exp(s) So g (sI max [In(S,/S0)] < In(5/S)), = 0) ds.
The general setup of the alternative approaches is summarized in the follow-
ing two propositions:
Proposition 3.1 The open approach to determine the optimal combina-tion of the peak size and the limit of an iceberg order can be represented bythe following optimization problem:
max Ss(ny doTs.t. Pe < ff fons (O) dt0So < §bp < bo
where P* is given exogenously.
5For example, imsl_d_int_fen_sing from the IMSL C-Library is such a routine.3.4 The Basic Setup 101
 
Proposition 3.2 The self-contained approach to determine the optimalcombination of the peak size and the limit of an iceberg order can be repre-sented by the following optimization problem:
max EG{#5}s.t. S < $bp < do;where EG is given byEG = )>PM=n=0
{h (n) 8 + [60 —h(n)][1 = ¥ (Go ~ h(n))] E (S| M =n) }= fa t) i do - [1 — U(bo)] E(Sr| M = 0)
Lf fon(t)dt — ob eo at] x
h(n) 5 + [do — b(n) {1 — W [do ~ h (n)]}E (S| M = n) }
jv
and E(Sr| M = 0) and E(S7|M =n) forn > 1 are given by
+
oo
E(Sr|M =0)In(/S0) _= / exp(s) Sog (s max [In(Su/So)] < In($/5o), n= 0) dsE(S7|M =n)
T fon(t )Q-" Fransi(s )ds)LOMA lo tL xi fon(u Jdu— fo fon+i(u) duIn(S/Stn)/ exp(s) Sig (s max |In(S,/S.)) < n(S/S)), n) as| dt.
—0°0102 The Navigation of an Iceberg
 
3.5 Modeling of the Drift Component
Up to now the drift of the best bid price has been assumed to be a constant.This section completes the theoretical framework by modeling explicitly theimpact of the peak size on the drift following the intuition that the disclosureof large order volumes has an adverse market impact. For this purpose we
will model the drift 4; as a function of the order imbalance B;.
Similar to Brown (1997) we define the imbalance B; of the order book asthe number of shares displayed on the bid side divided by the sum of sharesdisplayed on the bid side and the ask side. The imbalance coefficient isbounded by 1 (if no orders are stored on the ask side of the book) and by 0(if the bid side is empty). The parameter is 0.5 if the ask volume equals thebid volume. Whenever a new peak shows up in the order book the displayedask volume increases, which in turn reduces B,.
To keep the setup tractable for exposition, we assume the following simplifiedscenario: The best bid price exhibits a zero drift u, = —& = 0 prior to the
submission of the iceberg order (t < tg). Furthermore, suppose that B, = Bfor t < to.
As soon as the iceberg order is submitted to the market the symmetry ofthe order book starts varying. Suppose that each variation in the displayedvolume of the iceberg order ¢qp influences the order book symmetry. Thedisplayed volume of other orders remains constant over time. Thus, we areable to model the imbalance as a function of the displayed volume of theiceberg order only:
Bi (bap) = (3.8)
—£_d+ bap’where the parameters c (and d) denote the number of shares displayed onthe bid side (on the bid side and the ask side) before the submission of the
iceberg order.
The displayed volume ¢ap is equal to ¢p whenever a new peak is submittedto the order book. When the peak of the iceberg order receives a complete3.5 Modeling of the Drift Component 103
 
or partial fill the parameter ¢gp will be reduced. In our setup the displayedvolume ¢gp depends on the number of times the limit was already hit. It canbe calculated as
ap (nr)
min [40 — h(n),= Oe ~ {max [(at2s5 ~ | s%; |) (Ga + Gp) — a;0| }] ifn <n*
 
 
0 else.
If the displayed volume of submitted orders has some information content tothe market, one would expect a positive relationship between past levels ofB, and future returns. We define yu, by the recursion
Pvt = Hn t+ B+ (Binz, — Bi,) forn > 0with the initial valueMo = f+8- (By ~- 8B),such thatAun = BAB:,- (3.9)
Figure 3.5 illustrates this idea. Before time to the drift 4, is equal to thelong-term mean ji. At time to the first peak of the iceberg order appears inthe book and the drift y, is reduced. At times t3, ty, ts, and tg the firstpeak of the iceberg order receives partial fills, which goes along with smallupward jumps in the drift y,. At time ty the first peak becomes completelyfilled and the second peak appears in the order book, which again causes adownward jump in the drift y,. At time t* the iceberg order is completelyfilled and the drift 4, reverts towards its long-term mean ji.104 The Navigation of an Iceberg
 
 
 
to tz ty tste tr tio tii ti2 tig the
Fig. 3.5: Example for the alternating drift component in our model.
Note that py, is a deterministic function of the random variable n in ourframework. Thus, we can rewrite equation (3.4) for the case where the driftdepends on the displayed peak size:
for (t) = m/e) xa 2exp { — [In (5/55) = (us — 02/2) ] \ (3.10)
For the subsequent hitting times after the restart we get
exp { —[-In(1 =e) = (tna - 0/2) aa - (3.11)
2o0°t
This completes the introduction of the theoretical framework. We now turn
to the numerical implementation of the open approach.3.6 Numerical Results 105
 
3.6 Numerical Results
To exemplify the formal analysis of the previous section the open approachis implemented using the MAN dataset for 61 trading days. The results arepresented in the following.
3.6.1 Parameter Specification
Table 3.3: Estimated parameters for the MAN dataset for 61 trading days.
 
No. of obs. Mean| Std. dev.
 
@;\Average volume of all 40,888 868.2 1,257.4
transactionsga| Average displayed vol- 158,607 1,554.4 1,752.0ume of all best ask quotese | Average relative price dif- 57,290} 0.00090724 |0.00105471ference between the bestand the second best bidprice
c | Average number of shares 36,661} 166,465.87} 57,495.87displayed on the bid sideof the book
d | Average number of shares 36,661] 291,657.18] 84,722.34displayed on the bid sideand on the ask side of thebook
e
 
 
 
 
 
 
 
To implement the model a number of parameters need to be calibrated withorder book data. Table 3.3 summarizes the results for our clinical sample. Toestimate the parameters ¢,, ¢,, and € we consider all observed transactions,106 The Navigation of an Iceberg
 
and best and second best ask quotes with equal weights. For the calibrationof the parameters c and d we use the order book data collected at intervalsof 1 minute from 9:30 a.m. to 7:30 p.m. To estimate the parameter 8 weregress 60-minutes-ahead returns on changes of the order imbalance duringthe past 60 minutes, minute by minute. For this purpose we use best bidquotes and order book data collected at intervals of 1 minute from 9:30 a.m.
to 7:30 p.m. We do not consider overnight returns for our analysis.
Table 3.4 reports the results.
Table 3.4: Estimated regression coefficient for equation (3.9).
No. of obs.| Estimate| t-statistic
6 |60-minute fore- 29,341] 116.647882 22.45cast intervals
 
 
 
 
 
 
For the calibration of the volatility parameter we use best bid quotes collectedat intervals of 15 minutes from 9:00 a.m. to 8:00 p.m. and do not considerovernight price changes. The estimation for the volatility parameter yieldso = 0.7. Furthermore, we set Sg = € 28.55, which is the closing price atMarch 28, 2002, the last day of our sample period.
3.6.2 Numerical Implementation
The computation of fo,,(t) requires the calculation of an n-th iterated con-volution given by equation (3.6). In order to obtain fo,,(¢) one needs tocalculate (n — 1)-dimensional integrals. To the best of our knowledge, closedform expressions are not available. Thus, we apply numerical approxima-tions to these integrals. Employing conventional quadrature algorithms orMonte Carlo methods to compute high-dimensional integrals is very time
consuming and thus not suitable for the dimensions under consideration in3.6 Numerical Results 107
 
our framework. Therefore, we use interpolating cubic splines s,, for n > 2to approximate the convolutions in the following way:
fon (t) © Son (t) ,
such that
Ey . .son(ée) = [ sanaa(Ee —U) faaya(ts) de® fom(de),
where f,, denotes the equally spaced spline knots.
Alternatively, one can also invert the Laplace transform of the density func-tion.® The Laplace transform for the (defective) density function of the firsthitting time 7, is well known (see, e.g., Karlin and Taylor, p. 362), and isgiven by
in(éEexp(—An) = ex — Bwsro) 6/5) x
2\2 2(3) 7 (m-8)]}
As the following sequences of hitting times 7; for 7 > 2 are identically dis-
tributed their Laplace transform is given by
_in[s/S(—2)]
o2
o\? o(1 - 5) + 20? — (4 - =) \
The Laplace transform of the sum of the independent hitting times
Bexp(-An) = eof
th = 17 is equal to the product of the corresponding exponentialfunctions:In (S/SEBexp(—Xt,) = emp { {Sf x
SNumerical routines that do this job pretty fast are, for example,imsl_d_inverse_laplace from the IMSL C-Library or CO6LAF/CO6LBF from theNag Fortran-Library.108 The Navigation of an Iceberg
[v/(e0 ~ 92/2)? +202 ~ (tig ~ 0° /2)|“ -(= vag?) xVo (uaa — 02/2)? 4 202d — (ur 0?/2)]
3.6.3 Numerical Examples
at
For the first example, suppose that the investor wants to liquidate 10,000MAN shares (approximately 1-2% of daily turnover) within 10 hours. As-sume, furthermore, that ¢, has to be a multiple of 1,000 shares.
§ in Euro30.00
29,7529.5029.2529.00
28.75
 
$o=28.55
 
Fig. 3.6: Optimal limit § as a function of the probability P* that the iceberg orderreceives a complete fill. Other parameters: T = 10 hours, ¢9 = 10,000MAN shares.
Figure 3.6 represents the optimal limit S as a function of the probability P*that the iceberg order receives a complete fill. For P* < 59%, the optimallimit is a monotonic decreasing function of P*. The optimal peak size remainsat a constant level of 8,000 shares and is thus insensitive to changes of P*.Smaller or higher peak sizes reduce the value of the objective function, for3.6 Numerical Results 109
 
§ in Euro
29.50
29,25
29.00
28.75
S=28.55
 
Tinh© 10 20 36 40 50 60 70 80 90 100
Fig. 3.7: Optimal limit § as a function of the final time horizon T. Other para-meters: P* = 50%, oo = 10,000 MAN shares.
example by approx. 1% if P* = 30% and ¢, = 1,000 shares. If the limit is setto € 28.56, i.e. the smallest possible value in this example, the probabilityto observe a complete execution is still less than 60%.
The optimal peak size is significantly higher than peak sizes that were ob-served empirically in Section 3.3. Two reasons may explain the difference.First, one may argue that the model systematically underestimates the nega-tive price impact of displaying a large order volume in the book. There aregood reasons to believe that a variation of the order imbalance within en-tries close to the best quotes has a stronger impact on future returns thanchanges of the order imbalance caused by an entry of an order that possessesa more unfavorable price priority than the majority of other orders alreadystored in the book. A redefinition of the order imbalance by weighting or-der book entries differently, depending on their price priority, might solvethis problem. Second, one may argue that market participants overestimatethe informational impact of revealing large orders in an open book. Theempirical exploration of these issues is left for further research.110 The Navigation of an Iceberg
 
8,000 seeeensaerronenanevenenecnseenteooseseas
7,000
6,000
5,000
4,000 “sesveensconsnsennescnenonensaneroseesesnesenseneesen3,000
2,000
1,000
0 Tinh0 160 20 30 40 «5006 66006700680) 90 200
 
Fig. 3.8: Optimal peak size ¢, as a function of the final time horizon T. Otherparameters: P* = 50%, ¢9 = 10,000 MAN shares.
In the next example we investigate the relationship between the final timehorizon T and the optimal combination of limit and peak size. We setP* = 50% and ¢p = 10,000 shares. Figures 3.7 and 3.8 report the re-sults. If T < 6 hours, then the probability of receiving a complete fill is lessthan 50%, no matter which limit is assigned to the order. If T > 6 hourswe can observe two beneficial effects for the originator of the iceberg order.First, as the final time horizon increases, the optimal order limit increases aswell. Second, a longer time horizon allows for a reduction of the peak size.However, ¢, is not strictly monotonic decreasing in T. Instead we observe astep function. For T < 46 hours a peak size of 8,000 shares is optimal, forT > 46 the optimal peak size is 4,000 shares.
In the last example (see Figures 3.9 and 3.10) we analyze the relationshipbetween the initial position ¢ 9 and the optimal pairs of ¢, and S. We setP* = 25% and T = 100 hours. Figure 3.9 corroborates the hypothesisthat if more shares have to be liquidated within the same period of timethe limit has to be lowered to keep the execution probability at the samelevel. Furthermore, an increase in ¢ tends to result in higher peak sizes,3.6 Numerical Results lll
 
S in Euro30.5030.30.29.29.29.29.
28.S=28.
 
 
10,000 20,000 30,000 40,000 50,000 60,000
 
Fig. 3.9: Optimal limit § as a function of ¢9. Other parameters: P* = 25%,T = 100 hours.
be11,000 .
10,000 ‘ .3,000 . ‘
8, 000 ob nbeeesennes ce eee 67,000 . .
6,000 eee ee eee
5,000 wena
4,000 fe evee ee
3,000} «
2,0001,000
 
 
010,000 20,000 30,000 40,006 50,000 60,000
Fig. 3.10: Optimal peak size ¢p as a function of ¢ 9. Other parameters:P* = 25%, T = 100 hours.
as we can observe in Figure 3.10. However, in some cases the optimal peaksize decreases if the initial position is raised. At the first moment this may
seem somehow counterintuitive. The main reason for this phenomenon can112 The Navigation of an Iceberg
 
be found in the discrete setup of the order execution process. Wheneverthe limit S is hit, a fixed transaction size ¢, is processed. At the n*-th hitthe last part of the iceberg order, which is given by ¢9 — h(n* — 1), becomesexecutable. However, if ¢—h(n*—1) « ¢, a small reduction in ¢, would notchange n* but would increase the drift component js; and thus the probabilitythat t, <T.
3.7 Summary and Conclusion
This chapter introduces a setup that allows the determination of the optimalcombination of limit and peak size of an iceberg order, given a large positionin a security that should be liquidated within a finite time horizon. Theframework balances the direct advantage of a large peak size that leads to abetter time priority of an iceberg order and the adverse informational impactof revealing large order volumes in an open order book. Furthermore, itassesses the tradeoff between the order limit and the execution probabilityof the iceberg order. We have presented two approaches to incorporate theexecution risk of an iceberg order. The so-called self-contained approachassumes that the unexecuted part is liquidated by a market order. Theopen approach is far more flexible as it does not require any assumptionconcerning the liquidation of the unexecuted part. It identifies the optimalcombination of limit and peak size, given a minimum probability of completeorder execution. Using real-world order book data we illustrate how the openapproach can be implemented and explore major properties of the model by
modifying input parameters.
To our knowledge, this framework is the first analytical approach that inves-tigates the tradeoff between limit and peak size of an iceberg order, on theone hand, and the resulting execution probability, on the other. This chapteris written in search of a stylized model that is able to illustrate the interac-tion between observable market variables and order specific parameters that
are important to analyze iceberg orders as a trading instrument.3.8 Appendix 113
 
The modeling of the best bid price by a Brownian motion or the assump-tion of constant parameters for order imbalance, transaction size, order flowand the price difference between the best and the second best price are,of course, approximations as the standard deviations in Table 3.3 clearlyindicate. These simplifications allow us to keep the number of stochasticvariables to the minimum required to illustrate the discussed trade-off in asimple way. Further research may focus on introducing more freedom fromdeterminism by modeling more sources of risk, for example, in a simulation-based approach and comparing the empirical performance of the differentmodels. Furthermore, although certainly challenging from a technical pointof view, the investigation of dynamic approaches seems highly relevant froman empirical perspective, since many market participants pursue dynamic
instead of static limit-setting strategies as shown in Table 3.1 in Section 3.3.
3.8 Appendix: Conditional Density g of In (S7/S;,)
The conditional density g of In (S7-/S;,) is given by
g (s},max In(S./S.,) < In(5/S,,), 2)We (s,In (S/S:,))
P (1s (28x, S. / Su) <In (3/5..))
whereUe(e,y) = (A/o)exp | (u" - 02/2) 2/0?~ (ur — 07/2) (T = ty) /20” | 5(x/o,y/o),6(e,y) = [v(2@-t)”)-9((@-w)r-ty?) ] T= ty?,and
P (1 (1023, 5/5.) < In(S/S,,) )114 The Navigation of an Iceberg
 
a (f= tn)—exp [2 (u" — 0? /2) In (8/5:,) /o?] x5 (- In (8/51) — (wt — 0? /2) (T — “))oV(T — tn) ,
where ¢ (z) denotes the standard normal density function and © (z) the stan-
b (" (5/5) — (ut — 07/2) (T - ))
dard normal cumulative distribution function. If the drift is a constant, asassumed in Section 3.4, set w* = p. If the drift is modeled as a time-dependent variable, as proposed in Section 3.5, replace u* by wz. For the
derivation of the respective formulas, see, e.g., Harrison (1990), pp. 1-16.
References
Aitken, M. J., P. Brown, and T. Walter (1996): Infrequent trading and firmsize as explanation for the intra-day patterns in returns on SEATS, WorkingPaper, Securities Industry Research Centre of Asia-Pacific.
Aitken, M. J., H. Berkman, and D. Mak (2001): The use of undisclosed limitorders on the Australian Stock Exchange, Journal of Banking & Finance 25,1589-603.
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.
Brown, P., D. Walsh, and A. Yuen (1997): The interaction between orderimbalance and stock price, Pacific-Basin Finance Journal 5, 539-57.
Cho, J. W. and E. Nelling (2000): The probability of limit-order execution,Association for Investment Management and Research, September /October,28-33.References 115
 
Chordia, T. and A. Subrahmanyam (2002): Order imbalance and individualstock returns, Working Paper, Emory University, Atlanta, USA.
Copeland, T. E. and D. Galai (1983): Information effects on the bid-askspread, Journal of Finance 38, 1457-69.
D’Hondt, C., R. De Winne, and A. Francois-Heude (2003): Hidden orders onEuronext: Nothing is quite as it seems ..., Working Paper, FUCaM-CatholicUniversity of Mons, Belgium.
Dubil, R. (2002): Optimal liquidation of large security holdings in thinmarkets, Working Paper, University of Connecticut, Storrs, USA.
Harrison, M. J. (1990): Brownian Motion and Stochastic Flow Systems,Krieger Publishing Company, Malabar, Florida, USA.
Hisata, Y. and Y. Yamai (2000): Research toward the practical applica-tion of liquidity risk evaluation methods, Monetary and Economic Studies,December, 83-128.
Karlin, S. and H. M. Taylor (1975): A First Course in Stochastic Processes,Academic Press, San Diego.
Lo, A. W., A. C. MacKinlay, and J. Zhang (2002): Econometric models oflimit-order execution, Journal of Financial Economics 65, 31-71.
Ménch, B. (2003): Optimal liquidation strategies, Working Paper, GoetheUniversity, Frankfurt am Main, Germany.
Pardo, A. and R. Pascual (2003): On the hidden side of liquidity, WorkingPaper, Universidad de las Islas Baleares, Palma de Malorca, Spain.
Parlour, C. A. (1998): Price dynamics in limit order markets, Review ofFinancial Studies 11, 789-816.
Ranaldo, A. (2004): Order aggressiveness in limit order markets, Journal ofFinancial Markets 7, 53-74.116 The Navigation of an Iceberg
Wald, J. K. and H. T. Horrigan (2001): Optimal limit order choice, WorkingPaper, forthcoming 2006 in Journal of Business 79.
Zhang, P. G. (1998): Exotic Options, World Scientific Publishing, Singapore.Lecture Notes in Economicsand Mathematical Systems
For information about Vols. 1-459
please contact your bookseller or Springer-Verlag
Vol. 460: B. Fleischmann, J. A. E. B. van Nunen, M. G.Speranza, P, Stahly, Advances in Distribution Logistic. XI,535 pages. 1998.
Vol. 461: U. Schmidt, Axiomatic Utility Theory under Risk.XV, 201 pages. 1998.
Vol. 462; L. von Auer, Dynamic Preferences, Choice Mecha-nisms, and Welfare. XH, 226 pages. 1998,
Vol. 463: G. Abraham-Frois (Ed.), Non-Linear Dynamies andEndogenous Cycles. VI, 204 pages. 1998.
Vol. 464: A, Aulin, The Impact of Science on Economic Growthand its Cycles. IX, 204 pages. 1998.
Vol. 465: T, J, Stewart, R. C, van den Honert (Eds.), Trends inMalticriteria Decision Making. X, 448 pages. 1998,
Vol. 466: A. Sadrieh, The Altemating Double Auction Market.V1i, 330 pages. 1998.
Vol, 467: H. Hennig-Schmidt, Bargaining in a Video Ex-periment. Determinants of Boundedly Rational Behavior.XH, 221 pages. 1999.
Vol. 468: A. Ziegler, A Game Theory Analysis of Options. XIV,145 pages. 1999,
Vol. 469: M. P. Vogel, Environmental Kuznets Curves. XUE,197 pages. 1999.
Vol. 470; M. Ammann, Pricing Derivative Credit Risk. XII,228 pages. 1999.
Vol. 471: N. H. M, Wilson (Ed.), Computer-Aided TransitScheduling, XI, 444 pages, 1999,
Vol. 472: J.-R. Tyran, Money Hlusion and Strategic Com-plementarity as Causes of Monetary Non-Neutrality.X, 228 pages, 1999,
Vol. 473: S. Helber, Performance Analysis of Flow Lines withNon-Linear Flow of Material. IX, 280 pages. 1999,
Vol. 474: U. Schwalbe, The Core of Economies with AsymmetricInformation. IX, 141 pages. 1999.
Vo}. 475: L. Kaas, Dynamic Macroeconomics with ImperfectCompetition. XI, 155 pages. 1999.
Vol. 476; R. Demel, Fiscal Policy, Public Debt and theTerm Structure of Interest Rates. X, 279 pages. 1999,
Vol. 477; M. Théra, R. Tichatschke (Eds.), [l]-posed VariationalProblems and Regularization Techniques. VIII, 274 pages.1999,
Vol. 478: S. Hartmann, Project Scheduling under LimitedResources. XII, 221 pages, 1999.
Vol. 479: L. v. Thadden, Money, jnflation, and CapitalFormation. IX, 192 pages. 1999.
Vol, 480; M, Grazia Speranza, P. Stilly (Hds.), New Trendsin Distribution Logistics. X, 336 pages. 1999.
Vol. 481: V. H. Nguyen, J. J. Strodiot, P. Tossings (Eds.).Optimation. IX, 498 pages. 2000.
 
Vol. 482: W. B. Zhang, A Theory of International Trade. XI,192 pages. 2000.
Vol. 483: M. Kénigstein, Equity, Efficiency and EvolutionaryStability in Bargaining Games with Joint Production. XI,197 pages. 2000.
Vol. 484: D. D. Gatti, M. Gallegati, A. Kirman, Interaction andMarket Structure. VI, 298 pages. 2000.
Vol. 485: A. Gamaev, Search Games and Other Applications ofGame Theory, VII, 145 pages. 2000.
Vol. 486; M. Neugart, Nonlinear Labor Market Dynamics. X,175 pages. 2000.
Vol. 487: Y, Y. Haimes, R. B, Steuer (Eds.), Research andPractice in Multiple Criteria Decision Making. XVII,553 pages, 2000.
Vol. 488: B. Schmolck, Ommitted Variable Tests and DynamicSpecification. X, 144 pages. 2000
Vol. 489: T. Steger, Transitional Dynamics and EconomicGrowth in Developing Countries, Vill, 151 pages, 2000.
Vol. 490; S, Minner, Strategic Safety Stocks in Supply Chains.