Olivier Milla - Market Liquidity, Measures, Models and Best Execution

ESLSCA

Market Liquidity: Measures, Models

and Order Execution

by

Olivier Milla

A paper submitted in partial fulfillment to

obtain the Master 2 - Market Finance Diploma

in the

Pole Trading

December 2010

http://www.eslsca.fr)

Faculty Web Site URL Here (include http://)

ESLSCA

Abstract

Pole Trading

by Olivier Milla

This paper attempts to present several topics surrounding the concept of liquidity on

the financial markets. Liquidity is defined as the ability in time or price to trade assets.

Macro-economical liquidity and market liquidity are tied through money supply and

inflation. As markets evolved over the past centuries, banks took a preeminent place in

the creation of money, and more recently through leverage in investment banking. As

such, the 2007 subprime crisis can be seen as a liquidity shortage that hit the financial

markets, starting with credit and real estate.

At a smaller scale, liquidity on a single market is closely tied to the balance sheet of

the actors and the timing of the orders. This conclusion can also be extended to the

micro-structure of the financial markets. The May 6, 2010 ”flash crash” exhibits the

importance of timing and behavior to create liquidity on a financial market.

Market participants usually read liquidity through first or second order measures such

as volumes, open interests, bid/ask spreads, etc. While this lecture is the same for

everyone, each participant does not benefit from the same access to liquidity due to

legal, operational and technological costs.

Models exist to integrate these costs along with the costs of slippage into mark to

market models. The Almgren-Chriss model presented in this paper is one of them. We

also expose how it can be used to optimize the execution of orders to reduce slippage.

http://www.eslsca.fr)

Faculty Web Site URL Here (include http://)

Acknowledgements

I would like to thank my project advisor, Naji Freiha.

Special thanks go to Adrien Mouillon and David Arnaud for their conversation that lead

to parts of this report.

ii

Contents

Abstract i

Acknowledgements ii

List of Figures v

Symbols vi

1 Markets And Flows 3

1.1 What Is A Market, Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 A little Bit Of History . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Credit Intermediation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 The Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Controlling Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 Controlling The Money Supply . . . . . . . . . . . . . . . . . . . . 10

1.4 Evaluating Market Liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Definition Of The Measure . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Subjectivity Of The Measure . . . . . . . . . . . . . . . . . . . . . 12

1.4.3 Non-Market Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Market Liquidity Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.1 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.2 Market Width and Market Depth . . . . . . . . . . . . . . . . . . . 16

1.5.3 Market Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5.4 Late Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Three Liquidity Crisis 20

2.1 The 2008 Fall of the Shadow Banking System . . . . . . . . . . . . . . . . 20

2.1.1 Chronicle of the Crisis . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 The November 2000 Turkish Overnight Liquidity Crisis . . . . . . . . . . 25


2.2.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 The May 6, 2010 ”Flash-Crash” . . . . . . . . . . . . . . . . . . . . . . . . 27


2.3.2 Cross-market propagation . . . . . . . . . . . . . . . . . . . . . . . 31

iii

Contents iv

2.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Modeling Market Liquidity 35

3.1 The Almgren-Chriss Liquidity Asset Price Model . . . . . . . . . . . . . . 35

3.1.1 Price Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 The Definition Of A Trading Strategy . . . . . . . . . . . . . . . . 37

3.1.3 Cost Of Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Order Execution 40

4.1 Impact Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Linear Impact Functions . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.2 Exponential Impact Functions . . . . . . . . . . . . . . . . . . . . 43

4.1.3 Empirical Impact Functions . . . . . . . . . . . . . . . . . . . . . . 46

4.2 The Efficient Frontier Of Optimal Execution . . . . . . . . . . . . . . . . 47

5 Conclusion 50

Bibliography 51

List of Figures

1.1 Credit Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Maturity Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Liquidity Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Year on Year CPI vs Transaction Volume of the SPX Index . . . . . . . . 9

1.5 U.K. money supply and a combined capital market price index, 1950-1972. 10

1.6 Real Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 OTC Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1 Shadow Bank Liabilities vs. Traditional Bank Liabilities . . . . . . . . . . 21

2.2 The 2000 Turkish Liquidity Crisis . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 May 6,2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 E-mini S&P 500 Futures Volume and Price . . . . . . . . . . . . . . . . . 30

2.5 E-mini S&P 500 Futures Volume Market Depth . . . . . . . . . . . . . . . 30

2.6 Market Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7 E-mini SPY and Aggregated S&P 500 Stocks Buy-Side Market Depth . . 33

3.1 Order Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Random Walk with Order Execution . . . . . . . . . . . . . . . . . . . . . 37

4.1 Minimum Variance for Linear Impact Functions . . . . . . . . . . . . . . . 43

4.2 Minimum Expected Value for Exponential Impact Functions . . . . . . . . 45

4.3 Minimum Variance for Exponential Impact Functions . . . . . . . . . . . 45

4.4 Minimum Expected Value for Linear Impact Functions . . . . . . . . . . . 48

4.5 Minimum Variance for Linear Impact Functions . . . . . . . . . . . . . . . 49

4.6 Minimum Utility Function for Linear Impact Functions . . . . . . . . . . 49

v

Symbols

X Initial number of securities

T Time taken to sell the X securities

tk Discrete time k (0 . . . N)

Sk Security’s screen price at time k

Sk Security’s paid price at time k

xk Number of securities held at time k

nk Number of securities sold between time k and k − 1

g(ν) Permanent impact function

h(ν) Temporary impact function

U Utility function

C Cost of trading

σ Volatility of returns

µ Drift

τ Time between two orders

ν Average rate of trading nkτ

β Fixed cost of execution

γ Permanent impact function parameter

λ Temporary impact function parameter

λ equals λ− 12γτ

α Utility function parameter

vi

Introduction

The defiance witnessed between operators of the money and repo markets during the

Lehman Brothers crisis froze entire spans of the financial markets. When bid/ask spread

widened or quotes simply vanished from the screens (as it was the case for the equity repo

market for instance), assets lost their value by lack of consideration. Mark-to-market

becoming impossible, assets were marked as null in the banks’ balance sheets until

eventually the opportunity to unwind these positions shows up, which is still not the case

for several asset classes (think credit derivatives). Even now, two years later, numerous

banks see their quarterly results improve as some of their assets under management

regain value with the slow revival of their respective markets.

Along with the mechanical devaluation of assets caused by the rise of risk and uncer-

tainties, the crisis shed light on a component of the financial markets: liquidity.

The trouble is, what is liquidity? How can it be measured? How much does it contribute

to an asset valuation? How does it impact the mark-to-market of assets and the execution

of orders? Can liquidity be hedged?

This paper aims at reviewing some of the prevailing concepts to date that can help

clarifying the perception of liquidity both from practical and theoretical standpoints.

The paper will develop as follows:

Chapter 1 will review market liquidity at three different scales: the global macro-

economy, a local market and a market’s micro-structure. Three crisis of the corre-

sponding magnitude will picture the mechanics leading to a shortage of liquidity.

At the macro-economical scale, we will start by exposing the definition of a financial

market and the historical conditions foregoing its inception, underlying its role as

a macroeconomic engine for cash allocation. To remain close to our main topic, we

1

Chapter 1. Introduction 2

will simply aim at revealing the vital strings and stakeholders that link a markets’

life to the economy and which should be closely followed to assess its liquidity.

The 2008 Fall of the Shadow Banking System will serve as an illustration.

We will then expose liquidity at the scale a of a single, national market by studying

the Turkish overnight market and its 2000 crisis, showing how information and

behavior shape the events on a finite market.

Finally, at the scale of a market’s micro-structure, we will reveal how participants

add or remove liquidity, underlying the importance of technology and algorithms

in that process. The May 6, 2010 ”Flash-Crash” will serve as an illustration.

Chapter 2 will present how market liquidity is usually read by traders and risk man-

agers. We will list both generic and market-dependent measures.

Chapter 3 will expose a simple model developed by Robert Almgren and Neil Chriss

in 2000 to add execution pricing and fixed execution costs to mark-to-market.

Chapter 4 will briefly use the model of Chapter 3 to show it is possible to better the

execution of large orders on a market.

Chapter 5 will conclude the study.

Chapter 1

Markets And Flows

1.1 What Is A Market, Anyway?

A market is a virtual place. Two people meeting and exchanging an object or an

information spawn a one-time market that lives for the time of the transaction. Once

they are done, the market vanishes until two people meet and eventually trade the same

underlying again.

Over the last centuries, as people started to steadily trade a variety of financial contracts

and as exchange methods became more standardized, a certain idea of continuity and

tradition got attached to the activity of trading, spawning the idea of the financial

markets as a constant, unmovable activity.

For instance, the history of the Amsterdam Stock Exchange is telling.

1.1.1 A little Bit Of History

The stock market was born in Holland in 1602 when the Dutch East India Company

issued the first shares to finance its trading with Asia (dividends were delivered as

spices). As investors (both institutional and retail, already at that time) wanted to keep

informed about the company with other investors, they took the habit of meeting daily

at a certain spot in Amsterdam which soon became the Amsterdam Bourse, with an

official building.

3

Chapter 2. Markets And Flows 4

The concept grew all over Europe and national exchanges were rapidly created in all

the major capital cities. The number of quoted securities grew and many corporations

started to fund their activities through the capital markets. As investments throttled

and crisis occurred, contracts hedging against certain risks appeared, along with the

derivatives contracts (Futures, Options, etc.) that were introduced as contracts linking

their price to the evolution of one or more other contracts or market parameters. In 1978,

the European Option Exchange (EOE) was founded in Amsterdam, bringing complete

legitimacy and transparency to these new products.

Another milestone worth reporting is the addition of a stock market index to the place

in 1983 (the AEX) 1. This abstraction layer allowed for a direct appreciation of the

market-as-a-whole and to more developments regarding the study of the stock market

and the economy (and also to the development of technical analysis, behavioral analysis,

etc.)

In 2000 the Amsterdam Stock Exchange merged with its equivalents in Brussels and

Paris to form the Euronext exchange2.

Beyond the technical and financial creativity that sustained this growth, financial mar-

kets slowly entered the daily life of the crowd. In America, daily stock prices were first

published in the newspapers in 1795 by the New York Price Current which aimed at

listing daily goods’ prices in the city. In 1889, the Wall Street Journal began printing

as the first newspaper dedicated to the stock market and related affairs3.

In 1915, pension (to cover old age, death, disability, etc.) became part of the social

claims. Private - and later public - funds spawned, as well as insurances. Numerous of

these institutions used the financial markets to finance part or all of their returns.

All companies irreversibly joined the action with the institution of mark-to-market as

an accounting standard. In the late 1980s this accounting principle entered the U.S.

Generally Accepted Accounting Principles (G.A.A.P.). From this moment on, both

companies, banks, institutional and retail investors’ balance sheets were tied to market

values, with financial risks in their balance sheets.

1The Dow Jones Industrial Average - the first stock index - was created by Charles Dow in 1884.2For further details about the inception of exchanges and financial markets, [See 1].3Source: History Of Business Journalism, http://www.bizjournalismhistory.org/.


However how important financial markets became, they serve in essence the same pur-

pose as they were in their first days: put money to work. In practical terms, that means

matching borrowers and lenders through various contracts.

Trading was also institutionalized by another important wheel that needs to be presented

to understand the flows we see on the financial markets : banks and the most important

of them, the central banks.

1.1.2 Banks

Banks were introduced very early in the system. They first acted as custodians (safe-

keepers of the securities), then as consultants (giving advices to clients) and represen-

tatives (taking actions in name of other parties) and finally as intermediaries (allowing

any financial institution to reach any fund through the network of banks). This late step

completed the integration of finance in economies as it linked, compared and matched

all the reachable sources of cash and risk.

As noted by Zoltan Pozsar [2]: ”Relative to direct lending (that is lenders lending directly

to borrowers), credit intermediation provides savers information and risk economies

of scale by reducing the costs involved in screening and monitoring borrowers and by

facilitating investments in a more diverse loan portfolio” and therefore banks rapidly

became the core users of the financial markets.

This recycling of money by banks through the financial markets is called Credit Inter-

mediation.

1.2 Credit Intermediation

For a bank, credit intermediation has three main dimensions that represent the different

natures of investments and associated risks it can carry:

Credit Transformation That is when a bank enhances the credit quality of the debt

it issues on the financial markets. This is done via the use of equities as collaterals

and the definition of priority of claims4. Credit Intermediation creates a difference

4Priority of claims: The order following which claims against the bank will be treated if it becomesinsolvent.


in terms of credit quality between the cash flows in and out of the bank, the

remaining being the exposition which could result in a profit or a loss.

A good example of credit transformation is the issue of any note by a depository

bank. The credit quality of the note on the market depends on its seniority and

is backed by deposits that can be of far less credit quality. Another example are

Collateralised Debt Obligation (CDO)5 which are products available on the credit

market as having very good quality grades while they are backup by a pool of

much riskier assets.

Credit transformation throttled in the recent years and was at the root of the 2007

”subprime” crisis. Section 2.1 will exhibit some of the mechanisms that can lead

to a liquidity crisis on the credit market.

Figure 1.1: Credit Transformation

Maturity Transformation That is when a bank uses short-term deposits to fund long-

term loans. That operation creates a difference in duration between the cash flows

in and out of the bank, leaving it at risk. A detailed example of a liquidity crisis

involving maturity transformation constitutes the section 2.2 (The 2000 Turkish

overnight market crisis).

Figure 1.2: Maturity Transformation

5”CDOs are a type of structured asset-backed security whose value and payments are derived from aportfolio of fixed-income underlying assets.[...] The first CDO was issued in 1987.”(Source: Wikipedia)


Liquidity Transformation That is when a bank uses liquid deposits (say cash) to

fund illiquid assets (say any security, compared to cash). That operation creates a

difference in liquidity between the bank’s cash flows in and out. The bank, again,

carries the difference in the hope to gain from it. Section 2.3 will exhibit a subtle

liquidity crisis at the micro-structural level of a market (between S&P Futures and

the underlying Stocks/ETFs).

Figure 1.3: Liquidity Transformation

These three dimensions are the main risks that one faces when taking positions on a

financial market. Respectively Credit Risk, Duration Risk (also known as Curve Risk)

and Liquidity Risk. It is obvious that any loan mingles these three risks. For instance

the funding of a the construction of a private company headquarters (a 20 years loan on

a BBB-rated corporation for the construction of an illiquid commercial asset) by issuing

1-year rolling B-rated commercial papers.

One bank supervises and regulates the relationships between lenders and borrowers: the

central bank6. The central bank’s mandate is - since its inception in 1913 - to set the

most important economic parameters to allow for a suitable economic expansion. As

such it provides stability by acting as a lender of last resort7 and as a controller of the

inflation levels. It also stimulates the economy when needed by regulating the money

flowing in the system. All these operations are referred to as the monetary policy8.

6When talking about the central bank, we are talking about the Federal Reserve System - The ”Fed”- everywhere in this paper, except where expressively notified.

7A lender of last resort is a financial institution ready to lend money when nobody else will.8For more details, see [3].


1.3 The Monetary Policy

The Fed’s monetary policy is to amount the cost of money and credit in the U.S. economy

to achieve growth and price stability.

To achieve growth, money and credit must grow at a pace that allows economic activity

to expand at a sustainable rate without excessive price increases. If money and credit

grow too slowly, the cost of money (the interest rates) is too high and people and business

will not be able to afford it, slowing the pace of consumption. Conversely, if money and

credit increase too rapidly, inflation will rise (interest rates will drop in real terms).

This is most crucial as this sets the return expectations for the investors. In other

words, it sets how repealing or attractive market returns are compared to off-markets

placements (direct consumption, real estate, etc.) and therefore dictates the liquidity

(in the sense of ”usage”) of the financial markets.

1.3.1 Controlling Inflation

Inflation is ”a persistent increase in the level of consumer prices or a persistent decline

in the purchasing power of money, caused by an increase in available currency and credit

beyond the proportion of available goods and services”9.

In absolute terms, inflation hurts entrepreneurship and corporate management by mak-

ing price evolution and cost evaluation difficult (and therefore adds difficulty to pricing

future cash flows). It is also a very unfair process. For instance retirees, sheltered by

state money and workers, who negotiate their wage do not have the same elasticity

regarding inflation.

In relative terms, even though a high inflation well anticipated by the market gives

protection to both borrowers and lenders, it puts at a disadvantage all forms of money

not carrying interests (such as paper money). As such, the financial markets provide

in essence a protection against anticipated inflation by setting the terms of contracts

between borrowers and lenders.

9The American Heritage Dictionary of the English Language, Fourth Edition, Copyright c©2000Houghton Mifflin Company.


In the United States, the inflation rate for ordinary consumers is most commonly mea-

sured by the percentage rise in the Consumer Price Index (CPI)10 reported by the

Bureau of Labor Statistics (BLS). For an evaluation of the inflation of the economy as

a whole, the GDP deflator11 is more suited as it includes the prices of non-consumers

goods and excludes the prices of foreign-produced goods.

Inflation is directed by the money supply through the equation of exchange:

MV = Py

where M is the money supply (see below), V is the number of times per year the average

dollar turns over in transactions for goods and services, P is the general price level and

y denotes the economy’s real income (as measured, e.g., by real GDP)12.

Figure 1.4: Year on Year CPI vs Transaction Volume of the SPX Index

10”The CPI is a statistical measure of a weighted average of prices of a specified set of goods andservices purchased by wage earners in urban areas. It is a price index which tracks the prices of aspecified set of consumer goods and services, providing a measure of inflation.” (Source: wordIQ.com)

11”The GDP deflator is a price index measuring changes in prices of all new, domestically pro-duced, final goods and services in an economy.[...] A simple GDP deflator formula goes like this

GDP deflator in % = Current Year GDPBase Year GDP × 100.” (Source: wordIQ.com)

12For more details about Inflation, [See: 4]


1.3.2 Controlling The Money Supply

The money supply is ”the amount of money in the economy, measured according to

varying methods or principles.”13

Measures

The main measures are called M1, M2 and M3. M1 is a narrow measure of money’s

function as a medium of exchange. It sums currency and checking account deposits.

M2, a broader measure also reflects money’s function as a store of value as it adds some

types of saving deposits to M1. M3 is a still broader measure that covers items widely

regarded as a close substitute for money in addition to M2.

Figure 1.5: U.K. money supply and a combined capital market price index, 1950-1972.[Source: 5]

Over the years, changes in the relationship between the money supply and the economy

have complicated the Fed’s decisions and the correlation between the financial markets

and these measures.

For many years M1 was an accurate measure of the Gross Domestic Product (GDP)

until in the 1980s when banks started paying interest on checking accounts, people put

a lot of money into checking accounts. The rapid growth in M1 broke the relationship

between M1 and the GDP.

Similarly, the relationship between M2 and the economy broke down in the 1990s, when

the interest rates were very low and people pulled money out of saving accounts to

13The American Heritage Dictionary of the English Language, Fourth Edition, Copyright c©2000Houghton Mifflin Company.


put it into financial investments outside of banks, such as mutual funds (which are not

included in the M2 money supply measure).

As a result, the Fed tended to look more at direct economical data (as opposed to

monetary ones) such as the employment rate, prices indexes or commodities prices14.

Even though the relationship between financial markets and money supply distended, it

didn’t disappear and markets and money supply remain tied together. A loose monetary

policy may increase liquidity and encourage more investment by making margin loan

requirements less costly, and by enhancing the ability of dealers to finance their positions.

For instance, monetary expansion increases equity market liquidity and unexpected in-

creases (decreases) in the Federal Funds15 rate lead to decreases (increases) in liquidity

and increases (decreases) in stock and bond volatility. This establishes a link between

macro liquidity, or money flows, and micro or transactions liquidity[7].

The Fed’s Tools

To control the creation of money, the Fed has three main tools at its disposal:

Reserve Requirement It is the amount of cash that a deposit bank needs to keep,

the rest being allowed to be lent to other parties. The Monetary Control Act of

1980 sets the Reserve Requirement to stay between 8% and 14%. The last reserve

requirement move happened in April 1992, when the Fed lowered it to 10% (from

12%).

Discount Window This is the interest rate at which the Federal Reserve Banks make

very short-term loans to banks. It is a national rate, fixed since Frebruary, 18,

2010 to 0.75%. Since the beginning of the 1980s, the Fed uses the Federal Funds

rate as the key figure for its lending target.

Open Market Operations are the Fed buying or selling securities in the open market.

The Fed could do so to offset a seasonal need of cash from banks (Christmas

shopping season, etc.), an exceptional need of cash (the bank run of late 1999

14For more details about Money Supply, [See: 6].15”Federal Funds are overnight borrowings by banks to maintain their bank reserves at the Federal

Reserve.” (Source: Wikipedia)


when people worried about the possible 2000’s computer bug), or to add/remove

cash from the system by growing/reducing the bank’s balance sheets.

Finally, the Fed looks at the value of the dollar in terms of foreign currencies as inter-

national trade and financial activity picked up in the recent years.

In the end, both inflation and money supply are reflected through the financial markets.

This supply and demand of money in turn leads ”transaction liquidity” in the sense of

the availability of counterparties on markets. This market liquidity is visible through

different indicators.

1.4 Evaluating Market Liquidity

1.4.1 Definition Of The Measure

Generally, liquidity is broadly defined from two point of views16, price and time:

Price ”The degree to which an asset or security can be bought or sold in the market

without affecting the asset’s price.”

Time ”The speed at which an asset (resp cash) can be converted (invested) into cash

(an asset).”

It is important to see that this very dichotomy shows that the definition of liquidity

hasn’t yet converged to a sole, broadly accepted and clearly stated definition - such as

the definition of volatility for instance. In other words, liquidity remains an hidden,

implicit parameter of the market with no universal appreciation tool available.

1.4.2 Subjectivity Of The Measure

In addition, market liquidity mingles market risks and non market risks, both not being

equally accessible by the operators, giving them different access to liquidity. For example,

you want to enter a security at 100, screen price. By actually taking your position you

will go through costs that will impact the book value of the operation (broker, taxes,

16see http://www.investopedia.com/terms/liquidity.asp


etc.) and which will not be the same for another given operator. Any operator sees

in fact a modified bid/ask that includes these effects. The following diagram shows in

black a ”screen” bid/ask spread for a security and in red the ”effective” bid/ask if an

order were to be executed.

Figure 1.6: Screen price (mark to market price) vs effective price.

While some of these costs are known and stable (broker fees, taxes, etc.), some others

remain hidden and non-predictable.

1.4.3 Non-Market Costs

Transaction costs The transaction costs are an important factor to take into account.

There are commonly three:

Exchange/Clearing Fees which are taken by the exchange for their services.

Broker/Execution Fees which are taken by the broker. They can include fees

for the usage of execution algorithm or other services.

Fiscal Fees which can be taken on a per trade basis (such as the ”Federal Stamp”

in Switzerland).

Tax costs are a more intricate topic. They are usually thought of at the scale of

portfolios and evaluating each trade’s impact is not instrumental.

Operational costs are mostly seen on an annual basis and in term of throughput.

These costs can be broken down to each trade. They comprise all the investment

done to compute position and account statements (Net Asset Value, etc.).

Legal costs comprise all the costs that are needed to bring a legal stature to the com-

pany and all the legal costs induced by market activities.


More important, these hidden costs represent about 5/6 of the overall cost17 and give

to the different operators very different access to liquidity depending on their scale,

country, broker, technology, etc.

Considering that the overall cost is assumed to represent 150 billions dollars per year

just for the American market, which represents, according to BNPPAM18 from 1% to

15% of an investment’s value (with an average of about 2%), the question of assessing

these costs becomes key for any investment institution.

The other components of liquidity are shared by all operators and depend on market

conditions. Any trader or risk manager taking a position assess liquidity following a set

of indicators.

1.5 Market Liquidity Indicators

Market liquidity can be observed through several market variables which are independent

from the user’s point of view. While none of them mirror entirely the state of a market’s

liquidity, they together give a good picture of it at the time of the trade. Practitioners

tend to focus on a chosen set of indicators, depending on their underlying and their needs.

For instance, a commodity trader will closely follow the Open Interest (OI) evolution

while a bond trader will concentrate on the bid/ask spread. The same difference prevails

between a long-term and a short-term trader. Here is a review of the main indicators

currently looked at:

1.5.1 Flows

Volume represents the number of trades of a given security during a period of time. It

measures the activity going on in the security, as agents allocate or reallocate their

portfolios. The more volume, the more it is possible to get in/out of a position

fast and without affecting the price of the security.

Volume follows strong seasonality (see Book Depth) that are not without affecting

prices and how the mar-to-market of a position should be read.

17Result published by Plexus, a company specialized in the evaluation of transaction costs and quotedby Naji Freiha and Berge [8].

18quoted by Naji Freiha and Berge [8].


Technical analysis has derived two other indicators to link price trends to volume

trends:

On Balance Volume (OBV) is the cumulative total volume updated say every-

day (volumes are taken at each day’s market close or at midnight):

OBVtoday = OBVyesterday +

+volume if closetoday > closeyesterday

0 if closetoday = closeyesterday

−volume if closetoday < closeyesterday

This indicator underlines whether a trend in price is followed by a trend in

volume, or whether price is changing for other reasons than the availability

of the asset on the market.

Money Flow is typical price multiplied by volume.

typical pricetoday =hightoday + lowtoday + closetoday

3

money flowtoday = typical pricetoday ∗ volumetoday

This indicator attempts to evaluate roughly the cash value of a day’s trans-

actions.

Fund Flows They are surveys conducted by agencies such as TrimTabs19 that track

the cash allocation into mutual and hedge funds. As these funds allocate these

flows in securities, these surveys reveal large volume and allocation patterns that

can also be linked to market prices. See for instance Using Equity ETF Flows as

a Contrary Leading Indicator [9].

Order Imbalance is defined as the notional value of buys less the notional value of

sells over a time period, divided by the total value of buys and sells over that same

period of time:Value of buys−Value of sells

Value of buys + Value of sells

. It is still an aggregate account of the transactions but it adds informations

concerning the trade initiations. A trade can be initiated as a buy or a sell and if

the amount of buys (resp. sells) outnumbers the amount of sells (resp. buys), the

available liquidity to buy or sell a security differs.

19http://www.trimtabs.com.


1.5.2 Market Width and Market Depth

Bid/Ask spread On OTC markets, the bid/ask spread is the difference between the

prices at which market makers are ready to buy and sell a certain quantity (known

in advance) of a security. On listed markets, the same difference can be seen in the

order book, between the first offered price and the first bid price. It is important

to note that , first of all the prices between the bid price and the ask price are

totally illiquid. Second, that even though market makers operate in a best effort

fashion, liquidity outside this spread is unknown. Nevertheless, by continuously

publishing a spread and an order size, market makers offer a certain liquidity.

A market can also be deemed illiquid when either the bid (ask) price moves away

from a (virtual) market middle price. This effect exists for all markets and a shock

on the bid/ask spread in one market is followed by the same behavior in other

markets, as characterized for instance by [7]. This study also underlines a time

correlation between the US bond and US stock markets’ bid/ask spread of +28%

along with a magnitude correlation between a move of the bid/ask spread of the

10Y bonds and stock for the same issuer (when the bid/ask spread widens for the

bond, the stock’s bid/ask spreads by a relative 1/4 of that magnitude.)20.

Bid/Ask spread follows different seasonality. (see Book Depth)

Figure 1.7: Bid/ask spread of some currency pairs as published by the retail brokerfxpro.com on Oct.13, 2009

Book Depth is defined as the difference between the price of the highest and the price

of the lowest orders placed in an order book. This price is often smoothed as a

20This also underlines the credit risk of the issuer.


bell, centered around the current market price. It is used to evaluate the number

of people ready to trade at prices distant from the current market price. The more

people, the more orders one can hit, even if it reduces the return. As pointed out by

[7], book depth is also strongly linked to the tick size. As with the Bid/Ask spread,

book depth suffers from seasonality of different frequencies. These seasonality are

very market-dependent.

Daily frequency following the open trading hour and the overlapping of the dif-

ferent timezones.

Quarterly due to the numerous accounting variables linked to this frequency

(refinancing rate, company results publishing, dividend paying), the natural

seasons which impact most industries and the importance of Future contracts

which also follow this frequency (Euribor 3M contracts, Bund contracts, etc.).

Yearly also due to the numerous deadlines associated with that frequency (com-

panies’ yearly balance sheet and turnover, asset managers yearly target, etc.).

In addition, depths are lower around holidays, lower on Friday compared to the

other days of the week, higher in August and September relative to January and

slightly lower during daily lunch hours. Some markets also have specific seasonality

[see 7, Page 12]. US bond depth is relatively high in February, May and July

whereas the US stock depth is relatively low on Monday and high in March. Also

the stock depth decreases during bond market crisis and both markets depth are

correlated by +20% [7].

1.5.3 Market Size

Open Interest (OI) is used by Futures and options traders. It represents the number

of contracts alive at a certain point in time. As opposed to spot contracts, Futures

and options contracts have no fixed number. Thus, open interest represents the

size of the secondary market for these derivatives and it also impacts the bid/ask

spread. Associated with volume, open interest can reveal important details about

a contract. For instance, a relatively high number of derivatives contracts in

comparison to the daily volume underlines the possibility of a squeeze at maturity.

Level II quotes The NASDAQ is a computerized system created in 1971 that facili-

tates trading and provides price quotations on more than 5,000 of the more actively


traded over the counter stocks. It permits to see who is trading and how. The

three different types of players are:

Market Makers They provide liquidity as they are entailed to sell (resp buy)

when everybody is buying (resp selling). The most important market maker

is known as the ”Ax”. The Ax of a stock is the market maker who leads the

price action of a given stock.

Electronic Communication Networks (ECN) They are computerized order

placement systems. They act as proxies for institutional or retail investors.

Wholesalers They also act as proxies but only for retail traders. They are also

known as Order Flow firms.

Commitment of traders The Commitment of traders (COT) is a report issued by

the Commodity Futures Trading Commission (CFTC) enumerating the holdings

of participants in various Futures and options markets in USA along with their

natures: Hedger, Speculator, Other. It focus on commodities (agriculture, energy,

whether,...) and is updated every Friday at 3:30 Eastern Time.21

Some indicators are only available after an order execution:

1.5.4 Late Indicators

Slippage is the indicator that matches the closest the liquidity definition given above

but with the disadvantage of only being available after a trade’s execution. Slip-

page can either refer to price slippage (commonly called ”slippage”) and to time

slippage (commonly called ”Execution time”). Here is an example of slippage

(taken from Taleb [10]):

A fund manager needs to go long the JPY currency against the dollar. The 115

calls on the Chicago Mercantile Exchange are quoted 88/92 which means that the

middle market price in theory is at 90. He would then assume that for buying 4000

contracts, he would have to pay 92 for the first 1000, 93 for the next 1000 and up

to 98 for the balance of 2000 as he would drive the currency itself higher thanks to

option traders hedging their deltas. His weighted average will then be 95.25 and

he will count an overall slippage of 5.25 ticks for his execution. He should make

21Homepage of the COT: http://www.cftc.gov/marketreports/commitmentsoftraders/index.htm


the same allowance for his unwinding the trade, provided he picks the same time of

the day. Slippage would be more important when the market gets more volatile or

when overlapping of time zones reduces the total number of market participants.

Slippage is often wrapped into order execution conventions such as VWAP22 or

OHLC23.

Historical Volatility is a mixed indicator of a market’s liquidity. There exists count-

less studies linking liquidity and volatility using both empirical and mathematical

terms. All of them suggest that both parameters are driven by the same factors.

None of the studies can separate them or at least, differentiate them entirely. (See

for instance the paper by Leland which integrates liquidity as a modified volatility

parameter). In all cases, volatility is a key indicator to liquidity even though its

interpretation is intricate.

22Volume Weighted Average Price23Open High Low Close

Chapter 2

Three Liquidity Crisis

This chapter serves two purposes. First, to expose some of the mechanisms of credit,

maturity and liquidity transformation presented in section 1.2. Second, to illustrate what

are liquidity crisis at the global (section 2.1), local (section 2.2) and micro-structural

scales (section 2.3). The latter serving as an introduction to the last two chapters where

micro-structural liquidity and order execution will be studied.

The 2008 crisis will expose how credit intermediation is performed by some institutions

and how their activities collapsed in a gigantic liquidity crisis.

2.1 The 2008 Fall of the Shadow Banking System

Hopefully, macro-economical liquidity collapse are not so frequent (generally one or two

per century). The most famous one is the bank run of the nineteen-thirties. While the

creation of the Fed as lender of last resort in 1913 certainly reduced the occurrence of

bank runs, one happened very recently, more subtle in its causes but as dramatic in its

effects: The fall of the shadow banking system, which began in the summer of 2007 and

peaked following the failure of Lehman Brothers.

Actors of the Shadow Banking System are financial intermediate as presented in 1.2

with the only difference being that they do not have access to central bank liquidity or

public sector credit guarantees.

20

Chapter 3. Three Liquidity Crisis 21

Instead, they were supposedly backed by the private sector. As defined by McCulley who

forged the term ”shadow banking” in 2007: ”unregulated shadow banks fund themselves

with uninsured commercial paper, which may or may not be backstopped by liquidity

lines from real banks. Thus, the shadow banking system is particularly vulnerable to

runs - commercial paper investors refusing to re-up when their paper matures, leaving

the shadow banks with a liquidity crisis - a need to tap their back-up lines of credit with

real banks and/or to liquidate assets at fire sale prices.”

Indeed, they notably used supposedly AAA assets as collaterals for their operations. The

problem was that these assets were the product of a range of securitization and secured

lending techniques, that imply risks that were underevaluated such as the agency risk

or the correlation risk.

Figure 2.1: Shadow Bank Liabilities vs. Traditional Bank Liabilities in USD trillion[Source: 2]

2.1.1 Chronicle of the Crisis

There exists many timelines describing the events of the credit crisis of 2007. The Fed

of New York has published one underlying their actionsRyan [11] and the international


components of the crisis. Academics also have published detailed chronicles, such as [12]

or spontaneous consortium such as [13].

From 2001 to 2004, the number of mortage-backed securities 1 jumped as lending condi-

tions loosen and home prices increased following the economical growth and the zero-rate

policy of the dot-com bubble burst of 2001. Banks stuffed themselves with subprimes,

notably Fennie Mae and Freddie Mac. In 2006, home prices began a rapid decline be-

cause mortgage loans terms changed as interests rose and people couldn’t pay up. The

excess supply of homes put pressure on home prices.

Between February 2007 and April 2007, 25 major subprime lending firms declared

bankruptcy2. All major banks published losses in the subprime markets including HSBC,

Bearn Sterns, Merril Lynch, J.P.Morgan Chase, etc. BNP Paribas locked two of its funds

as it couldn’t value the assets in them, owing to a ”complete evaporation of liquidity”.

During the summer of 2007, most banks amounted their exposure to the subprimes

market. In September, Northern Rock faced a big problem as liquidity was cut off and

it couldn’t finance its positions. It appeared that even as a deposit bank, Northern Rock

relied mostly on the markets to finance its mortgage lending activity. A traditional bank

run followed until the Bank of England provided the bank with an emergency funding.

The first figures of the banks’ exposure were revealed. Most major banks (UBS, Citi,

Merril Lynch, etc.) started to announce their exposure to the subprimes. Citigroup

losses already summed up to $40 billions within the last six months. The Fed attempted

a global, coordinated plan with the five major central banks to offer billions of dollars in

loans to banks. The ECB auctioned $500bn to help commercial banks over the Christmas

period.

In January 2008, the financial markets suffered the largest fall since September 11, 2001

on fears that the recession might become global. The scope of the financial crisis just

started to be grasped. The G7 amounted the possible losses due to the subprimes to

$400 bn. Series of interest cuts were started by central banks. U.S. home prices felt

most in 25 years. Foreclosures throttled. In March 2008, Bear Sterns was purchased by

J.P.Morgan Chase at $2 a share. A year earlier share prices reached $170 a share.

1Mortgage-backed securities were created in 1983 by Salomon Brothers and First Boston. They aretradable securities which payments come from fixed-income underlying assets.

2Ben Bernanke announced at that time that growing number of mortgage defaults will not seriouslyharm the US economy.


In September 2008, Fannie Mae and Freddie Mac were taken over by the U.S. govern-

ment. They revealed owning over $5 trillions of mortgage-backed securities. Lehman

Brothers had incurred billions of dollars in losses due to the mortgage crisis and couldn’t

find a buyer. All markets got affected. All Lehman debts, credit lines and market posi-

tions incurred losses to most of the financial institutions worldwide. Even money markets

jumped, revealing that even short term debts could tumble. Bank failures continued.

In October 2008, Eastern Europe was hit by a currency crisis. People were doing carry

trade by borrowing in Eastern Europe currencies and investing abroad. When returns

fell, people closed their lines, selling Eastern-European currencies and facing huge debts.

This fueled the crisis in Europe, particularly in Germany.

The TARP (Troubled Assets Relief Package) was released in the U.S. The government

injected $700 billions into banks’ balance sheets against preferred shared. Consolidation

of the financial sector continued. The IMF started to provide help to Eastern-European

countries. China announced a $586 billion stimulus package. Worldwide central banks

cut their interest rates. Iceland government collapsed.

A $787 billion stimulus package was released by newly elected President Obama followed

by quantitative easing from the Fed amounting to over $800 billions.

2.1.2 Conclusion

We can outline a liquidity spiral at the macro-bank-level as follows ([14]) :

Bank’s balance sheets deteriorate All banks see exceptional losses in their activi-

ties. What was priced as an idiosyncratic risk appears to be a systemic risk: a

whole set of lendings deteriorate. That could come from mortgage, consumption

or investment lending. In our case it came from mortgages in the U.S.

Banks de-lever, selling assets This is amplified by the fact that banks tend to have

a pro-cyclical leverage, meaning that when things are going goods (balance sheets

grow), they lend even more, and when balance sheets are smalls, banks do not

leverage as much[15]. Volumes rise considerably as agents still think it is a conjec-

tural problem and therefore the slide appears as a good buying opportunity.


Risk management tighten, lending reduced, counterparty exposures minimized.

A beginning of defiance between operators appears. Prerequisites for lending cash,

reverse-repos, etc. become more strict. Volumes are reduced to levels below the

beginning of the crisis as prices keep on sliding.

Margins increase At this point, people may think they’re safe. Some ”crisis” can stop

at this point if the lendings at the root of the problem find their real value or that

extra cash incomes balance the deterioration. In such case, volumes can go back

to normal, pre-crisis levels.

Liquidity vanishes This is the point where market liquidity problems become signif-

icants. Prior to the crisis, market prices made sense compared to each others as

idiosyncratic risks were mirrored by relevant spreads and where exceptional dete-

riorations could be priced in and be diluted in the entire market. In our case, the

entire mortgage-backed securities market had to be reevaluated.

At this point, most parties ”find themselves caught on the same side” and adverse

interest is reduced by far. Assets loose value both as their inner value is reevaluated

but also because of lack of consideration: no other party is ready to take these

assets against cash.

If assets still find no solid bid, markets are deserted. Volumes dive, prices become

extremely volatiles. Assets are marked-to-value in the books: they are given a

value (can be zero) in the balance sheets.

Traders face funding liquidity risk as they become unable to fund their positions

and are forced to unwind. Some OTC governmental deal can appear where public

institutions take parts of these assets against cash to protect the bank’s balance

sheets.

It is also important to underline that a fair amount of cash available do not entail

liquidity. Decisions as to whether rescue plans by governments and central banks helped

restoring liquidity are still questioned.

We will now assess the liquidity flows at a smaller scale through another crisis: The

November 2000 Turkish overnight crisis.


2.2 The November 2000 Turkish Overnight Liquidity Cri-

sis

Figure 2.2: The Turkish Overnight Rate from 1998 to 2001. [Source: Bloomberg]

A liquidity crisis hit Turkey on November 20, 2000 that spanned for about 15 days. At

the peak of the crisis, overnight rates reached 2000%, forcing the intervention of the

Turkish central bank, an emergency loan by the IMF and the takeover or bankruptcy of

several banks of the country.

A close study of the events shows that the rationale behind the crisis lay between macroe-

conomic liquidity and market microstructure. The relatively small size of the Turkish

overnight market ( 180 financial institutions executing about 1000 trades per day) allows

for a close understanding of the timing of the crisis and of the liquidity problems.


During the 1990’s, Turkey was an active emerging market with an inflation rate close

to 100% and an annual real GDP growth rate of 6.8%3. It signed numerous agreement

3Global Finance Magazine - http://www.gfmag.com/gdp-data-country-reports/157-turkey-gdp-country-report.html


with the IMF stipulating that local rates should remain floating, that the government

had to keep off the overnight market, that banks were limited to 20% of their total assets

in foreign currencies.

Many banks used ”off-balance sheet” transactions to exceed this ratio using bonds as

collateral. At the end of 2000, yields increased, reducing the value of these collateral and

banks faced margin calls. These operators often used the overnight market as a source

of funds to cover these margin calls, which in turn pushed short term yields higher,

decreasing bonds value, generating margin calls and so fueling more needs for cash to

cover margin calls. As a result, a vicious loop between short term yields and the cash

market appeared.

In the second half of 2000, the Turkish yield curve ended up inverted and some banks who

were on the edge of bankruptcy were aggressively borrowing on the overnight market,

including the biggest actor on that market: Demirbank. In November, banks tried

unsuccessfully to dump assets, stocks as well as local T-bills (1-year maturities bonds).

At that time, market operators started to question the solvency of each others and

solvent banks reduced their exposure to institutions rumored to be in trouble. Foreign

creditors started to withdraw their credit lines, rapidly selling the local currency and

forcing the central bank to provide liquidity by buying back the local currencies (it still

didn’t intervene on the overnight market though).

On November 30, the central Bank stopped providing funds to the domestic market as

it reached its Net Domestic Assets target. The crisis culminated the very next day (Dec.

1), when the overnight interest rate reached 2000%.

On December 5, the IMF announced an emergency loan to Turkey which stabilized

the country’s economy but didn’t save numerous financial institutions from bankruptcy,

starting with Demirbank which stopped all banking activities that day. On December

6, the government took over Demirbank (which eventually was sold later on to HSBC).

The total outflow during the crisis was summed to USD 6 billions, eroding about 25%

of he foreign exchange reserves of the Turkish central bank.

Also note that on November 22, in the midst of the crisis, Standard & Poors upgraded

the rating of Demirbank to B+ long-term and B short-term, completely overlooking the

liquidity needs of the bank.


2.2.2 Conclusion

The liquidity shortage cycle is the same as the one exposed in the previous section.

However this crisis is also interesting as there was no real need to reevaluate assets in

a way similar to the asset-backed securities. Banks were simply over-leveraged through

off balance sheets techniques and the overnight market, tying together the entire rate

curve of the country. Another main difference is the scale of the market, which allows

for agents to clearly read it and assess the troubles that may face other participants.

These informations allowed for numerous agents to withdraw from the market in time.

A last scale will be considered to assess liquidity at the execution level : minutes or

hours. To underline this, we will take as an example the financial markets crash of May

6, 2010.

2.3 The May 6, 2010 ”Flash-Crash”

Figure 2.3: Intraday quotation of the iShares S&P 500 Index Fund on May 6, 2010.[Source: 16]


On May 6, 2010, the prices of many U.S.-based equity products experienced a seldom

seen price decline and recovery within a time frame of about 20 minutes. This ”free

fall” hit both equity indexes, equity ETFs, stocks and their related derivatives. Some

equities lost up to 15% in a few minutes before recovering most of the loss. Over 20,000

trades were executed at prices 60% away (up or down) from the values they had a few

minutes before, at prices ranging from one penny to $100,000.

On September 30, 2010, the U.S. Commodity Futures Trading Commission and the

U.S. Securities & Exchange Commission jointly released a memorandum regarding the

events of that day as seen both from the exchange aggregate flows and the various market

operators, market-makers, funds, etc.[16]

The report reveals the mechanisms that lead to a reduction of 99% of the market depth

for several contracts during few minutes.


May 6 started as a turbulent day over the European debt crisis, leading European CDS

up, U.S. equities slightly down and Euro currency down against the USD and the JPY.

At 1:00 PM, the bearish momentum started to lead to above average automatic execution

volumes, triggering a mechanism on the New York Stock Exchange (NYSE) known as

Liquidity Replenishment Point4 at an unusual rate.

At 2:30 PM, the VIX Index, tracking the implied volatility of the S&P 500 options, was

up 22.5 percent from the opening level, U.S. Treasuries were flying and the Dow Jones

was down about 2.5%. At that time, both the E-mini Futures contracts and the ETFs

tracking the S&P 500 Index had their book order depth reduced by respectively 55%

and 20%.

At 2:32 PM, a mutual fund initiated a sell program to sell a total of 75 000 E-mini

Futures contracts (valued at $4.1 billion) as an hedge to an existing equity position.

This sell program was operated via an automated execution algorithm programmed to

target an execution rate of 9% of the volume of the preceding minutes (sliding), without

regard to price or time.

4Liquidity Replenishment Points are moments during which automated execution is paused and thenresumed a few second later, allowing other, non-automated participants to catch up with the flow


This execution resulted in the largest net change in daily position of any trader in the

E-mini since the beginning of the year. The price of the E-mini went down by 3% in

just four minutes as the other participants couldn’t or didn’t want to provide buy-side

liquidity. At that time the market depth of the E-mini fell to 1,050 contracts ($58

million) which is less than 1% of the depth level from that morning. With virtually no

participants left, the contracts price fell an additional 1.7% in 15 seconds to its intraday

low of 1056 at 2:45:28 PM.

Between the beginning of the sell program and that point, the sell algorithm had pushed

35,000 E-mini contracts in the market. During the same time frame, reacting to the

events, all the other traders combined sold more than 80,000 contracts and bought

50,000 which represent levels respectively 15 times and 10 times bigger than any 13

minutes interval during the past three days. In other words, volumes spiked, fueling

even more the execution rate of the sell algorithm.

At that moment, a circuit breaker of the Chicago Mercantile Exchange (where the E-

mini Futures are quoted) triggered a safety pause for five seconds, allowing buy-side

interest to increase. When trading resumed, prices stabilized and shortly after the E-

mini began to recover. The sell algorithm kept operating until 2:51 PM but prices were

quickly rising.

Only two sell programs of similar size were executed in the E-mini within the 12 months

prior to May 6. These two other programs used a combination of manual trading and

several automated execution algorithms taking price, time and volume into account to

execute 75,000 contracts in over 5 hours. However, on May 6, as the markets were

already under stress and the sell algorithm only targeted volume, the sell program took

just 20 minutes to fill the orders.

Due to the relationships between the E-mini Futures, equity ETFs and stocks, (they

share the same underlyings), what could have been a local crisis to the E-mini Futures

also became a market-wide crisis, affecting all contracts. An important parameter is the

way market participants operate depending on their strategies and how they provide

liquidity to each others. Taking a closer look to these mechanisms will for instance

elucidate why both the market depth on the buy and the sell sides fell, why the volumes

spiked when no market depth was left and how some stocks happened to be traded at

irrational prices.


Figure 2.4: E-mini S&P 500 Future Volume and Price on May 6, 2010. [Source: 16]

Figure 2.5: E-mini S&P 500 Future Market Depth on May 6, 2010. [Source: 16]


2.3.2 Cross-market propagation

We segregate the different market participants depending on their role and strategies:

Exchanges Their role is to match orders and to publish data (volumes, prices, interests,

etc.) to the other participants.

Traders They are the ”clients” of the market. They seek exposure to market moves

through various products, using management strategies and research analytics to

bet on price directions or gain dividends from their investments.

HFTs / Arbitrageurs They primarily focus on benefiting from cross-market prices

discrepancies but with limited or no exposure to subsequent price moves in those

products. They can operate on the futures, ETFs or the underlying securities.

They trade significant volumes (they account for over 30% of a day’s volume) but

keep very limited exposure.

Market-makers Their activity is very similar to HFTs and Arbitrageurs except that

it is mandatory for them to keep publishing bid/ask quotes to the market. Their

business is not to try to benefit from product prices differences but to gain the

bid/ask spread while keeping as little exposure as possible.

When the sell program began to operate, the E-mini Futures became relatively cheap

compared to ETFs or the underlying securities. HFTs and arbitrageurs were naturally

the first to react and started to buy aggressively the E-mini to sell ETFs or the securities.

They traded nearly 140,000 E-mini contracts (33% of the volume) between 2:41 PM and

2:44 PM, while maintaining an overall exposure of less than 3,000 contracts at all time.

The Sell Algorithm responded to that increase in volume by increasing the rate at which

it was feeding the orders in the market even though the previous orders were obviously

not fully absorbed by fundamental buyers. As HFTs and Arbitrageurs reached their

exposure limit they started to act as liquidity-consumers instead of liquidity-providers

to reduce their exposure. At the same time numerous HFTs also stopped operating as

their systems triggered security pauses (taking into consideration prices moves, prices

integrity, volume, risk limits, etc.).


Figure 2.6: U.S. Equity Market Participants.

At 2:45 PM, volume spiked as HFTs and Arbitrageurs were passing each others the

contracts (”hot-potato” effect) as no sufficient buyer appeared and the sell algorithm

accelerated. Between 2:45:13 and 2:45:27 PM, HFTs traded over 27,000 contracts (49%

of the total trading volume) while only buying only about 200 additional contracts net.

That is when buy-side market depth in the E-mini fell to 1% of its depth from that

morning level and that the E-mini dropped by an additional 1,7% to reach its intraday

low.

For stocks and ETFs market-makers, who trade by submitting non-marketable ”rest-

ing” limit order and capturing a bid-ask spread, rapid price movements and volatility

triggered both a reduction of the number of shares offered as well as a widening of their

quotes. Some market-makers completely left the market thus publishing mandatory

stub quotes5. Eventually, when liquidity vanished, some of these stub quotes were hit

by market orders.

It also appeared that many of the securities experiencing the most severe price disloca-

tions on May 6 were equity-based ETFs.

5Stub quotes are quotes at unrealistic prices. They are submitted to respect the obligation of quotespublication but are not expected to be hit.


There are two ways of quoting an ETF. Either by providing quotes to the market as if

it was a stock, letting the price flow with the buy and sell interests. Either by using the

underlying securities to price th ETF. While the first category of market-makers was

not severely injured by the crisis, the latter faced pricing problems when the volatility

increased, thus widening their quotes and stopping to provide liquidity to the underlying

securities.

Figure 2.7: E-mini, SPY and Aggregated S&P 500 Stocks Buy-Side Market Depthon May 6, 2010. [Source: 16]

The cross-market propagation is clearly visible on the figure 2.7 where the E-mini leads

the fall, followed first by the SPY ETF6 and later on by the underlying securities. Note

that the recovery is also lead by the E-mini. In fact market participants generally

acknowledge that the E-mini leads the ETFs and the securities and not the other way

around.

2.3.3 Conclusion

This crisis also underlines the problem of assessing liquidity. For instance, measuring

liquidity through volumes during the crisis would have proven to be completely wrong

6The SPY is the S&P ETF with the highest volume.


(which measure was used by the sell program). One would have had to look at bid/ask

spread for market-makers and market depth on the exchanges.

The next Chapter will present models used to assess liquidity. The following chapter

will present the principles behind execution algorithms such as the Sell Program of this

chapter.

Chapter 3

Modeling Market Liquidity

This section presents the Almgren-Chriss Liquidity Asset Price Model (ACLPM) which

tries to mirror the market behavior using an asset price model generated from a discrete

Arithmetic Brownian Motion (ABM) and fixed transaction costs. It was published by

Robert Almgren and Neil Chriss in December 2000[17].

The reader may find a more comprehensive review of the literature surrounding the topic

by reading Mitton [18, Page 134].

The section ”Symbols” (1) at the beginning of the document lists the letters and symbols

used in this chapter to facilitate the reading.

3.1 The Almgren-Chriss Liquidity Asset Price Model

3.1.1 Price Dynamics

We assume a security price with an initial price S0. At time t0 we hold XS0. The

security price evolves naturally according to two factors: volatility (σ) and drift (µ)

following the arithmetic random walk:

Sk = Sk−1 + µτ + σwk

with k going from 1 to N and wk being a Brownian wk ∼ N(0, τ), that is wk = εk√τ with

εk a draw from independent random variables. Note that in extremely volatile markets

35

Chapter 4. Modeling Market Liquidity 36

or over a long period, it would be important to consider the geometric instead of the

arithmetic Brownian motion. Nevertheless, for our present situation, the arithmetic

motion should be enough.

We add to this behavior our own impact on the security through two parameters: a

temporary impact, caused by the surplus of supply caused by our order, and a permanent

impact, which implies a change in the equilibrium price of the security for a time at least

equal to the liquidation time. The permanent impact is thereafter named g(ν) with ν

being the average rate of trading nkτ (in units/time) between times tk−1 and tk.

Figure 3.1: Example of a generated buy order execution happening every 5 ticks.

In the end, we will consider the security’s price to evolve following:

Sk = Sk−1 + µ τ + σ√τεk − τg(

nkτ

) (3.1)

with µ = 0 at first to imply that we have no information about the direction of future

price movements, hence not permitting to optimize the execution using the knowledge

of an existing trend.

The temporary impact represents a slippage which is resorbed before passing the next

order. It is the paid price (Sk) on a give order, which depends on the previous price but

which will not remain until the next order (Sk+1). This difference, thereafter named

h(ν), depends on the average rate of trading during one time interval. Hence the actual


Figure 3.2: Example of a generated walk with a buy order execution happening every5 ticks.

price received on the k’th sale is

Sk = Sk−1 − h(ν) (3.2)

and h(ν) represents the difference between the quoted price and the paid price.

3.1.2 The Definition Of A Trading Strategy

Suppose we hold a block of X units of a security that we want to liquidate before time T .

We divide this interval into N periods of equal length τ = TN . We define subsequently

the discrete time tk = k · τ with k = 0, 1, ...N .

A trading trajectory is a list of holdings x0, x1, . . . , xN where xk is the number of units

that we plan to hold at time tk. Our initial holding is x0 = X and we must obtain

xN = 0. (We take the case of a sell program. The construction stands the same for a

buy program.)

We also define a strategy by its trade list n1, n1, . . . , nN where nk = xk−1 − xk, that is

the number of units sold between the times tk−1 and tk. The relation between xk and


nk therefore writes:

xk = X −k∑j=1

nj =N∑

j=k−1nj with k=0,. . . ,N.

A trading strategy is a rule determining the nk depending on the informations available

at time tk−1. Such a strategy can either be static or dynamic. In a static strategy, the

rule for determining each nk depends on the information available at time t0, that is

prior to execution:

t0 → nk

while in a dynamic strategy, each nk depends on the information up to tk−1:

t0

t1...

tk−1

→ nk

3.1.3 Cost Of Trading

What we really want is to find the cost (C) of a complete liquidation (from X to 0)

which is the difference between the total paid price and the quoted price observed just

before the liquidation. That is,

C = XS0 −∑

nkSk

withN∑k=1

nkSk = XS0︸︷︷︸(a)

+

N∑k=1

σ√τεkxk︸︷︷︸

(b)

−N∑k=1

τg(ν)xk︸︷︷︸(c)

−N∑k=1

nkh(ν)︸︷︷︸(d)

with (a) the initially quoted price, (b) the effect (positive or negative) of volatility, (c)

the loss of our total position and (d) the price drop suffered for the nk units sold during

the period tk. Finally, the total cost of trading is:

C =

N∑k=1

(σ√τεk −

N∑k=1

τg(ν))xk −N∑k=1

nkh(ν) (3.3)


which is a random variable centered around

E(x) =

N∑k=1

τxkg(ν) +

N∑n=1

nkh(ν) (in dollars) (3.4)

and of variance

V (x) = σ2N∑k=1

τx2k (in dollars squared) (3.5)

When a trader must execute an order, he has to take a decision according to the level of

risk he is willing to take. A defensive trader will try to reduce V (x) as much as possible,

thus carrying no uncertainty, while a trader willing to take some risk will try to minimize

E(x) given a maximum V (x).

Chapter 4

Order Execution

The directive of the European parliament dated of September 25, 2003 specifies the duty

of the market agents to operate as best as possible and thus, to quantify, report, and

optimize an order price.

Today, investors rely on simple averaging methods to get a ’clean price’ of a portfolio

transaction. Such methods encompass VWAP1 or OHCL2. In all cases, the hidden part

(the ’dirty price’) of the execution lies within the market impact of the order and the

risk carried during its execution.

According to Joe Ratterman, C.E.O of BATS Exchange, Inc.3, ”the average execution

size in the displayed markets today is under 200 shares. And there’s so much investment

in technology to take some firms’ large interest in moving either buying or selling a

security, and making sure that that’s get put into the market in a way that doesn’t

move the market adversely against them”[19].

Best market execution is simply the optimization of bots pushing orders in the market

following the principles exposed in the previous chapter.

1calculated as the average price weighted by the volumes during the length of the execution2calculated as the median price of the highest, the lowest, the opening and the closing price of a day3Third largest exchange in the world by volume behind the NYSE and NASDAQ.

40

Chapter 5. Order Execution 41

4.1 Impact Functions

To see practical results of section 3.1, we have to define the two impact functions. We

will take here two important cases of g(ν) and h(ν). In the first case, the permanent and

temporary impacts are a linear function of the volume while in the second case, they

are an exponential function of the volume.

4.1.1 Linear Impact Functions

By taking g(ν) = γν (with γ a real absolutely positive number), the selling of n units will

depreciate the security’s price by γν, whatever how long it takes to sell them. Similarly,

taking h(ν) = λν (with λ a real absolutely positive number), we get a proportional

effect. The fixed costs of selling can also be taken into consideration as they have a

simple yet important impact. This amount is taken as a constant per order, that we

name β. Finally,

h(ν) = β sign(nk)︸︷︷︸constant

+ λν︸︷︷︸linear

Rewriting equations 3.1 and 3.2 with these new impact functions we get:

Sk = S0 + σ

k∑j=1

√τεj − γ(X − xk) (4.1)

Sk = Sk−1 − βsign(nk) +λ

τnk (4.2)

which leads to rewriting equations 3.3 and 3.4 as:

C =N∑k=1

(σ√τεk − τγ

nkτ

)xk −N∑k=1

nk(β sign(nk) +λ

τnk)

=

N∑k=1

σ√τεkxk −

N∑k=1

γnkxk −N∑k=1

nk(β sign(nk) +λ

τnk)

=N∑k=1

σ√τεkxk −

1

2γ(X2 −

N∑k=1

n2k)−N∑k=1

nk(β sign(nk) +λ

τnk)

=

N∑k=1

σ√τεkxk −

1

2γ(X2 −

N∑k=1

n2k)−N∑k=1

nk(β +λ

τnk) as we are selling


and

E(x) =1

2γX2 + β

N∑k=1

|nk|+λ

τ

N∑k=1

n2k with λ = λ− 1

2γτ

=1

2γX2 + β |X|+ λ

τ

N∑k=1

n2k as we are selling all the way

while the variance 3.5 remains unchanged.

So, given these impact functions, we can either want minimum variance by selling ev-

erything at once, or minimum impact, by selling at a constant rate.

Minimum Variance In this case we liquidate everything with a single order: n1 =

X,n2 = n3 = · · · = nN = 0 (implying x1 = x2 = · · · = xN = 0), which leads to:

E = Xh(X

τ) = βX + λ

X2

τ

V = 0

As expected the variance is null and the expected value does not depend on γ as there

is a single order. On the other hand, the loss can be arbitrarily large and depends on

the fixed cost β and the ratio λτ .

Minimum Impact In this case we liquidate at a constant rate nk = XN (implying

xk = (N − k)XN ) which leads to:

E =1

2XTg(

X

T)(1− 1

N) +Xh(

X

T)

=1

2γX2 + βX + λ

X2

T

V =1

3σ2X2T (1− 1

N)(1− 1

2N)


Figure 4.1: Plot of V (N) for X = 10000,σ = 20%,τ = 0, 01

We can compare this expected value with the previous one. By replacing λ by its

expression and T by τN we get:

E =1

2γX2 + βX +X2(

λ

τN− γ

2N)

then we add and remove λX2

τ :

E =1

2γX2 + βX +X2(

λ

τN− γ

2N)− λX2

τ+λX2

τ

and we factorize to get:

E = βX +λ

τX2︸︷︷︸

Expected value of minimum variance

+ (1− 1

N)X2[

1

2γ − λ

τ]︸︷︷︸

a

(4.3)

It is now clear that there is a risk as the expected value can be greater/smaller than the

previous one depending on whether γ2 is greater or smaller than λ

τ .

4.1.2 Exponential Impact Functions

While linear impact functions constitute an intuitive reference, studying exponential

impact functions can be also lead to valuable conclusions. Indeed, even though definitive

proper impact functions for a given market should be calibrated empirically, exponential


impact functions are the simplest continuous, non-liear forms of behaviour. We take:

g(ν) = eγν (4.4)

h(ν) = β sign(nk) + eλν (4.5)

which leads to rewriting equations 3.1 and 3.2 as:

Sk = S0 + σk∑j=1

√τεj − τ

k∑j=1

eγnkτ (4.6)

Sk = Sk−1 − βsign(nk) + eλnkτ (4.7)

equations 3.3 and 3.4 become:

C =N∑k=1

(σ√τεk − τ

N∑k=1

eγnkτ )xk −

N∑k=1

nk(βsign(nk) + eλτnk)

=

N∑k=1

(σ√τεk − τ

N∑k=1

eγnkτ )xk −

N∑k=1

nk(β + eλτnk) as we are selling

and

E(x) =N∑k=1

τeγnkτ xk + β |X|+

N∑k=1

nkeλnkτ (4.8)

while the variance 3.5 remains unchanged.

It is still obvious that we can get minimum variance by selling everything at once:

Minimum Variance In this case we liquidate everything with a single order: n1 =

X,n2 = n3 = · · · = nN = 0 (implying x1 = x2 = · · · = xN = 0) and which leads to:

E = Xh(X

τ) = X(β + eλ

Xτ )

V = 0

As expected the variance is null. On the other hand, the loss can be arbitrarily large.

Minimum Impact Finding the minimum of equation 4.8 is not trivial. Nevertheless,

we can note a few things. First, that without any impact (γ = λ = 0)we have E =

X(τ N−12n + β + 1) which represents the floor of the expected value.


Now, if we suppose that we sell at a constant rate nk = XN (implying xk = (N − k)XN )

as in the minimum variance solution for linear impact functions we get:

E =X

2τeγ

XN (N − 1) + βX +Xe

λτXN

= X(N − 1

2τeγ

XN + β + e

λτXN )

and V = σ2X2τ(N + 1)(2N + 1)

6

Figure 4.2: Plot of E(N) for X = 10000, σ = 20%, τ = 0, 01, γ = 1.10−6, λ = 1.10−6

The minimum is found for N = 15.

Figure 4.3: Plot of V (N) for X = 10000, σ = 20%, τ = 0, 01, γ = 1.10−6, λ = 1.10−6


4.1.3 Empirical Impact Functions

The linear and exponential impact functions allow us to grasp the market’s behavior.

In practice, different market may have different behaviors, and one market can have

impact functions with more complicated expressions, including gaps, finite liquidity, etc.

Moreover, the two impact functions above display additional problems: First, trying to

calibrate the parameter (say using a linear regression) can lead to important errors for

certain points. Second, if we were to try to calibrate ”in real time”, we would have

to recalculate the λ and γ parameters using all the observed data, which is very time-

consuming. Therefore, we can try to build a better solution, which could be easily

exploited and which could reduce the average as well as the maximum error. The idea

is to use the linear impact functions with different parameters according to ν. Formally,

it is:

We fix limits to ν: ν ∈ {0, . . . , νmax} Now, we build a partition of M subsets ν ∈

{ν1, . . . , νM} on this set. For each set, we have:

g(νk) = γk

h(νk) = β sign(ν) + λk

with k ∈ {1, . . . ,M}. Finally, we can have a single expression for all possible values of

ν:

g(ν) =

M∑k=1

δνk,ν γk (4.9)

h(ν) = β sign(ν) +M∑k=1

δνk,ν λk (4.10)

with δνk,ν the Kronecker symbol such as:

δνk,ν = 1 if ν ∈ νkδνk,ν = 0 if ν 6∈ νk


4.2 The Efficient Frontier Of Optimal Execution

A trader is entailed to support a limited risk. This risk deploys over time and is rep-

resented by the variance of its strategies V (x). The optimal strategy is therefore the

strategy that minimizes execution costs E(x) in respect to a given limit V∗ for V (x).

This reduces to minimizing the function

U(x) = E(x) + αV (x)

where α is a number specifying the level of accepted variance. Given 3.4 and 3.5, the

general expression of U(x) writes:

U(x) =N∑k=1

τxkg(ν) +N∑n=1

nkh(ν) + ασ2N∑k=1

τx2k (4.11)

As it is convex for any α ≥ 0, the minimum of U(x) is found by setting it’s partial

derivative to zero:

∂U

∂xk= τ

[g(νk)− νk+1 g

′(νk)]+h(νk+1)− h(νk)

τ+h′(νk+1)

νk+1 − νkτ

+2ασ2τxk (4.12)

for k = 1, . . . , N . Then setting ∂U∂xk

= 0 leads to:

τ[g(νk)− νk+1 g

′(νk)]

+h(νk+1)− h(νk)

τ+ h′(νk+1)

νk+1 − νkτ

= −2ασ2τxk (4.13)

If we use the linear impact functions of the previous section we have:

g(ν) = γ ν g′(ν) = γ

h(ν) = β sign(ν) + λ ν h′(ν) = λ

By replacing in 4.13, we get:

λ

τ(xk−1 − 2xk + xk+1) = ασ2τxk (4.14)

This is a recurrence equation, which if solved can generate the list of xk and nk.


According to [17], the solution may be written as a combination of exponentials e±κtj

where κ satisfies:2

τ2(cosh(κτ)− 1) =

ασ2

λ

and which gives, for x0 = X and xn = 0:

xk =sinh(κ(T − tk))

sinh(κT )X

and

nk =2sinh(12κτ)

κTcosh(κ(T − (k − 1

2)τ))X

For small steps of τ , [17] gives the following expected value and variance:

E(x) =1

2γX2 + βX + λX2 tanh(12κτ)(τsinh(2κT ) + 2Tsinh(κτ))

2τ2sinh2(κT )

V (x) =1

2σ2X2 τsinh(κT )cosh(κ(T − τ))− Tsinh(κτ)

sinh2(κT )sinh(κτ)

which is not a trivial solution.

Figure 4.4: Plot of E(N) for X = 10000, σ = 20%, τ = 0, 01, γ = 1.10−6, λ = 1.10−6,α = 0.01


Figure 4.5: Plot of V (N) for X = 10000, σ = 20%, τ = 0, 01, γ = 1.10−6, λ = 1.10−6,α = 0.01

Figure 4.6: Plot of U(N) for X = 10000, σ = 20%, τ = 0, 01, γ = 1.10−6, λ = 1.10−6,α = 0.01

Chapter 5

Conclusion

Market liquidity is a difficult parameter to evaluate as it encompasses the desire that

people have to trade along with their constraints and their behaviors. As such, most

of the measures of liquidity in fact grasp historical liquidity and there is no way to

date to predict future liquidity (as opposed to the implied volatility extracted from

options prices). Different proposals have been put forward to cover that problem such

as American forward contracts (contracts allowing to buy/sell securities at any point in

time at screen price, see [18]).

The growing size of the financial markets sure helped liquidity a lot and say stock

liquidity today is miles away from what it was only a few years ago (as an example,

the NYSE’s average speed of execution fell to 0.07 second in 2009 from 10.1 seconds in

2005[21]). Nevertheless, when markets freeze, assets can loose all their values and create

a systemic crash.

A good part of the bettering of liquidity is also due to technology and the arrival of high-

speed trading and dark pools. For some, like George Sauter, chief investment officer at

Vanguard Group Inc.1, high frequency trading is good as it has reduced transaction

costs by 50 percent over the past 10 years[21]. For regulators, such techniques make

financial markets more opaque (as price manipulation can be hidden more easily) and

add systemic risk as they speed up the rate of execution and can be the source of costly

mistakes and avalanches[20].

1The biggest U.S. mutual fund manager with $1.4 trillion in assets.

50

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