<s>  Lecture Notes in Economicsand Mathematical Systems
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553Burkart MG6nch
Strategic Tradingin [liquid Markets
2) SpringerAuthor
Burkart Ménch
Goethe University Frankfurt
School of Economics and Business AdministrationFinance Department, Derivatives Group
Uni-PF 77
60054 Frankfurt am Main
Germany
E-mail: b.:moench@econ.uni-frankfurt.de
Library of Congress Control Number: 2005922554
ISSN 0075-8442ISBN 3-540-25039-5 Springer Berlin Heidelberg New York
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Printed on acid-free paper 42/3130Di 543210Acknowledgments
 
This work arose from three papers that have been written while I have beena research assistant at Johann Wolfgang Goethe-University, Frankfurt amMain. It was accepted as my doctoral thesis by the Faculty of Economics andBusiness Administration in October 2004. I am deeply indebted to the school,the department, and the staff working there for support and encouragement.
First of all I would like to thank my Ph.D. advisor, Prof. Christian Schlag.He taught me much of the foundations on which the research of this thesishas been based. Despite his busy schedule, he always found time to providemany extremely useful suggestions and provided me with a friendly and safeatmosphere in which to grow professionally. He has granted me broad free-dom to pursue all of my ideas, and I am grateful for his show of confidence.
I would also like to thank the committee members Prof. Raimond Maurer,Prof. Reinhard H. Schmidt, and Prof. Dieter Nautz for reviewing my thesisand for their interest in my work.
My special thanks go to Dr. Angelika Esser without whose mathematicalsupport I would not have got this far. Angelika was my officemate for threeyears and our close neighborhood and friendship resulted in two joint papers,which in turn were taken as Chapters 1 and 3 of this thesis.
Thanks are also due to Dr. Nicole Branger and Dr. Christoph Benkert forbeing a source of constant inspiration and for sharing their knowledge andideas with me. I thank Micheal Belledin for teaching me how to programVI Acknowledgments
in C and Micong Klimes for her patience when constructing the order booksand for a number of muffins, each one a delicious masterpiece. I owe thanksto Dr. Matthias Birkner and Dr. Roderich Tumulka for providing valuablehelp with some of the mathematical background for this thesis.
I would also like to thank Raisa Beygelman, Eva Schneider, and NormanSeeger for the stimulating and pleasant working atmosphere and for takingover some of my tasks during the last weeks before my defense, and oursecretary Sigrid Peschel for all the administrative help. Thanks go to JeffreyBoys and Sabine Steiner for copyediting and proofreading my thesis.
This book is about liquidity in financial markets and it would not exist with-out the liquidity that was generously provided by DFG, the German ResearchFoundation. Financial support by DZ Bank-Stiftung for the publication ofthis monograph is gratefully acknowledged. Furthermore, I am grateful to theTrading Surveillance Office of Deutsche Bérse AG for providing an excellentlimit order book data set.
Finally, my warmest thanks go to my dear parents for all the inexhaustiblesupport and good advice over the years. They have been instrumental infurthering my education and helping me to get to this point.
Burkart Monch, Frankfurt am Main February 2005Contents
 
Acknowledgments
List of Figures
List of Tables
Introduction
1 Modeling Feedback Effects with Stochastic Liquidity
1.1 Introduction... 0... 0000002000000 0.2000.1.2 The Deterministic Liquidity Model ............1.3 Is Market Liquidity Indeed Stochastic? ..........14 The Stochastic Liquidity Model .. 0... 0.0.00...1.5 The Effective Price Process in the Stochastic Liquidity Model15.1 Impact on Correlation .............0..1.5.2 Impact on Volatility .............00.1.5.2.1 Positive Feedback Strategy .......
1.5.2.2 Contrarian Feedback Strategy ..... .
1.6 Numerical Results... 0.2... 00... 0.004. 0000.4
XI
XII
17 
VIII Contents16.1 Parameter Specification ©... .........00... 23
1.6.2 Positive Feedback Strategy... ........-000.4 251.6.3 Contrarian Feedback Strategy .........00..2. 26
1.7. Application: Liquidity Derivative ................ 271.7.1 Pricing Formulas ...............2..000. 28
1.7.2 Example... . 0... 00.0.0... eee eee 331.7.3 Sensitivity Analysis. 2... 0... eee 341.7.4 Market Price of Liquidity Risk. ............. 37L7.4.1 Overview 2... ee 37
17.4.2 Intuition. ... 2... 0.2.0.0... 0008. 38
1.7.4.3 Pricing a European Put on an Illiquid Stock . 40
18 Conclusion... 0... 0. 0 ee 4119 Appendix .... 0... ee 421.9.1 Proof of Proposition ll ..............00. 42
1.9.2 Functional Form of d(S,p)... 0.0.0.0. eee 43References... ee 442 Optimal Liquidation Strategies AT2.1 Introduction. 2.0... es AT2.2 Related Literature 2... 2... ee 502.3 Description of the Market and Dataset ............. 5224 The BasicSetup.... 2... 0. ee eee 532.5 Modeling the Impact Function... ............... 572.5.1 U-shape of Stock Market Liquidity ........... 58
2.5.2 Power Price Impact Function .............. 59Contents IX
 
2.5.3 The Resiliency of the Order Book... ........, 592.6 Numerical Results... 2.0... 2 ee ee en 612.6.1 Parameter Specification ...........20.0004. 612.6.2 Numerical Examples ..............20004 642.6.3 The Distribution of Liquidation Revenues ....... 702.7 Conclusion and Issues for Further Research... 0.2.2.0... 712.8 Appendix: Derivation of EG and var(G) ............ 73References 6 0. ee 75
3 The Navigation of an Iceberg:
The Optimal Use of Hidden Orders 793.1 Introduction. 2... ee ee 793.2 Related Literature 2... ......00.00.0... 00000. 853.3 Description of the Market and Dataset ............. 883.4 The Basic Setup... . 2... 0.0000 eee ee ee 913.4.1 General Idea and Dynamics ...........-...0. 913.4.2 Execution of the Iceberg Order ............. 943.4.3 Liquidation Value... 0.2... eee 953.5 Modeling of the Drift Component ................ 1023.6 Numerical Results... 0. ee ee 1053.6.1 Parameter Specification ..............04. 1053.6.2 Numerical Implementation ...............0. 1063.6.3 Numerical Examples ..............-.2.5. 1083.7 Summary and Conclusion .......0.0.00-.-000.804 1123.8 Appendix: Conditional Density g of In(S7r/S;,) ........ 113
References... ee 114List of Figures
 
Fig.
Fig.Fig.
Fig.Fig.
Fig.
Fig.
Fig.Fig.Fig.Fig.Fig.Fig.
11
1.21.3
141.5
1.6
17
2.12.22.32.42.52.6
Scatterplot for best bid prices and percentage liquiditydiscounts 2. ee
Positive and contrarian feedback strategy ........
Sample paths for the stock price in the Black-Scholesmodel and the stochastic liquidity setup .........
Contrarian feedback strategy versus Black~Scholes
Best bid prices and percentage liquidity discounts in theorder book sample... 2... 0... ee ee
Price of the liquidity derivative as a function of differentparameters 2... ee
Price of the liquidity derivative as a function of differentparameters 2.0. te
General model setup... ..........0 000004Relative liquidity discounts in the order book sample . .Estimated price impact function .............Fitted price impact function... ..........4..Resiliency of the order book... 2... 0. De ee
Expected liquidation revenues as a function of the num-ber of liquidation intervals for the benchmark case 
XII List of FiguresFig. 2.7. Time that should elapse between subsequent trades... 66Fig. 2.8 Expected liquidation revenues as a function of the num-
ber of liquidation intervals for different initial positions . 67Fig. 2.9 Expected liquidation revenues as a function of the num-
ber of liquidation intervals for different transaction costs 68Fig. 2.10 Values of the objective function as a function of the
number of liquidation intervals for different levels of risk
AVETSION . ee 69Fig. 2.11 Duration of the liquidation process for different levels of
risk aversion 2... ee 69Fig. 2.12 Approximated density of simulated liquidation revenues 71Fig. 3.1. Absolute frequencies of order volumes of all observed
iceberg sell orders in the order book sample ....... 89Fig. 3.2 Ratio of initial order volume to peak size of all different
iceberg sell orders in the order book sample ....... 89Fig. 3.3 Absolute price difference between the chosen limit and
the current best bid price... ee 90Fig. 3.4 General model setup... 0. oe 92Fig. 3.5 Example for the alternating drift component ...... 104Fig. 3.6 Optimal limit as a function of the execution probability 108Fig. 3.7. Optimal limit as a function of the final time horizon . . 109Fig. 3.8 Optimal peak size as a function of the final time horizon 110Fig. 3.9 Optimal limit as a function of the initial position . . . . 111Fig. 3.10 Optimal peak size as a function of the initial position. . 111List of Tables
 
Table 1.1Table 2.1Table 2.2Table 3.1
Table 3.2Table 3.3Table 3.4
Liquidity-related parameters in different models... . . 20Estimations for different parameters ........... 62Resiliency of the order book... ............05 63Execution or deletion of iceberg sell orders in the order
book sample 1... 0... ee ee 91Notation used in the liquidation model... ....... 93Estimations for model parameters ............ 105
Estimations for model parameters ............ 106Introduction
 
The Area of Research and the Object of Investigation
In this thesis we will investigate trading strategies in illiquid markets from amarket microstructure perspective. Market microstructure is the academicterm for the branch of financial economics that investigates trading and theorganization of security markets, see, ¢.g., Harris (2002).
Historically, exchanges evolved as a location, where those interested in buy-ing or selling securities could meet physically to transact. Thus, traditionallysecurity trading was organized on exchange floors, where so-called dealersarranged all trades and provided liquidity by quoting prices at which theywere willing buy or sell. Consequently, the initial surge of the market mi-crostructure literature focused predominantly on this type of market design,which is often referred to as quote-driven.
Nowadays, the interest is shifting towards order-driven markets. Beginningwith the Toronto Stock Exchange in the mid 1970s and increasing in fre-quency and scope, this market structure has emerged as the preeminent formof security trading worldwide. In order-driven markets, exchanges arrangetrades by matching public orders, often by employing automatic execution
systems.2 Introduction
 
A major difference between a quote-driven and an order-driven market arisesfrom the transparency pre- and post-trade. The pre-trade transparency con-cerns the question whether the order book is visible to the keeper only, orwhether it is open to the public. The post-trade transparency concerns the is-sue whether the details of recent trades are immediately reported to the pub-lic or not. Traditionally, most of the quote-driven markets were opaque, i.e.they offered only a low pre- and post-trade transparency, since the knowledgeof the order flow could expose market makers to undue risk as they wantedto unwind positions. However, nowadays many order-driven markets, whichare operated electronically, provide an order book that is completely or par-tially open to the public. In this thesis we will investigate the question ofhow investors can exploit the pre-trade transparency in order to derive op-timal trading strategies. By choosing a market or a limit order, selecting alimit price, and by eventually specifying further contingencies under whichcircumstances the order should be executed, a market participant enjoys ac-cess to a wide range of strategies that trade off execution certainty againstthe expected execution price. Specifically, we will address the following fourissues:
Whether to break up orders.e How to time order submissions.
e How to chose an reasonable limit.
Whether to restrict the display of limit orders.
Our analysis will focus on large institutional investors (like insurance com-panies or pension funds) whose transaction sizes in a single stock representa significant fraction of this stock’s daily trading volume. These are particu-larly interesting, since acquiring or unwinding large positions in one securitycan incur significant costs which directly influence the return on the invest-
ment.Introduction 3
 
Imagine, for example a market order. Small market orders can usually beexecuted with little or no effect on prices and they pay one-half of the bid—ask spread for the opportunity to immediately trade. However, large marketorders are more difficult to execute than small ones as they run throughthe order book until they are completely filled. The effect of executing amarket order on prices is called price impact. It increases with the volumeof the market order. Often, the price impact is the most significant cost oftrading a large order. Hence, a large trader who uses a market order payssignificantly more for immediacy than a retail investor who just wants totrade a few shares. Note that the price impact of a certain market orderdepends crucially on current market conditions. Since market conditionschange quickly and unpredictably, traders who will use market orders atsome point in the future cannot be sure about the prices they will receive.We will introduce an innovative liquidity derivative that offers protectionagainst varying price impacts for a certain period of time.
Instead of using a market order, the large trader might want to employ alimit order. Investors who submit limit orders offer liquidity. Their ordersgive others the ability to trade when they want to trade. The originator ofa limit order should keep in mind that displaying large orders in an openbook may reveal his or her motives for trading. For small volumes, this factmay be negligible. However, institutional investors often break up large limitorders into smaller packages if they fear that showing their full sizes wouldcause the market to move away from them.
Iceberg orders facilitate these trading practices by executing such businessautomatically in the order book. They allow market participants to submitan order with only a certain portion of the order publicly disclosed — just asthe iceberg’s peak is the only visible portion of a huge mass of ice. Once thedisclosed volume of an iceberg order has received a complete fill, a new peakappears in the book, with the volume equal to the initially disclosed amount.This loop is repeated until the whole iceberg is completely satisfied. At thispoint you may already suspect that the use of iceberg orders also comesat a cost, otherwise everybody would use them exclusively instead of limit4 Introduction
 
orders and the order book would be completely opaque. In order to discoverthe differences between pure limit and iceberg orders one needs to considerthe way orders are queued and processed in the order book. In almost anycase this is governed by the price~time-priority rule. Price priority givesprecedence to the lowest sell and the highest buy orders over all other ordersstored on the same side of the book. Time priority means that among ordersat the same price, the order placed earliest takes precedence. To control fortime priority every order receives a time stamp when it appears in the book.This implies that a limit order enjoys a better overall time priority, as it hasjust one initial time stamp than an iceberg order where every peak is givena new time stamp when it shows up in the book.
Note that a limit order is basically a special case of an iceberg order wherethe peak size coincides with the overall order volume.
So far we have discussed the general area of research and we have presentedthe object of investigation: a large investor who is trading in an order-drivenmarket that provides pre-trade access to order book data. Furthermore, wehave introduced the basic order types available to the large trader: market,limit, and iceberg orders. There follows a brief overview of issues that willbe analyzed in the next three chapters.
Outline of the Thesis
In the first chapter we model the interaction between the trading activitiesof a large investor, the stock price, and liquidity. Our framework generalizesthe constant liquidity model of Frey (2000), and an extension by Frey andPatie (2001), where liquidity is made a deterministic function of the stockprice by introducing a stochastic liquidity factor. This innovation has twoimplications. First, we can analyze trading strategies for the large investorthat are affected by changes in market depth. Second, the sensitivity of stockprices to the trading strategy of the large investor can vary due to changesin liquidity. The main features of our model are demonstrated using MonteIntroduction 5
 
Carlo simulation for different scenarios. The flexibility of our frameworkis illustrated by an application that deals with the pricing of a liquidityderivative. The claim under consideration compensates a large investor whofollows a stop loss strategy for the liquidity risk that is associated with a stoploss order. The payoff of this claim can be best described by comparing astop loss strategy in different regimes of liquidity.
If the market is perfectly liquid, as in the setup of Black and Scholes (1973),the investor will always receive the stop loss price for the asset. However, ifthe market becomes illiquid, the trader will receive a price less than the stoploss limit. If the degree of the illiquidity in the market does not change, asin Frey (2000), or can be interpreted as a function of the asset price, as inFrey and Patie (2001), this discount due to illiquidity is deterministic. Thisin turn means that the investor can adjust the stop loss limit in advance,so that he or she will always receive a deterministic amount for the assets.However, if order book depth is stochastic, the large investor faces liquidityrisk.
The liquidity derivative matures when the stock price falls below the stoploss limit for the first time and then pays the price difference between theasset price immediately before and after the execution of the stop loss order.The setup to price the liquidity derivative is calibrated for one example usingreal-world limit order book data to give an impression of the magnitude ofthe liquidity effect. A number of empirical studies provide strong evidencethat investors care about liquidity risk, which implies that this type of riskis priced into asset returns. Since liquidity is a non-traded risk factor themarket price of liquidity risk appears in the pricing formula of our liquidityderivative. Similar to models with stochastic volatility it has to be estimatedfrom other traded instruments that are exposed to this source of risk. Wepresent a pragmatic approach to determine the market price of liquidity riskfrom traded European put options.
In the second chapter we will present a new pragmatic approach to determineoptimal liquidation strategies if an investor uses market orders to unwindlarge security positions in an illiquid market. To keep the setup tractable we6 Introduction
 
take a step backward and assume a deterministic relationship between thevolume of a market order and the resulting price impact. Under the assump-tion that the liquidation horizon is given exogenously, the investor maximizesan objective function that considers the expected liquidation revenues andthe respective standard deviation. The basic structure of our setup is closelyrelated to approaches proposed by Bertsimas and Lo (1998), Almgren andChriss (2000), Hisata and Yamai (2000), and Dubil (2002). While theseauthors focus mainly on theoretical aspects with the intention of derivingclosed-form solutions for special types of market impact functions, we willpropose a framework that is able to capture important empirical phenomenain the stock market. Specifically, the model contains:
e a U-shape for intraday stock market liquidity,power price impact functions,
e periods that allow the order book to be rebuilt, as boundary conditionsfor the time between subsequent trades.
Furthermore, it allows us to incorporate fixed transaction costs, as fixedcharges incurred by the exchange or opportunity costs for handling the trans-actions in the front and back offices. The new model is very flexible since itallows for liquidation intervals of varying length and forgoes the assumptionof a constant speed of trading. Examples with real-world order book datademonstrate how the setup can be implemented numerically and provide adeeper insight into relevant properties of the model.
As already discussed briefly in the previous section, market participants withlarge orders to execute are often reluctant to expose these to an open orderbook in their entirety in order to avoid a potential adverse market impact.In these situations investors often use iceberg orders.
In the third chapter we analyze the rationale for the use of this order type byassessing the costs and benefits of this trading instrument. At least to ourknowledge, this is the first analytical approach that allows the determinationIntroduction 7
 
of the optimal limit and the optimal peak size of an iceberg order. We assumethat the investor who has to liquidate a large position in a stock within afinite time horizon follows a static strategy, i.e. once the limit and the peaksize of the iceberg order are chosen, the trader sticks to this strategy over afixed period. In our setup we will balance the relative advantage of hidingthe actual order volume against the overall time priority of the iceberg orderthat deteriorates the smaller the peak size that is chosen.
Note that unless an iceberg or a limit order is immediately executable, i.e. thelimit is set so aggressively that it is actually a market order, the probabilityof receiving a complete fill within a finite time horizon is strictly smaller thanone. We propose two different approaches to incorporate the execution riskinto our model. The first one assumes that the investor is forced to trade theremaining shares with a market order if the iceberg order fails to receive acomplete fill within the prespecified time horizon. This setup is referred to asthe self-contained approach. The second framework considers the executionprobability as a boundary condition, i.e. only those combinations of peak sizeand limit are admissible that ensure a certain execution probability within aprespecified time horizon. This model is referred to as the open approach.
Once again we use a clinical order book data sample to explore importantfeatures of the model.
References
Almeren, R. and N. Chriss (2000): Optimal execution of portfolio transac-tions, Journal of Risk 3 (Winter 2000/2001), 5-39.
Bertsimas, D. and A. W. Lo (1998): Optimal control of execution, Journalof Financial Markets 1, 1-50.
Black, F. and M. Scholes (1973): The pricing of options and corporateliabilities, Journal of Political Economy 81, 637-54.8 Introduction
 
Dubil, R. (2002): Optimal liquidation of large security holdings in thinmarkets, University of Connecticut, Storrs, USA.
Frey, R. (2000): Market illiquidity as a source of model risk in dynamichedging, in Model Risk, ed. by R. Gibson, RISK Publications, London, 125-36.
Frey, R. and P. Patie (2001): Risk management for derivatives in illiquidmarkets: a simulation study, RiskLab, ETH-Zentrum, Ziirich, Switzerland.
Harris, L. (2002): Trading and Exchanges, Oxford University Press.
Hisata, Y. and Y. Yamai (2000): Research toward the practical applica-tion of liquidity risk evaluation methods, Monetary and Economic Studies,December, 83-128.1 Modeling Feedback Effects withStochastic Liquidity
1.1 Introduction
Aspects of market liquidity include the time involved in acquiring or liqui-dating a position and the price impact of this action. In this chapter we focuson the second issue. For large institutional investors in particular, many ofthe existing pricing models are only of limited use since they assume per-fectly elastic supply and demand functions for the asset under consideration.This assumption is often violated if the trading activity of a large investoraccounts for a significant fraction of the overall turnover in an asset. In thiscase, ignoring liquidity issues can result in a serious underestimation of the
risk that is inherent in a certain investment strategy.
We present a framework that incorporates the liquidity risk arising for a largeinvestor, whose trading volume cannot be absorbed by the market withouta significant price change. Our model has two main ingredients. On theone hand, the stock price process is influenced by the trading activity of thelarge investor, whereas the impact of the trading strategy on the stock priceis modeled using a stochastic liquidity (henceforth SL) factor. On the otherhand, the stock price and the liquidity factor can have an impact on thetrading strategy of the large investor.10 1 Modeling Feedback Effects with Stochastic Liquidity
 
In principle one could describe this scenario in two ways. First, one canbuild an equilibrium setup to explain the machinery of the market. Such anapproach is sensible if one intends to analyze the motivation for trading or toinvestigate strategies that the large investor can use in order to exploit thepower to move prices in a certain direction, see, e.g., Kyle (1985). However,for pricing purposes such complex frameworks are often unsuitable, as theyare difficult to calibrate. In this chapter we follow a second approach bydirectly modeling the asset price dynamics that result if the large traderfollows a certain trading strategy. We assume that the liquidity in the market
is given exogenously.
There is a growing theoretical literature that investigates the interaction ofliquidity and trading strategies of large investors. Part of this literatureconsiders optimal liquidation strategies for large portfolios. Dubil (2002),Hisata and Yamai (2000), Almgren and Chriss (2000), and Bertsimas andLo (1998) are just a few examples. Another branch of this literature in-vestigates how large traders can manipulate stock prices. Jarrow (1992),Allen and Gale (1992), and Schénbucher and Wilmott (2000) can be men-tioned in this line. Recent research focuses more and more on the mod-eling and hedging aspects that are introduced by illiquidity and the pres-ence of one or more large traders. Cvitanié and Ma (1996), Cuoco andCvitanié (1998), Sircar and Papanicolaou (1998), Frey (1998, 2000), Schén-bucher and Wilmott (2000), Kampovsky and Trautmann (2000), Frey andPatie (2001), Liu and Yong (2004), and Bank and Baum (2002) are someprominent examples.
Our approach generalizes both the model of Frey (2000), where liquidity isconstant, and an extension by Frey and Patie (2001), where liquidity is adeterministic function of the stock price. Modeling liquidity as a stochasticfactor first of all enables us to incorporate random changes in market depth.Furthermore, we can significantly generalize existing models by introducingthe concept of liquidity feedback effects. The presence of liquidity feedbackeffects implies that (i) trading strategies of large investors are affected by1.1 Introduction 11
 
the degree of illiquidity and (ii) the sensitivity of stock prices to the tradingstrategies of large investors can vary due to changes in liquidity.
The objective of our model is to provide a large investor with a flexibleframework that allows us to evaluate the liquidity risk associated with dif-ferent types of trading strategies. We use a stop loss order as an exampleand analyze the effects of market liquidity for this type of trading strategyin detail. To get a flavor of the problem, imagine a pension fund. The fundmanagement has to limit the downside risk of the fund, and in order to doso, assume that it follows a simple stop loss strategy. When the price of thesecurity falls below a certain level the position in this security is liquidatedcompletely and immediately by placing a market order. If the market isperfectly liquid the fund will always receive the stop loss price for the asset.However, if the market becomes illiquid, the investor will receive a price lessthan the stop loss limit. If the degree of the illiquidity in the market does notchange or can be interpreted as a function of the asset price, this discountdue to illiquidity is deterministic. This means that the investor can adjustthe stop loss limit in advance, so that he or she will always receive a deter-ministic amount for the assets. However, if order book depth is stochastic, alarge investor faces liquidity risk.
We propose a liquidity derivative compensating for this liquidity discount.We show how the setup can be calibrated with market data and present asimple and innovative approach to determine