<s> Lecture Notes in Economicsand Mathematical SystemsFounding Editors:M. BeckmannH. R KunziManaging Editors:Prof. Dr. G. FandelFachber ei ch Wi r tschafts wi s sens chaftenFemuniversitat HagenFeithstr. 140/AVZ II, 58084 Hagen, GermanyProf. Dr. W. TrockelInstitut fur Mathematische Wirtschaftsforschung (IMW)Universitat BielefeldUniversitatsstr. 25,33615 Bielefeld, GermanyEditorial Board:A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kursten, U. Schittko
553
Burkart Monch
Strategic Tradingin Illiquid Markets
Spriringer
AuthorBurkart MonchGoethe University FrankfurtSchool of Economics and Business AdministrationFinance Department, Derivatives GroupUni-PF7760054 Frankfurt am MainGermanyE-mail: b.moench@econ.uni-frankfurt.de
Library of Congress Control Number: 2005922554
ISSN 0075-8442ISBN 3-540-25039-5 Springer Berlin Heidelberg New YorkThis work is subject to copyright. All rights are reserved, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, re-use ofillustrations, recitation, broadcasting, reproduction on microfilms or in any other way,and storage in data banks. Duplication of this publication or parts thereof is permittedonly under the provisions of the German Copyright Law of September 9,1965, in itscurrent version, and permission for use must always be obtained from Springer-Verlag.Violations are liable for prosecution under the German Copyright Law.Springer is a part of Springer Science+Business Mediaspringeronline. com© Springer-Verlag Berlin Heidelberg 2005Printed in GermanyThe use of general descriptive names, registered names, trademarks, etc. in thispublication does not imply, even in the absence of a specific statement, that suchnames are exempt from the relevant protective laws and regulations and thereforefree for general use.Typesetting: Camera ready by authorCover design: Erich Kirchner, HeidelbergPrinted on acid-free paper
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Acknowledgments
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 freedom 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 program
VI
_
Acknowledgments
in C and Micong Klimes for her patience when constructing the order booksand for a number of muffins, each one a deUcious 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 without 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 Borse 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 2005
Contents
Acknowledgments
V
List of Figures
XI
List of Tables
XIII
Introduction
1
1
Modeling Feedback Effects with Stochastic Liquidity
9
1.1
Introduction
9
1.2
The Deterministic Liquidity Model
12
1.3
Is Market Liquidity Indeed Stochastic?
13
1.4
The Stochastic Liquidity Model
16
1.5
The Effective Price Process in the Stochastic Liquidity Model
17
1.5.1
Impact on Correlation
20
1.5.2
Impact on Volatihty
21
1.5.2.1
Positive Feedback Strategy
22
1.5.2.2
Contrarian Feedback Strategy
23
1.6
Numerical Results
23
VIII
1.7
2
Contents
1.6.1
Parameter Specification
23
1.6.2
Positive Feedback Strategy
25
1.6.3
Contrarian Feedback Strategy
26
Application: Liquidity Derivative
27
1.7.1
Pricing Formulas
28
1.7.2
Example
33
1.7.3
Sensitivity Analysis
34
1.7.4
Market Price of Liquidity Risk
37
1.7.4.1
Overview
37
1.7.4.2
Intuition
38
1.7.4.3
Pricing a European Put on an Illiquid Stock , 40
1.8
Conclusion
41
1.9
Appendix
42
1.9.1
Proof of Proposition 1.1
42
1.9.2
Functional Form of 0(5, p)
43
References
44
Optimal Liquidation Strategies
47
2.1
Introduction
47
2.2
Related Literature
50
2.3
Description of the Market and Dataset
52
2.4
The Basic Setup
53
2.5
Modeling the Impact Function
57
2.5.1
U-shape of Stock Market Liquidity
58
2.5.2
Power Price Impact Function
59
Contents
IX
2.5.32.6
The Resiliency of the Order Book
Numerical Results
61
2.6.1
Parameter Specification
61
2.6.2
Numerical Examples
64
2.6.3
The Distribution of Liquidation Revenues
70
2.7
Conclusion and Issues for Further Research
71
2.8
Appendix: Derivation of E G and var (G)
73
References3
59
75
The Navigation of an Iceberg:The Optimal Use of Hidden Orders
79
3.1
Introduction
79
3.2
Related Literature
85
3.3
Description of the Market and Dataset
88
3.4
The Basic Setup
91
3.4.1
General Idea and Dynamics
91
3.4.2
Execution of the Iceberg Order
94
3.4.3
Liquidation Value
95
3.5
Modeling of the Drift Component
102
3.6
Numerical Results
105
3.6.1
Parameter Specification
105
3.6.2
Numerical Implementation
106
3.6.3
Numerical Examples
108
3.7
Summary and Conclusion
112
3.8
Appendix: Conditional Density ^^ of In (5'r/S't„)
113
References
114
List of Figures
Fig. 1.1
Scatterplot for best bid prices and percentage liquiditydiscounts
15
Fig. 1.2
Positive and contrarian feedback strategy
24
Fig. 1.3
Sample paths for the stock price in the Black-Scholesmodel and the stochastic liquidity setup
25
Fig. 1.4
Contrarian feedback strategy versus Black-Scholes . . .
27
Fig. 1.5
Best bid prices and percentage Hquidity discounts in theorder book samplePrice of the liquidity derivative as a function of different
34
parameters
35
Fig. 1.6
Fig. 1.7
Price of the liquidity derivative as a function of differentparameters
36
Fig. 2.1
General model setup
55
Fig. 2.2
Relative liquidity discounts in the order book sample . . 58
Fig. 2.3
Estimated price impact function
60
Fig. 2.4
Fitted price impact function
62
Fig. 2.5
Resihency of the order book
63
Fig. 2.6
Expected liquidation revenues as a function of the number of liquidation intervals for the benchmark case . . .
65
XII
List of FiguresFig. 2.7
Time that should elapse between subsequent trades . . .
Fig. 2.8
Expected liquidation revenues as a function of the number of liquidation intervals for different initial positions . 67
Fig. 2.9
Expected liquidation revenues as a function of the number of liquidation intervals for different transaction costs 68
Fig. 2.10 Values of the objective function as a function of thenumber of liquidation intervals for different levels of riskaversion
66
69
Fig. 2.11 Duration of the hquidation process for different levels ofrisk aversionFig. 2.12 Approximated density of simulated liquidation revenuesFig. 3.1Fig. 3.2Fig. 3.3
6971
Absolute frequencies of order volumes of all observediceberg sell orders in the order book sample
89
Ratio of initial order volume to peak size of all differenticeberg sell orders in the order book sample
89
Absolute price difference between the chosen limit andthe current best bid price
90
Fig. 3.4
General model setup
92
Fig. 3.5
Example for the alternating drift component
104
Fig. 3.6
Optimal limit as a function of the execution probability 108
Fig. 3.7
Optimal hmit as a function of the final time horizon . . 109
Fig. 3.8
Optimal peak size as a function of the final time horizon 110
Fig. 3.9
Optimal limit as a function of the initial position . . . . I l l
Fig. 3.10
Optimal peak size as a function of the initial position . . I l l
List of Tables
Table 1.1
Liquidity-related parameters in different models
20
Table 2.1
Estimations for different parameters
62
Table 2.2
Resiliency of the order book
63
Table 3.1
Execution or deletion of iceberg sell orders in the orderbook sample
91
Table 3.2
Notation used in the liquidation model
93
Table 3.3
Estimations for model parameters
105
Table 3.4
Estimations for model parameters
106
Introduction
The Area of Research and the Object of InvestigationIn 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, e.g., Harris (2002).Historically, exchanges evolved as a location, where those interested in buying 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 microstructure 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 frequency 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 executionsystems.
Introduction
A major difference between a quote-driven and an order-driven market arisesfrom the transparency pre- and post-trade. The pre-trade transparency concerns 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 issue whether the details of recent trades are immediately reported to the public 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 partially open to the public. In this thesis we will investigate the question ofhow investors can exploit the pre-trade transparency in order to derive optimal trading strategies. By choosing a market or a hmit order, selecting alimit price, and by eventually specifying further contingencies under whichcircumstances the order should be executed, a market participant enjoys access 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.• How to time order submissions.• How to chose an reasonable limit.• Whether to restrict the display of limit orders.
Our analysis will focus on large institutional investors (like insurance companies or pension funds) whose transaction sizes in a single stock representa significant fraction of this stock's daily trading volume. These are particularly interesting, since acquiring or unwinding large positions in one securitycan incur significant costs which directly influence the return on the investment.
Introduction
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 bidask 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 hquidity 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 limit
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 earhest 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,hmit, and iceberg orders. There follows a brief overview of issues that willbe analyzed in the next three chapters.
Outline of the ThesisIn the first chapter we model the interaction between the trading activitiesof a large investor, the stock price, and hquidity. Our framework generalizesthe constant hquidity model of Prey (2000), and an extension by Prey andPatie (2001), where hquidity is made a deterministic function of the stockprice by introducing a stochastic hquidity factor. This innovation has twoimplications. Pirst, 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 Monte
Introduction
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 hquidity 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 hquidityrisk.The hquidity 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 hquidity risk appears in the pricing formula of our hquidityderivative. 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 ilhquid market. To keep the setup tractable we
Introduction
take a step backward and assume a deterministic relationship between thevolume of a market order and the resulting price impact. Under the assumption 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:• a U-shape for intraday stock market liquidity,• power price impact functions,• 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 transactions in the front and back ofiices. 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 beneflts of this trading instrument. At least to ourknowledge, this is the first analytical approach that allows the determination
Introduction
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.
ReferencesAlmgren, R. and N. Chriss (2000): Optimal execution of portfolio transactions, Journal of Risk 5 (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 corporate
liabilities. Journal of Political Economy 81, 637-54.
Introduction
Dubil, R. (2002):
Optimal liquidation of large security holdings in thin
markets, University of Connecticut, Storrs, USA.Frey, R. (2000):
Market illiquidity as a source of model risk in dynamic
hedging, in Model Risk^ ed. by R. Gibson, RISK Pubhcations, London, 12536.Frey, R. and P. Patie (2001): Risk management for derivatives in iUiquidmarkets: a simulation study, RiskLab, ETH-Zentrum, Zurich, Switzerland.Harris, L. (2002): Trading and Exchanges^ Oxford University Press.Hisata, Y. and Y. Yamai (2000):
Research toward the practical apphca-
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 liquidating 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 perfectly 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 therisk 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 modehng the asset price dynamics that result if the large traderfollows a certain trading strategy. We assume that the liquidity in the marketis given exogenously.There is a growing theoretical literature that investigates the interaction ofliquidity and trading strategies of large investors. Part of this literatureconsiders optimal hquidation 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 investigates how large traders can manipulate stock prices. Jarrow (1992),Allen and Gale (1992), and Schonbucher and Wilmott (2000) can be mentioned in this line. Recent research focuses more and more on the modeling and hedging aspects that are introduced by illiquidity and the presence of one or more large traders.
Cvitanic and Ma (1996), Cuoco and
Cvitanic (1998), Sircar and Papanicolaou (1998), Frey (1998, 2000), Schonbucher 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 hquidity isconstant, and an extension by Frey and Patie (2001), where liquidity is adeterministic function of the stock price. Modehng 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 imphes that (i) trading strategies of large investors are affected by
1.1 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 hquidity.The objective of our model is to provide a large investor with a flexibleframework that allows us to evaluate the liquidity risk associated with different 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 hmit 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 hquid 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 ilhquidity 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 deterministic 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 the market price of liquidityrisk from the prices of traded European plain vanilla put options.The chapter is organized as follows: Section 1.2 summarizes the main ideasof Prey's model. Section 1.3 motivates the introduction of Hquidity as anautonomous source of risk. In a clinical study we show that modeling liquidityas a constant parameter or a deterministic function of the stock does notseem appropriate. Our general framework is presented in Section 1.4. InSection 1.5 we derive the effective dynamics for the underlying asset withSL. We use these results to compare the stock price dynamics in our SLmodel with the benchmark cases of geometric Brownian motion (as in thecase of Black and Scholes (1973) (henceforth BS)) and constant liquidity (as
12
1 Modeling Feedback Effects with Stochastic Liquidity
in Frey (2000)). In Section 1.6 we exemphfy the effects of SL by simulatingsample paths of the asset price. Section 1.7 describes the general setup fora liquidity derivative, to illustrate how the SL model can be used to buildpricing tools. The model is calibrated for an example using real-world limitorder book data so that one gets an impression of the magnitude of theliquidity effect. Furthermore, we present an innovative approach to estimatethe market price of liquidity risk from the prices of European put options.The chapter concludes in Section 1.8 with a brief summary and a discussionof issues for further research.
1.2
The Deterministic Liquidity Model
We now describe the setup proposed by Frey (2000). Assume there exists arisky asset S (the stock) and a risk-free investment earning a zero