to liquidate 50,000 MAN shareswithin one trading day (approximately 5-10% of daily turnover) and thateach transaction incurs fixed costs of €50.00. To solve the optimizationproblem, the sequential quadratic programming algorithm e04ucc of the NAGC Library was used. It is designed to minimize an arbitrary smooth functionsubject to simple bounds on the variables, linear constraints, and smooth
nonlinear constraints. To find the optimal sales trajectory, the following
problem was solved for each n = 1,...,n™*:max EG{71,A¢i}st. Ag, > 0Ad, = 9i=1T% > BA,To > 0n=lSn < Tm.i=02.6 Numerical Results 65
 
1,424,000
EG
1,423,000
1,422, 000
1,421,000
 
1,420,0000 20 40 60 n 80
Fig. 2.6: Expected liquidation revenues as a function of n for optimal 7; and A¢;,.
Then, the optimal n* that provides the highest value of the objective func-tion was identified. Figure 2.6 illustrates the result. The optimal sales tra-jectory comprises 23 intervals and 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 liquidated 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 linear constraint (2.10)controlling 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 liquidation value and if one observes an intradayU-shape pattern in liquidity discounts, one may expect an inverse U-shapepattern in the liquidation amounts. However, each transaction of the large66 2 Optimal Liquidation Strategies
Time of day
    
anon woWowSUSINSSSRVSSISHAOARSOSS:
46
0 2,000 4,000 6,000 8,000
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 liquidation 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 effect also, since the slope of the intraday liquidity curveis flat around noon.
To compare this example with other parameterizations, the duration D ofthe liquidation process is calculated. It is defined as
 
 
 
 
A AD= nt (nn) 8 tt (ty bet Ta) oe2 (4) Ad,= % i (2.11)E(2)4
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 respective2.6 Numerical Results 67
 
 
EGgq=20,000 EGeq-70, 000570, 000- on cpeceteentestratnateerenre | 71,992,000we fa: ta ~1,988, 000569,000-| ma. wa ~1, 984,000568, 000 1,980, 0000 10 20 30 40 n 50
Fig. 2.8: Expected liquidation revenues for ¢9 = 20,000 (left scale) and ¢p =70,000 (right scale) shares as a function of n, if 7% and Ad;, 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 liquidity is comparatively high. Later on, one can see that thedurations of the optimal liquidation strategies decline as the investor becomes
more risk averse.
In the next example the initial stock holdings ¢o are varied. Figure 2.8 showsthe results. As a rule of thumb one can state that the larger ¢, the more oftenthe investor should trade. In the case where ¢9 = 20,000 the portfolio shouldbe liquidated in five packages. If ¢9 = 70,000, the investor should trade 41times in order to realize the highest expected liquidation value. Comparedwith the first example, the duration of the liquidation process changes onlyslightly to D = 3h, 58min, 49sec for ¢9 = 20,000 and to D = 4h, 6min, 16secfor ¢9 = 70,000.68 2 Optimal Liquidation Strategies
 
 
570,500EG ee ee ka,a k=20
570,300}eR ey570,100} * * # , “eS2 ks50*¢ *569, 900 + *2x70,+569, 700 ’+569,5002 4 6 8 10 ««120—~Ct«d ans 16
Fig. 2.9: Expected liquidation revenues for k = 20, k = 50, and k = 70 asafunction of n, if 7 and Ag,, are optimally chosen (¢9 = 20,000).
The amount of fixed transaction costs influences the optimal number of trad-ing intervals as well. Figure 2.9 provides an insight into this issue. Assumefor this example that the initial stock holdings are ¢9 = 20,000. If fixedtransaction costs are increasing, the optimal number of intervals decreases.For k = 20, the optimal strategy consists of 9 intervals, for k = 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 liquidated in smaller packages if theaversion towards risk increases. An investor who trades more often reducesthe price risk of the liquidation process, since the probability 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, respec-tively. The impact of the parameter a on the liquidation strategy becomes
apparent in more detail if the duration of the liquidation process is investi-2.6 Numerical Results
69
 
 
 
1,425,900.“ 20.25EG-aSTD (G) pensnnen eeenennnensnca Tans net tetinas cesenasnaasosee t, a=0.5: sn esenbeees Tene nenerbesseeneneneneecnneseney”at1,415,000 aw‘‘ ”eni"1,410,000 “.'1,405,0000 20 40 n 60
Fig. 2.10: Values of the objective function for a = 0.0, 0.25, 0.5, and1.0 as afunction of n, if 7% and Ag;; are chosen optimally (¢o = 50,000). Thearrows point to the optimal number of liquidation intervals for a given
value of a.
04:00:00
02:00:00
01:00:00
00:00:00 -
 
0.0 0.5 1.0 a 2.0
Fig. 2.11: Duration of the liquidation process as a function of a, if 7 and Ad;,
are chosen optimally (¢9 = 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 earlier the portfolio should be sold.Two cost components reduce the expected liquidation value in this case.First, fixed transaction costs are rising, since the investor trades more of-ten. 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 reve-nues, 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 toliquidate ¢y = 50,000 MAN shares with k = 50. Thus, the value of the objec-tive 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, EG = 1,419,672.98, and STD(G) = 5,458.01.
Figure 2.12 shows the results. For the simulated liquidation value G the grayline shows the approximated density obtained by applying standard kernelestimation procedures. The black dots represent the estimated Gaussiandensity function for = 1,419,672.98 and o = 5,458.01. Obviously, thedifferences between the two distributions are negligible. The hypothesis thatthe two distributions are different is tested via the Kolmogorov-Smirnoy 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 of2.7 Conclusion and Issues for Further Research 71
0.00008 5“Density of simulated. “ distributionDensity aya * Points located on the, 4 density function of} ‘ N(p=1, 419, 672.98 o = 5, 458.01)0.00006 oeP|? \i0.00004 }Hié’j’:ii;:
0.90002:: \::
|01,380,000 1,400,000
 
1,420,000 1,440,000 g 1,460,000
Fig. 2.12: The gray line shows the approximated density of the simulated liquida-tion value for ¢9 = 50,000, k = 50, and a = 1.64. The black dots repre-
sent some values of the Gaussian density function for = 1,419,672.98and o = 5,458.01, the parameter values that were obtained for the
optimal 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 Research
This chapter operationalizes the concept of intraday market liquidity pat-terns in the determination of optimal liquidation strategies for large securityholdings. Innovative features of this setup include the explicit modeling ofan intraday U-shape pattern, the consideration of the resiliency of the or-der book through a time constraint in the objective function of the large72 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 liq-uidation 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 liquidity factor. Theimplementation of the model with real-world order book data shows that themodel is computationally tractable and able to provide intuitively reasonable
results.
A sensitivity analysis shows that if the number of initial stock holdings isincreased, the optimal number of liquidation 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, increas-ing liquidity discounts due to an earlier liquidation of the portfolio have tobe balanced against the risk of the liquidation value. One can observe a
decreasing duration of the liquidation process in this situation.
In contrast to the papers by Almgren and Chriss (2000), Hisata and Ya-mai (2000), and Dubil (2002) the price impact is not separated into a perma-nent and a temporary component. Instead it is assumed that the whole priceimpact is permanent and influences the liquidation revenues in the followingliquidation 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 ex-ample until the end of the trading day. This definition may be acceptable
for very large transactions. However, for medium-sized trades that occur2.8 Appendix 73
 
more frequently, further research toward a comprehensive method has to beconducted, so that one can separate overlapping temporary price impacts
resulting 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 liquidity model proposed by Esser and Ménch (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 S, given the new starting point after a jump downwards. Using(2.2) one can write S;, asSy = WSK= % Sy. exp [-1/207(t — ta) + o(Wa — Was)
since S;,_ is log-normally distributed. Iterating this procedure yields
il
St; M1 % Sip exp [—1/207[(ti — ta) +... + (tr ~ to) I]exp [o(Wi, — Wi, 1) +... + o(M, — W)]= 91--- i Sig XP [-1/207(t; - to) exp [o(Wi, - W,,.1)]...exp [o(Wi, — Wiy)] - (2.12)
This yields the following expectation value using the independenceof the increments of the Brownian motion and the fact thatEexp(oW,) = exp(0.50%t):
ES, = Yi... %Sto-74 2 Optimal Liquidation Strategies
 
Plugging this into (2.4) one getsEG = —-nk+ 52 A, 5: M1 +++ Vi Sto-t=1Using (2.3) the variance of G can be written and decomposed as follows:var(G) = var (> Adi, 5: s.)i=l
(Ady, 6:)° var (5t,)
1
n
n+2 Ade, 5; Adu, 5; cov (St, S1,)-i<jThus, all that remains is the computation of variances and covariances. Basedon the relationshipsvar(S;,) = ES? —(ES,,)°cov(Si,,5i,) = E(S:,5:,) — (ES:,)(ESi,)it suffices to compute ES? and E(S;,5;,). Based on (2.12), the second mo-ment can be written asSho= yp...9P S? exp [-07(t; — to)] exp [20(W, — Wi,_,)]...exp [20(W;, -— W,,)]-By using Eexp(2cW,) = exp(207t) one getsES; = ¥...77 Si, exp [07(ti - to)] ,implyingvar(Si.) = 9... Sf, (exp [07(t: — to)] — 1).The covariance terms can be calculated in an analogous fashion. SinceS52, = Se : areas 2 Wexp[—1/20°(t; — to)] exp[—1/20°(t; — to)]exp[20(W,, — W;,_,)]...exp[2o(Wi, — Wi,)]|
1).exp[o(W,, 7 Wi;1) tee expl[o(Wi,,, _ W,,)]References 75
 
one obtains
E(S:,51;) = Si: : Ws He WGexp[—1/20°(t, — to)] exp[—1/20(t; — to)]exp[20?(t; — to)] exp[1/207(t, — ta)]= Srvt...-yie.-%y explo(ti — to).
So, the covariance can be written as
cov(Si,S1;) = VE. 2 Fer 14 Sz, (exp [o?(ti — to)] — 1).
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3.1 Introduction
The rapid development in technology over the last couple of years has per-mitted 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 con-tinuous trading in real time. Usually the limits, the accumulated order vol-umes 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 mar-ket 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 limit 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 IOB), 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 adverse
informational impact of disclosing the actual order volume. However, the3.1 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 astrategy 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 is
unlikely 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 optimization
problem.
Unless an iceberg sell order is immediately executable, i.e. the limit is so lowthat it is actually a market sell order, the probability 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 risk
into 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 liquidity from the book until they arecompletely filled. Consequently the investor has to bear a liquidity discount,so that he or she gets penalized for every share that could not be sold via
the iceberg order. However, in our opinion such a rigorous assumption may
1The 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 significant
price change.
In this case, investors typically follow an adapt