market impact models. This underscores the importance of taking into account the motivation behind a large trade in order to optimise it properly, as we already emphasised in Navigating Liquidity 6. We used different intraday analytics to work out what happened: pattern-matching techniques, market impact models, order flow imbalances and PnL computations of potential stat. arb intraday strategies. After looking at open interests of derivatives on these stocks, we conclude that repetitive automated hedging of large-exposure derivatives lay behind this behaviour. This is an opportunity to understand how a very crude trading algorithm can impact the price formation process ten times more than is usually the case.

FIGURE 1: SAWTOOTH PATTERNS ON COCA-COLA, MCDONALD'S, IBM AND APPLE ON 19 JULY 2012

92

92.5

93

93.5

09:3110:35

11:4012:45

13:5014:55

16:00

McDonald's

194

194.5

195

195.5

196

196.5

197

09:3110:35

11:4012:45

13:5014:55

16:00

IBM

608

610

612

614

616

09:3110:35

11:4012:45

13:5014:55

16:00

Apple

76.4

76.6

76.8

77

77.2

77.4

77.6

77.8

09:3110:35

11:4012:45

13:5014:55

16:00

Coca Cola

The market impact is one of the main factors of the price formation process

Strange patterns were observed on four large cap US stocks (Coca-Cola, McDonald's, IBM and Apple) on 19 July 2012. A few days later the “Knight Algorithm Went Crazy” issue eclipsed this strange phenomenon, by raising the usual concerns about the current market microstructure, the integrity of the price formation process, etc.

We have been hearing such complaints for years now (during the public hearing in MiFID Review in September 2010, after the SEC issued a concept release seeking comment on the structure of equity markets in January 2010). What has changed? Not much: we continue to observe glitches on US markets, and the MiFID review in Europe has been under discussion for months, with no clear agenda about elements as crucial as the tick size.

We look at the events of 19 July to learn more about the price formation process. Our intention here is not to merely say that the way a (reputed) fair price is formed, thanks to matching of supply and demand, is simple. The roots of its mechanism are not very difficult to understand. Its complexity stems from the way they combine.

Charles-Albert LEHALLE Matthieu LASNIER Global Head of Quantitative Research [email protected] (33) 1 41 89 80 16

Market microstructure specialist, Quantitative Research, Paris

Paul BESSON Hamza HARTI Weibing HUANG Nicolas JOSEPH Lionel MASSOULARD Index and Portfolio Research Specialist, Quantitative Research, Paris

Trading specialist, Quantitative Research, Paris

PhD student, Quantitative Research, Paris

Performance analysis specialist, Quantitative Research, London

Trading specialist, Quantitative Research, New-York

CA CHEUVREUX QUANTITATIVE RESEARCH

QUANT NOTE

28th of August, 2012

www.cheuvreux.com

What does the saw-tooth pattern on US markets on 19 July 2012 tell us about the price formation process The saw-tooth patterns observed on four US securities on 19 July provide us with an opportunity to comment on common beliefs regarding the market impact of large trades; its usual smoothness and amplitude, the subsequent “reversal” phase, and the generic nature of market impact models. This underscores the importance of taking into account the motivation behind a large trade in order to optimise it properly, as we already emphasised in Navigating Liquidity 6. We used different intraday analytics to work out what happened: pattern-matching techniques, market impact models, order flow imbalances and PnL computations of potential stat. arb intraday strategies. After looking at open interests of derivatives on these stocks, we conclude that repetitive automated hedging of large-exposure derivatives lay behind this behaviour. This is an opportunity to understand how a very crude trading algorithm can impact the price formation process ten times more than is usually the case.

FIGURE 1: SAWTOOTH PATTERNS ON COCA-COLA, MCDONALD'S, IBM AND APPLE ON 19 JULY 2012

92

92.5

93

93.5

09:3110:35

11:4012:45

13:5014:55

16:00

McDonald's

194

194.5

195

195.5

196

196.5

197

09:3110:35

11:4012:45

13:5014:55

16:00

IBM

608

610

612

614

616

09:3110:35

11:4012:45

13:5014:55

16:00

Apple

76.4

76.6

76.8

77

77.2

77.4

77.6

77.8

09:3110:35

11:4012:45

13:5014:55

16:00

Coca Cola

The market impact is one of the main factors of the price formation process

Strange patterns were observed on four large cap US stocks (Coca-Cola, McDonald's, IBM and Apple) on 19 July 2012. A few days later the “Knight Algorithm Went Crazy” issue eclipsed this strange phenomenon, by raising the usual concerns about the current market microstructure, the integrity of the price formation process, etc.

We have been hearing such complaints for years now (during the public hearing in MiFID Review in September 2010, after the SEC issued a concept release seeking comment on the structure of equity markets in January 2010). What has changed? Not much: we continue to observe glitches on US markets, and the MiFID review in Europe has been under discussion for months, with no clear agenda about elements as crucial as the tick size.

We look at the events of 19 July to learn more about the price formation process. Our intention here is not to merely say that the way a (reputed) fair price is formed, thanks to matching of supply and demand, is simple. The roots of its mechanism are not very difficult to understand. Its complexity stems from the way they combine.

Charles-Albert LEHALLE Matthieu LASNIER Global Head of Quantitative Research [email protected] (33) 1 41 89 80 16

Market microstructure specialist, Quantitative Research, Paris

Paul BESSON Hamza HARTI Weibing HUANG Nicolas JOSEPH Lionel MASSOULARD Index and Portfolio Research Specialist, Quantitative Research, Paris

Trading specialist, Quantitative Research, Paris

PhD student, Quantitative Research, Paris

Performance analysis specialist, Quantitative Research, London

Trading specialist, Quantitative Research, New-York


28 August 2012

CA CHEUVREUX QUANTITATIVE

RESEARCH Saw-tooth effect on the Us market

18 www.cheuvreux.com

Q Repetitive delta hedging seems to be the most plausible explanation

Options stucturers and market-makers need to delta hedge their positions in order to offset their market exposure. So-called "dynamic hedging" flows can create strong selling or buying pressure, which might impact market prices. These flows can reduce or increase stock volatility depending on the hedgers' positions. If market-makers are selling vanilla options to long-only managers, then dynamic hedging will tend to increase stock volatility. This phenomenon can be particularly strong when the options get close to maturity. Intuitively, this result can be understood by considering for example that a call option seller who delta hedges his position will have to buy stocks in order to deliver them to the option buyer. He will thus positively impact the underlying stock price. This behaviour cannot logically explain a price reversal. Conversely, when market makers buy vanilla options from long only managers (through call overwriting for example), then dynamic hedging tends to reduce stock volatility. This phenomenon is called "pinning" for short-term options. Nevertheless, when open interests and the gamma are very large, hedging flow impacts can be very significant.

FIGURE 18: SAW-TOOTH ON COCA-COLA AND 77.50 CALL

76.5

76.7

76.9

77.1

77.3

77.5

09:21 10:33 11:45 12:57 14:09 15:21

High Bound (77.5 $ Strike) 50% delta

Low Bound : 14% delta

Source: Crédit Agricole Cheuvreux Quantitative Research

19 July 2012: the derivative hedging scenario

Consider Coca-Cola on 19 July (FIGURE 18 and FIGURE 19) the VWAP was 77.24. Large open interests (20 556) on the 77.5 July Call and on the 75 July Put (13 477) were observed. For a 1% price change on Coca-Cola, delta hedging of the total number of open interests would imply a trading of 1.1 million shares, 12% of the volume on 19 July. This particularly high number was due to three elements: the size of the open positions, the fact that the stock trades next to the option strike, and the 1-day remaining maturity. On that particular day the price fluctuated 10 times, nearly every thirty minutes, with a 0.8% average move. Also, the traded volume amounted to 9.0 million shares. This means that the total hedging of those options would have represented nearly 100% of the total daily volume. This shows that delta-hedging can constitute a very large share of Coca-Cola's trading volume. It is important to note that the total open interest on listed options does not necessarily imply that delta hedging will occur on this total underlying notional. The net delta hedging notional will correspond to the net volatility positions of the long only players. Usually long only funds will sell volatility through call overwriting, they could also buy volatility through calls or puts to take advantage of their directional views. In qualitative terms, these counter-cyclical hedging flows can create a strong impact on prices. This situation has also been observed for McDonald's, IBM and Apple where sizes of open interests in listed options are large.

28 August 2012




We present a more detailed explanation in the table below (FIGURE 21), which illustrates the delta-hedging pressure on the four stocks at each peak of the saw-tooth pattern. We assume that for the two strikes closest to the money Calls and Puts of July 2012, the share of the delta hedged options stands at 100% of the open interest. This gives us theoretical maximum delta hedging flows. Thanks to the volume analysis carried out initially that showed a 20-30% volume participation, we estimate that the real fraction of the hedged options of the four stocks ranged from 30% to 50%. For Apple, dynamic hedging seems to constitute a smaller portion of the volumes. This is consistent with the Apple's less-pronounced saw-tooth pattern.

FIGURE 21: PERIODIC DELTA HEDGING PROCESS ON 4 STOCKS ON JULY THE 19

100% Option hedging Coke Mac Donald's IBM AppleSawtooth Timetable Price chg Gamma Price chg Gamma Price chg Gamma Price chg Gamma

From To (%) (Shares to trade) (%) (Shares to trade) (%) (Shares to trade) (%) (Shares to trade)

9:30 AM 10:00 AM -1.2% 1 277 419 -0.4% 172 786 -0.8% 535 032 -0.7% 653 595

10:00 AM 10:30 AM 0.8% -836 500 0.8% -303 688 1.0% -718 870 0.7% -657 895

10:30 AM 11:00 AM -0.8% 853 816 -0.6% 258 342 -1.0% 711 563 -0.3% 326 797

11:00 AM 11:30 AM 0.9% -1 003 911 0.8% -303 359 0.6% -449 294 0.3% -327 869

11:30 AM 12:00 PM -0.8% 852 713 -0.5% 215 054 -0.6% 446 429 -0.2% 163 399

12:00 PM 12:30 PM 0.9% -1 002 604 0.9% -345 946 0.9% -629 012 0.5% -490 998

12:30 PM 1:00 PM -0.8% 851 613 -0.8% 300 107 -0.8% 534 351 -0.3% 285 016

1:00 PM 1:30 PM 0.7% -715 215 0.6% -259 179 0.8% -538 462 0.4% -449 163

1:30 PM 2:00 PM -0.4% 426 357 -0.8% 300 429 -0.8% 534 351 -0.4% 406 504

2:00 PM 2:30 PM 0.5% -570 687 0.5% -216 216 0.8% -538 462 0.4% -408 163

2:30 PM 3:00 PM -0.4% 425 806 -0.4% 172 043 -0.8% 534 351 -0.4% 365 854

3:00 PM 3:30 PM 0.4% -427 461 0.3% -129 590 0.5% -358 974 0.4% -367 197

3:30 PM 4:00 PM -0.1% 141 935 -0.2% 86 114 -0.5% 357 143 -0.2% 162 602

Total (Absolute nb of shares) 9 386 039 3 062 853 6 886 293 5 065 052

Hedging / dai ly volume (%) 104% 41% 66% 32%Source: Crédit Agricole Cheuvreux Quantitative Research

When the gamma is very large and the hedging process very crude, stock pinning can even forge a saw-tooth effect On a very short-term maturity, large delta hedging concentrated around specific strikes can impact prices. When some delta hedgers represent a large share of the open interest, this phenomenon can be very significant. This is what academics call "pinning", it should drive prices back to their strikes. Nevertheless, when delta hedging is combined with a simplistic execution process "pinning" can degenerate into "saw-tooth" effects. Simplistic hedging of large gamma options is a plausible explanation for the "saw-tooth" trading pattern. This explanation is consistent with the main features of this phenomenon: timing, aggressiveness, impact, predictability and information leakage which is what characterises those "saw-tooth" patterns. Fortunately, large option positions are most often managed dynamically in a continuous way, and discrete archaic hedging processes have almost disappeared in modern-day markets.

28 August 2012




Q Repetitive delta hedging seems to be the most plausible explanation

Options stucturers and market-makers need to delta hedge their positions in order to offset their market exposure. So-called "dynamic hedging" flows can create strong selling or buying pressure, which might impact market prices. These flows can reduce or increase stock volatility depending on the hedgers' positions. If market-makers are selling vanilla options to long-only managers, then dynamic hedging will tend to increase stock volatility. This phenomenon can be particularly strong when the options get close to maturity. Intuitively, this result can be understood by considering for example that a call option seller who delta hedges his position will have to buy stocks in order to deliver them to the option buyer. He will thus positively impact the underlying stock price. This behaviour cannot logically explain a price reversal. Conversely, when market makers buy vanilla options from long only managers (through call overwriting for example), then dynamic hedging tends to reduce stock volatility. This phenomenon is called "pinning" for short-term options. Nevertheless, when open interests and the gamma are very large, hedging flow impacts can be very significant.

FIGURE 18: SAW-TOOTH ON COCA-COLA AND 77.50 CALL

76.5

76.7

76.9

77.1

77.3

77.5

09:21 10:33 11:45 12:57 14:09 15:21

High Bound (77.5 $ Strike) 50% delta

Low Bound : 14% delta


19 July 2012: the derivative hedging scenario

Consider Coca-Cola on 19 July (FIGURE 18 and FIGURE 19) the VWAP was 77.24. Large open interests (20 556) on the 77.5 July Call and on the 75 July Put (13 477) were observed. For a 1% price change on Coca-Cola, delta hedging of the total number of open interests would imply a trading of 1.1 million shares, 12% of the volume on 19 July. This particularly high number was due to three elements: the size of the open positions, the fact that the stock trades next to the option strike, and the 1-day remaining maturity. On that particular day the price fluctuated 10 times, nearly every thirty minutes, with a 0.8% average move. Also, the traded volume amounted to 9.0 million shares. This means that the total hedging of those options would have represented nearly 100% of the total daily volume. This shows that delta-hedging can constitute a very large share of Coca-Cola's trading volume. It is important to note that the total open interest on listed options does not necessarily imply that delta hedging will occur on this total underlying notional. The net delta hedging notional will correspond to the net volatility positions of the long only players. Usually long only funds will sell volatility through call overwriting, they could also buy volatility through calls or puts to take advantage of their directional views. In qualitative terms, these counter-cyclical hedging flows can create a strong impact on prices. This situation has also been observed for McDonald's, IBM and Apple where sizes of open interests in listed options are large.

Chevreux: sawtooths caused by options hedging28 August 2012




FIGURE 19: LONG VOLATILITY DELTA HEDGING

Coke: July Call 77.5 prices

0.00

0.25

0.50

76.00

76.25

76.50

76.75

77.00

77.25

77.50

77.75

78.00

Stock

Spo t price : 76.75Optio n D elta : +14%

P o sit io n hedge: -14%

Spo t price :77.5 Optio n D elta : +50%


F ro m 76.75 to 77.5, hedgerssell -36% fro m -14% to -50%

Lo w bo und 76.75 $

Lo w bo und 77.50 $


The reverting dynamic explained by a periodic operational process

While many farfetched explanations can be advanced for the "saw-tooth effect" on Coca-Cola, McDonald's, Apple and IBM. Adhering to the Occam's Razor principle, we favour this simple explanation: the sudden use of a rigid repetitive operational process for delta hedging these four stocks. Suppose a trader has to follow an existing option book without having full real-time access to his monitoring and execution tool, or that he cannot dedicate all his time to this task and therefore chooses to monitor this book every thirty minutes. Repetitive Delta hedging process

x Step 1) t0 = (initial time)

o Price the options, compute the delta on the new spot price

o Evaluate the total amount of stocks needed for the delta hedge

o Send the orders to the market.

x Step 2): TWAP execution over 30 min

x Back to Step 1) t1 = (initial time+30min) For a very large open interest position encompassing a large gamma, a significant move in the stock price will have disastrous effects for a basic rudimental hedger such as the one described above.

28 August 2012




FIGURE 19: LONG VOLATILITY DELTA HEDGING

Coke: July Call 77.5 prices

0.00

0.25

0.50

76.00

76.25

76.50

76.75

77.00

77.25

77.50

77.75

78.00

Stock

Spo t price : 76.75Optio n D elta : +14%


Spo t price :77.5 Optio n D elta : +50%


F ro m 76.75 to 77.5, hedgerssell -36% fro m -14% to -50%

Lo w bo und 76.75 $

Lo w bo und 77.50 $


The reverting dynamic explained by a periodic operational process

While many farfetched explanations can be advanced for the "saw-tooth effect" on Coca-Cola, McDonald's, Apple and IBM. Adhering to the Occam's Razor principle, we favour this simple explanation: the sudden use of a rigid repetitive operational process for delta hedging these four stocks. Suppose a trader has to follow an existing option book without having full real-time access to his monitoring and execution tool, or that he cannot dedicate all his time to this task and therefore chooses to monitor this book every thirty minutes. Repetitive Delta hedging process

x Step 1) t0 = (initial time)

o Price the options, compute the delta on the new spot price

o Evaluate the total amount of stocks needed for the delta hedge

o Send the orders to the market.

x Step 2): TWAP execution over 30 min

x Back to Step 1) t1 = (initial time+30min) For a very large open interest position encompassing a large gamma, a significant move in the stock price will have disastrous effects for a basic rudimental hedger such as the one described above.

(we show how to do this better)


Our solutions with proportional cost

37

Ideal Black-Scholes hedge

Actual hedge holding

pursuit

Gârleanu & Pedersen:investment with proportional cost

✓t = � h�(T � t)

�·�Xt � target

�

Temporary impact: hedge strategyPermanent impact: effect on underlying


stationarymodifiedvolatility

Γ<0: hedger is short the option1+νΓ<1: overreaction, increased volatilityΓ>0: hedger is long the option

1+νΓ>1: underreaction, reduced volatility

Zt = �⌫(Xt �X0) + (Pt � P0)dZt = � dWt

What happens to price process?

same as in Pt

Pt = P0 +1

1+ ⌫��⌫Yt + Zt

�

= P0 +�

1+ ⌫�

Wt + ⌫�

Z t

0e�(1+⌫�)(t�s)dWs

!

/ 1/p


“Signature plot”

39

= (1+ ⌫�)

Δt

Volatility measuredon time interval

Δt�

�1+ ⌫� (� > 0)

�1+ ⌫� (� < 0)

1/

unconditional (F0)

1s � tE

�Ps � Pt

�2 =

=✓ �

1+ ⌫�

◆2

1+ (2+ ⌫�)⌫� 1� e�(s�t)(s � t)

!


Nondimensional parameter

40

1 + ⌫�

⌫ = price change due to market impact

shares executed

� = shares needed to adjust hedge

market price change

Problems unless |νΓ| ≲ 1


Conclusions

Many interesting math modelscalculus of variationsdynamic programming

Models are very approximatemarkets are messy

Can give insight nonethelesstrade fast vs trade slowpermanent and temporary impact

41

Market Microstructure and Algorithmic Tradingcfrei/PIMS/Almgren4.pdfPIMS Summer School 2016...

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