provides an expected liquidation valueof € 1,423,734.72. Reducing or increasing the number of intervals decreasesthe expected liquidation revenues, for example by €2,686.81 or 0.2% if theportfolio is otherwise optimally hquidated in three steps. Selling the wholeamount of 50,000 shares all at once at the opening of the exchange reducesthe expected liquidation revenues by € 15,944.72 or 1.1% to € 1,407,740.00.Figure 2.7 summarizes the optimal strategy. The investor should start theliquidation process at 12:40 p.m. and then increase the packages that are soldstep by step as market liquidity is improving. The hnear constraint (2.10)controlhng the time needed to rebuild the order book is binding in almostall cases, so that the length of the liquidation intervals is increasing as well.At this point the question may arise of why the amounts the large investorshould liquidate in each interval are strictly increasing. If the large traderconsiders only the expected hquidation value and if one observes an intradayU-shape pattern in hquidity discounts, one may expect an inverse U-shapepattern in the liquidation amounts. However, each transaction of the large
66
2 Optimal Liquidation Strategies
Time of day13:20:41 fe13:16:53 ^13:13:50 ^13:11:11 m13:08:4'13:06:313:04:13:02:13:00:
3]
112:58:12:57:12:55:39 "'-PME'm*>»as*__J112:54:0412:52:32 mwm^f'" '112:51:02 ^EMEEM^^12:49:35 1 _ _ ,112:48:10112:46:4812:45:2712:44:07 T:]12:42:50 §m?mMwAm^-Uij12:41:33 !'fT;s?g'?g?;piri12:40:180
2,000
4,000
6,000
3,000
A0
Fig. 2.7: Fitted function to model the time in minutes that should elapse betweensubsequent trades.trader has a negative price impact that adversely influences the liquidationrevenues of the following hquidation intervals. Taking this into consideration,the large trader should liquidate a substantial part of the overall position inthe last interval, since the resulting negative price impact of this transactionno longer affects his or her liquidation revenues. In the example this seemsto be the dominant eflPect also, since the slope of the intraday Hquidity curveis flat around noon.To compare this example with other parameterizations, the duration D ofthe hquidation process is calculated. It is defined as
D = n(2.11)It identifies the average weighted time needed to liquidate the position andmust not be confused with the overall time needed to complete the liquidationprocess. The latter measure does not weight the intervals with the respective
2.6 Numerical Results
67
EG0Q=2O,OOO
^^(f)Q=10,000
570,000-
-1,992,000
-1,988,000569,000-1,984,000
1,980,000
568,00010
20
30
40
n
50
Fig. 2.8: Expected liquidation revenues for 00 = 20,000 (left scale) and (/)o =70,000 (right scale) shares as a function of n, if n and A^^. are optimallychosen.
liquidation volume, it just sums them up. In the example one obtains D =4h, 3min, 12sec. Thus, if the large trader does not care about the risk of theliquidation value (a = 0), he or she should sell the portfolio around noon,when market hquidity is comparatively high. Later on, one can see that thedurations of the optimal hquidation strategies decUne as the investor becomesmore risk averse.In the next example the initial stock holdings 00 are varied. Figure 2.8 showsthe results. As a rule of thumb one can state that the larger 0o, the more oftenthe investor should trade. In the case where 0o = 20,000 the portfoHo shouldbe liquidated in five packages. If 0o = 70,000, the investor should trade 41times in order to realize the highest expected liquidation value. Comparedwith the first example, the duration of the hquidation process changes onlyslightly to D = 3h, 58min, 49sec for 0o = 20,000 and to Z^ = 4h, 6min, 16secfor 00 = 70,000.
2 Optimal Liquidation Strategies
68
570,500EG
A
A
A
•570,100
A
A
A
A
k=20
570,300
•
*
•
••
k=50
•
•
*•
•
569,900 |k=70569,700
569,50010
12
14
n
16
Fig. 2.9: Expected liquidation revenues for k = 20, k = 50, and A; = 70 as afunction of n, if Ti and Acpa are optimally chosen (0o = 20,000).
The amount of fixed transaction costs influences the optimal number of trading intervals as well. Figure 2.9 provides an insight into this issue. Assumefor this example that the initial stock holdings are (po = 20,000. If fixedtransaction costs are increasing, the optimal number of intervals decreases.For fc = 20, the optimal strategy consists of 9 intervals, for fc = 50, of5 intervals, and for k = 70, of only 4 intervals. Again, the duration ofthe liquidation process is quite insensitive to a change of fixed transactioncosts. For k = 20, k = 50, and k = 70, one obtains D = 3h, 59min, 07sec,D = 3h, 58min, 49sec, and D = 3h, 58min, 40sec respectively.In the last example the parameter a is varied. Figure 2.10 corroborates theinitial guess that the portfolio should be hquidated in smaller packages if theaversion towards risk increases. An investor who trades more often reducesthe price risk of the Hquidation process, since the probabihty of observing alow stock price at a single point in time is higher than the joint probability ofobserving low stock prices several times a day. For a = 0.0, 0.25, 0.5, and 1.0the optimal number of trading intervals is given by 23, 35, 37, and 48, respectively. The impact of the parameter a on the liquidation strategy becomesapparent in more detail if the duration of the liquidation process is investi-
2.6 Numerical Results
1,425,000
69
1
4. ^-.......*....,,,,,,,^^^^a=0^ 0
• • • • • • • • • • A
i
***********
EG-aSTD(G)
a=0 25i
• ••••
1,415,000
*
a=0 5
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • # 4 • ••<
a=l 0
4/
••*.••••
1,410,000 1
•
'
•
•
1,405,00020
40
n
60
F i g . 2.10: Values of the objective function for a = 0.0, 0.25, 0.5, and 1.0 as afunction of n, if r^ and A(/)t^ are chosen optimally {(po = 50,000). Thearrows point to the optimal number of liquidation intervals for a givenvalue of a.
04:00:00
02:00:00
01:00:00
00:00:00
0.0
0.5
1.0
2.0
Fig. 2.11: Duration of the liquidation process as a function of a, if TJ and A^t^are chosen optimally {4>o = 50,000).
70
2 Optimal Liquidation Strategies
gated. Figure 2.11 illustrates this result. The more a increases, the moreimpatient the investor becomes and the earher the portfoho should be sold.Two cost components reduce the expected Uquidation value in this case.First, fixed transaction costs are rising, since the investor trades more often. Second, the trader has to accept higher liquidity discounts due to theU-shape of intraday market liquidity.For the examples presented above it was assumed that the large investorsolves for the optimal trading strategy at the beginning of the trading day.However, this assumption is not critical and can easily be relaxed by adjustingthe parameters e and g in equation (2.8) if the decision on how to liquidatethe portfolio has to be made at another point in time.
2.6.3
The Distribution of Liquidation Revenues
In order to get an impression of the distribution of the liquidation value andto justify the implicit assumption of normally distributed liquidation revenues, 1,000,000 outcomes of the liquidation value G are simulated. For thispurpose the stochastic process given in equation (2.1) is discretized using anEuler scheme. The parameter a in the objective function is set to 1.64 andit is assumed that the large investor follows the optimal trading strategy tohquidate 0o = 50,000 MAN shares with k = 50. Thus, the value of the objective function can be interpreted as the lower 5% quantile of the liquidationvalue, if G was normally distributed. For the given choice of parameters theoptimization yields n* = 47, E G = 1,419,672.98, and STD(G) = 5,458.01.Figure 2.12 shows the results. For the simulated hquidation value G the grayline shows the approximated density obtained by applying standard kernelestimation procedures. The black dots represent the estimated Gaussiandensity function for fi = 1,419,672.98 and a = 5,458.01. Obviously, thedifferences between the two distributions are negligible. The hypothesis thatthe two distributions are different is tested via the Kolmogorov-Smirnov test,and the null cannot be rejected at the 5% level. Comparing the lower 5%quantile of the simulated distribution 1,410,729.35 with the optimal value of
2.7 Conclusion and Issues for Further Research
71
—Density of simulateddistribution• Points located on thedensity function ofN{/J = 1, 419, 672.98; a = 5, 458.01)
0.00002
1,380,000
1,400,000
1,420,000
1,440,000
G
1,460,000
Fig. 2.12: The gray Hne shows the approximated density of the simulated liquidation value for 00 = 50,000, k = 50, and a = 1.64. The black dots represent some values of the Gaussian density function for /x = 1,419,672.98and a = 5,458.01, the parameter values that were obtained for theoptimal liquidation strategy.the objective function 1,410,695.36 one can state that the downside risk ofthe strategy is slightly overestimated if the value of the objective functionis interpreted as a value of risk. However, for most practical purposes thisdifference is negligible and the model seems well suited for VaR calculations.
2.7 Conclusion and Issues for Further ResearchThis chapter operationalizes the concept of intraday market liquidity patterns in the determination of optimal Hquidation strategies for large securityholdings. Innovative features of this setup include the explicit modeling ofan intraday U-shape pattern, the consideration of the resihency of the order book through a time constraint in the objective function of the large
72
2 Optimal Liquidation Strategies
investor, and the introduction of a permanent price impact function thatfits empirical order book data very well. The proposed model is far moreflexible than other approaches suggested in the literature, as it allows for liquidation intervals of varying length and relaxes the assumption of a constantspeed of trading. Due to the formulation of the objective function the modelis especially suited for VaR-regulated institutions, which want to extend theshort-term management of market risk by an exogenous hquidity factor. Theimplementation of the model with real-world order book data shows that themodel is computationally tractable and able to provide intuitively reasonableresults.A sensitivity analysis shows that if the number of initial stock holdings isincreased, the optimal number of hquidation intervals will also rise. If fixedtransaction costs are reduced, the large investor obtains more flexibility andthe optimal number of liquidation intervals increases. By varying the agent'spreference towards risk it becomes apparent that the investor can use theflexibility to limit the risk of the liquidation revenues if he or she tradesmore often. Additionally, if the trader becomes more risk averse, increasing liquidity discounts due to an earlier liquidation of the portfolio have tobe balanced against the risk of the liquidation value. One can observe adecreasing duration of the liquidation process in this situation.In contrast to the papers by Almgren and Chriss (2000), Hisata and Yamai (2000), and Dubil (2002) the price impact is not separated into a permanent and a temporary component. Instead it is assumed that the whole priceimpact is permanent and influences the liquidation revenues in the followinghquidation intervals. The reason for this rather strong assumption is the lackof an appropriate procedure to separate the permanent and the temporaryimpact empirically from electronic order book data.
Existing approaches
(e.g. the one suggested by Holthausen, Leftwich, and Mayers (1987)) thatsplit the price impact into a permanent and a temporary component assumethat the temporary price impact lasts for a certain period of time, for example until the end of the trading day. This deflnition may be acceptablefor very large transactions. However, for medium-sized trades that occur
2.8 Appendix
73
more frequently, further research toward a comprehensive method has to beconducted, so that one can separate overlapping temporary price impactsresulting from different transactions.When applying the proposed framework one should be aware that the onlysource of randomness modeled in this setup is the stock price. Both themarket depth and the time needed for recovery of the order book are assumedto be deterministic, while in practice one may recognize significant variationsof these variables through time. Further research should therefore focus onextensions of the approach presented in this chapter, e.g. along the lines ofthe stochastic hquidity model proposed by Esser and Monch (2002).
2.8 Appendix: Derivation of EG and var (G)A simple way to compute the expectation is to use an iterative decompositionof the path of 5, given the new starting point after a jump downwards. Using(2.2) one can write 5^. as
=
7i Su_, exp [-l/2a\U
- U-i) + a{Wu - m , _ J ]
since Sf.- is log-normally distributed. Iterating this procedure yieldsSt,
=
7i---7i5toexp[-l/2a2[(t,-i,_i) + ... + (ii-io)]]exp \a{Wt, - Wu_,) + . . . + G{Wt, - VK,J]
=
71 • • • 7i St, exp [-l/2a2(ii - t^)\ exp [a(W,, - Wt,_,)]...expKWt,-H^^J].
(2.12)
This yields the following expectation value using theof
the
increments
of
the
Brownian
E5t,
=
motion
Eexp(crWt) = exp(0.5cr2^):7i---Ti'S'to-
and
independence
the
fact
that
74
2 Optimal Liquidation Strategies
Plugging this into (2.4) one getsn
E G = -nk + Y^ A4>t, 5i 71 • • • 7i St,.i=l
Using (2.3) the variance of G can be written and decomposed as follows:var(G)
= yax( J2^(l>u Si sAn
=
^(A0,,<5,fvar(5,Ji=ln
+ 2 J2 ^<t>u Si ^<t>t, Sj cov {St,, St,).i<j
Thus, all that remains is the computation of variances and covariances. Basedon the relationships
var(5tj coY{St„St,)
ESl-{EStf
= E(5t,5,.) - ( E 5 t J ( E 5 , . )
it suffices to compute ^S^. and E(S't.5'tj.). Based on (2.12), the second moment can be written assi
=
^l..^fSlexp[-a\U-to)]exp[2a{Wt,-Wt,_,)]...exv[2a{Wt,-Wt,)].
By using 'Eexp{2aWt) = exp{2a'^t) one getsESl=
^l..^fSlexp[a'iU-to)],
implying
var(5,J =
^l..j^Sl{exp[a\U-to)]-l).
The covariance terms can be calculated in an analogous fashion. Since
StA,
=
sl^l...^hi+i---iiexp[-l/2a2(ti - to)] exv[-l/2a'^{tj
- io)]
exp[2a(W^t, - Wt,_,)]... exp[2a(W^,, - Wt,)]exp[a(W,. - Wt,_,)].. .exp[a(iy,,^, - Wt,)]
References
75
one obtains
exp[-l/2a^{Uexp[2a\ti=
- to)] exp[-l/2a^{tj
- to)] exp[l/2a\tj
Stoli . . . ihi+i "'I3
- to)]
- ti)]
exp[a^(^^ - to)].
So, the covariance can be written asC0Y{StA,)
= 7? . •. 7 ' 7i+i.. • 7i '^'o (e^P i^^i^i - ^0)] - 1).
ReferencesAlmgren, R. and N. Chriss (2000): Optimal execution of portfolio transactions, Journal of Risk 3 (Winter 2000/2001), 5-39.Amihud, Y. and H. Mendelson (1982): Asset price behavior in a dealershipmarket, Financial Analysts Journal 42, 50-9.Bank, P. and D. Baum (2002): Hedging and portfolio optimization in financial markets with a large trader, Humboldt University, Berlin, Germany.Bertsimas, D. and A. W. Lo (1998): Optimal control of execution. Journalof Financial Markets 1, 1-50.Bessembinder, H. (1994): Bid-ask spreads in the interbank foreign exchangemarkets. Journal of Financial Economics 35, 317-48.Cuoco, D. and J. Cvitanic (1998): Optimal consumption choices for a largeinvestor. Journal of Economic Dynamics and Control 22, 401-36.Cvitanic, J. and J. Ma (1996):
Hedging options for a large investor and
forward-backward SDE's, Annals of Applied Probability 6, 370-98.
76
2 Optimal Liquidation Strategies
Dubil, R. (2002):
Optimal liquidation of large security holdings in thin
markets, University of Connecticut, Storrs, USA,Esser, A. and B. Monch (2002): Modehng feedback effects with stochasticliquidity, Goethe University, Frankfurt am Main, Germany.Foster, F.D. and S. Viswanathan (1993): The effect of pubhc informationand competition on trading volume and price volatility. Review of FiancialStudies 6, 23-56.Frey, R. (1998): Perfect option rephcation for large traders, Finance andStochastics 2, 115-42.Frey, R. (2000):
Market ilUquidity 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 illiquidmarkets: A simulation study, RiskLab, ETH-Zentrum, Zurich, Switzerland.Harris, L.E. (1986):
A transaction data study of weekly and intradaily
patterns in stock returns. Journal of Financial Economics 16, 99-118.Hasbrouck, J. (1991): Measuring the information content of stock trades,Journal of Finance 46, 179-207.Hisata, Y. and Y. Yamai (2000):
Research toward the practical applica-
tion of hquidity risk evaluation methods. Monetary and Economic
Studies,
December, 83-128.Holthausen, R. W., R. W. Leftwich, and D. Mayers (1987): The effect oflarge block transactions on security prices. Journal of Financial
Economics
19, 237-67.Huang, R. D. and R. W. Masuhs (1999): FX spreads and dealer competitionacross the 24-hour trading day. Review of Financial Studies 12, 61-94.
References
77
Jain, P. and G. H. Joh (1988): The dependence between hourly prices andtrading volume, Journal of Financial and Quantitative Analysis 23, 269-84.Kampovsky, A. and S. Trautmann (2000): A large trader's impact on priceprocesses, Johannes-Gutenberg-Universitat, Mainz, Germany.Kirchner, T. and C. Schlag (1998): An explorative investigation of intradaytrading on the German Stock market, Finanzmarkt und Portfolio Management 12, 13-31.Lin, J., G. C. Sanger, and G. G. Booth (1995): Trade size and componentsof the bid-ask spread. Review of Fiancial Studies 8, 1153-84.Linnainmaa, J. (2003):
Who makes the hmit order book?
Implications
for contrarian strategies, attention-grabbing hypothesis, and the dispositioneffect, University of California, Los Angeles, USA.Liu, H. and J. Yong (2004): Option pricing with an illiquid underlying assetmarket,
Washington University, USA and University of Central Florida,
Orlando, USA.Lyons, R. K. (1995):
Tests of microstructural hypotheses in the foreign
exchange market. Journal of Financial Economics 39, 321-51.Mclnish, T.H. and R. A. Wood (1990): A transaction data analysis of thevariability of common stock returns during 1980-1984, Journal of Bankingand Finance 14, 99-112.Mclnish, T. H. and R. A. Wood (1992): An analysis of intraday patterns inbid-ask spreads for NYSE stocks, Journal of Finance 47, 753-64.Ranaldo, A. (2000):
Intraday trading activity on financial markets: The
Swiss evidence, Ph.D. thesis, Universite de Fribourg, Switzerland.Schonbucher, P. J. and P. Wilmott (2000): The feedback effect of hedgingin illiquid markets, SIAM Journal of Applied Mathematics 61(1), 232-72.
78
2 Optimal Liquidation Strategies
Sircar, K. R. and G. Papanicolaou (1998): General Black-Scholes modelsaccounting for increased market volatility from hedging strategies, AppliedMathematical Finance 5, 45-82.Wei, P. H. (1992): Intraday variations in trading activity, price variabilityand the bid-ask spread. Journal of Financial Research 15, 265-76.Wood, R. A., T. H. Mclnish, and J. K. Ord (1985):
An investigation of
transaction data for NYSE stocks, Journal of Finance 40, 723-39.
3
The Navigation of an Iceberg:The Optimal Use of Hidden Orders
3.1
Introduction
The rapid development in technology over the last couple of years has permitted many stock exchanges to transfer trading from open outcry markets,where market makers or specialists act as intermediaries, to screen-basedelectronic markets. Typically, electronic trading platforms provide marketparticipants with information on an anonymous open order book during continuous trading in real time. Usually the hmits, the accumulated order volumes of each limit, and the number of orders in the book at each limit aredisplayed, so that traders can assess the altering order flow and the marketliquidity.What does the existence of an open order book imply for investment firmswho want to submit limit orders, the total volume of which is large relative toothers in the market? No doubt, exposing large limit orders in an open orderbook may reveal the investor's motives for trading and may raise suspicionthat the originator of the large order has access to private information aboutthe true value of the security under consideration. Consequently, other market participants change their own order submission strategy, which in turnlowers the probability that the large order will be executed at the prespec-
80
The Navigation of an Iceberg
ified limit. The investment firm then has to choose a less favorable limit ifit wants to increase the probability of execution and thus suffers losses fromthe indirect adverse price impact of its large exposure in the order book.A possible solution is not to submit one large limit order but to split theorder into several smaller hmit orders, which are submitted over time. Forthis reason many electronic trading platforms introduced so-called icebergorders. Euronext, the Toronto Stock Exchange, the London Stock Exchange(with its order-driven services SETS, SETSmm, and lOB), and XETRA arejust some prominent examples. Iceberg orders allow market participants tosubmit orders with only a certain portion of the order publicly disclosed.The metaphor alludes to the fact that in nature an iceberg's biggest partfloats unobservable under the water. Only one-ninth of the mass of ice isseen above the water surface.An iceberg order is specified by its mandatory limit, its overall volume, anda peak volume. The peak is the visible part of the iceberg order and isintroduced into the order book with the original time stamp of the icebergorder according to price/time priority. As soon as the disclosed volume ofan iceberg order has received a complete fill and a hidden volume is stillavailable, a new peak is entered into the book with a new time stamp. Thenew peak behaves in an identical manner to a conventional limit order. Fromthis point of view a pure limit order is basically a special case of an icebergorder where the peak volume coincides with the total order volume.However, it is important to note that iceberg orders exhibit a less favorabletime priority compared with pure limit orders. After the peak of an icebergorder is completely matched, all visible limit orders at the same limit thatwere entered before the new peak is displayed take priority, i.e. they musthave received a complete fill before the new peak comes to execution.When submitting an iceberg order to the market, several issues have to beconsidered. Imagine, for example, that the management of a mutual fund hasto close a large position in a single stock within one trading day. Using aniceberg order with only a small peak size allows it to minimize the adverseinformational impact of disclosing the actual order volume. However, the
3.1 Introduction
81
smaller the peak size the less favorable the time priority of the overall order.Thus, choosing a peak size that is too small seems suboptimal.
Such a
strategy would significantly lengthen the time to complete execution or wouldmake a complete fill unlikely. Moreover, the fund managers have to choosea reasonable limit for the order. If the limit is too low, one may miss sometrade opportunities, i.e. one would give away the chance to participate inraising stock prices. Otherwise, if the limit is too ambitious, the order isunlikely to receive a complete fill.In this chapter this tradeoff is modeled analytically in a continuous timesetup where a large position in a single stock is to be liquidated within afinite trading window.^ We assume that the investor uses an iceberg orderand follows a static strategy, i.e. once the limit and the peak size of theiceberg order are chosen, the trader sticks to this strategy over a fixed period.We then determine the optimal peak size and the optimal order limit bymaximizing the expected payoff of the liquidation strategy under certainassumptions concerning the execution risk of the iceberg order. Note thata pure limit order would be also an admissible solution to our optimizationproblem.Unless an iceberg sell order is immediately executable, i.e. the hmit is so lowthat it is actually a market sell order, the probabihty of receiving a completefill within a finite time horizon is strictly smaller than one. In principle atleast two alternative approaches would be able to incorporate execution riskinto a liquidation model.First, one may assume that the investor is forced to trade the remainingshares with a market order if the iceberg order fails to receive completeexecution. We call this setup the self-contained approach. Market ordersare executed immediately. They use hquidity from the book until they arecompletely filled. Consequently the investor has to bear a hquidity discount,so that he or she gets penalized for every share that could not be sold viathe iceberg order. However, in our opinion such a rigorous assumption may^The analysis for a purchasing strategy is symmetric.
82
The Navigation of an Iceberg
not always be justified in practice, especially if the remaining order volumeunder consideration cannot be absorbed by the market without a significantprice change.In this case, investors typically follow an adaptive strategy, i.e. they reviewtheir orders frequently and adjust them if the market moves away from theprespecified order limit. For this reason we also propose a different approachthat considers the execution probability as a boundary condition, i.e. onlythose combinations of peak size and limit are admissible that ensure a certainexecution probability within a prespecified time horizon. We call this modelthe open approach.Compared with the first one the latter framework is rather flexible and doesnot require any assumption concerning the liquidation of the unexecuted