)dt
>T-t)
P((^n<T)A(^n+l>T))/ o , n ( 0 ( l - IQ~^ fn,n+l{s)dsj
dt
JQ /0,n(^) du - J^ / o , n + l ( ^ ) du
Armed with this result we are able to write the conditional expectation ofST
as'E{ST\M
= n)
= J r^^{t)Et(^r ^ ^ ^ ( O E , ( 5 T | max {Su) < S ] dt
100
The Navigation of an Iceberg
I
T /o,nW(l-/o^
Vn,n+l(5)d5)
0 /o / o , n ( ^ ) d ^ - / o /o,n+l(^)d^/j
exp(5) St X
g { 5| m^ax^[ln(5J5,)] < ln(5/5,), n ) ds dt.
(3.7)
Note that the integral with respect to t in equation (3.7) has a singularity atthe upper end point of the integration range. Thus, for numerical integrationone should use a quadrature routine that can handle functions with end-pointsingularities.^Conditional on the event that the limit is not hit before T, the conditionalexpectation of ST simplifies toE ( S ' r | M = 0)n\n{S/So)
=
/
/
_
\
exp(5) ^ 0 ^ 5| max [ln(5^/S'o)] < ln(S'/5o), n = 0 ds.
J-oo
\
0<^<^
/
The general setup of the alternative approaches is summarized in the following 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
00 S
{4>j>.s]T
S.t
P*
<
//o,n*(t)dt0_
So
<
S
where P* is given exogenously.^For 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 optimal
combination of the peak size and the limit of an iceberg order can be represented by the following optimization problem:max
EG
s.t.
So
<
S
where EG is given byoo
EG = J^P(M = n)xn=0
{/i (n) S+[(l)o-hT
(n)] [1 - ^ {<Po - h (n))] E {ST\ M = n)}1
• - J fo,i{t)dt\<^o • [1 - *(<^o)] E{ST\ M = 0)0J'^*-l r pT
+ Y,
pT
/ kn(t)At- j
[h{n)S-^[(l)o-h
1
/o,n+lWdd X
(n)] {1 - * [00 - /^ (n)]}E ( 5 T | M = n) }
T
)dt
+
(poS,
and E {ST\ M = 0) and E {ST\ M = n) for n>lE{ST\M
are given by
= 0)
Hs/So)
I
exp(s) Sog[s\
m ^ [ln(S'„/S'o)] < ln(5/S'o), n = 0 ) ds
E {ST\ M = n)T /o,n(i) ( l - JQ~* /„,„+! (s)dsj
L0
/o /0,n('?^)dii-/Q
/o,n+l(^)dti
ln(5/5t,)
/
exp(s) 5(5 ( si m^J,\n{SJSt)]
< HS/St),
n ) ds
At.
102
The Navigation of an Iceberg
3.5 Modeling of the Drift ComponentUp 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 wewill model the drift fit ^s a function of the order imbalance Bt.Similar to Brown (1997) we define the imbalance Bt 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 Bt.To keep the setup tractable for exposition, we assume the following simplifiedscenario: The best bid price exhibits a zero drift //^ = /i = 0 prior to thesubmission of the iceberg order {t < to). Furthermore, suppose that Bt = 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 (pdp 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:
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 theiceberg order.The displayed volume (pdp is equal to (f)p whenever a new peak is submittedto the order book. When the peak of the iceberg order receives a complete
3.5 Modeling of the Drift Component
103
or partial fill the parameter (l)dp will be reduced. In our setup the displayedvolume (pdp depends on the number of times the limit was already hit. It canbe calculated as,(n)
I min 00 — /i(n),
^p - {^^^ [{A - [A\) (^«+^p) - ^- o]} if n < 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 ofBt and future returns. We define /i„ by the recursionA^n+i =
IJ^n-^P' {Btn+i - Bt^) forn > 0
with the initial value/^o
/2 + / 3 . ( 5 , , - 5 )
such thatA/x„ - f3ABt,
(3.9)
Figure 3.5 illustrates this idea. Before time to the drift fit is equal to thelong-term mean pt. At time to the first peak of the iceberg order appears inthe book and the drift fit is reduced. At times ^3, ^4, ^5, and tQ the firstpeak of the iceberg order receives partial fills, which goes along with smallupward jumps in the drift fif At time t^ the first peak becomes completelyfilled and the second peak appears in the order book, which again causes adownward jump in the drift fif At time t* the iceberg order is completelyfilled and the drift fit reverts towards its long-term mean p,.
104
The Navigation of an Iceberg
^t
to
t3
t4
tste t7
tio til ti2
ti3 tn*
Fig. 3.5: Example for the alternating drift component in our model.
Note that /in 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:
/o,i (t)
In (S/So)
[ln(5/5o)-(Mo-aV2)^]^exp •
(3.10)
For the subsequent hitting times after the restart we getfn-l,n \i)
-ln(l-e)Ty/2^
exp
[-ln(l-£)-(/i„-i-aV2)tf2aH
(3.11)
This completes the introduction of the theoretical framework. We now turnto the numerical implementation of the open approach.
3.6 Numerical Results
105
3.6 Numerical ResultsTo 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.
(t>s Average volume of alltransactions
40,888
868.2
1,257.4
(pa Average displayed vol-
158,607
1,554.4
1,752.0
ume of all best ask quotes
e Average relative price dif-
57,290 0.00090724 0.00105471
ference between the bestand the second best bidpricec Average number of shares
36,661 166,465.87
57,495.87
36,661 291,657.18
84,722.34
displayed on the bid sideof the bookd Average number of sharesdisplayed on the bid sideand on the ask side of thebook
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 (/>5, (/J^, and e we consider all observed transactions,
The Navigation of an Iceberg
106
and best and second best ask quotes with equal weights. For the cahbrationof 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 P 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.(3 60-minute forecast intervals
Estimate t-statistic
29,341 116.647882
22.45
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 yieldsa = 0.7. Furthermore, we set 5*0 = €28.55, which is the closing price atMarch 28, 2002, the last day of our sample period.
3.6.2 Numerical ImplementationThe computation of /o,n(0 requires the calculation of an n-th iterated convolution given by equation (3.6). In order to obtain /o,n(0 ^^^ needs tocalculate (n — l)-dimensional integrals. To the best of our knowledge, closedform expressions are not available. Thus, we apply numerical approximations to these integrals. Employing conventional quadrature algorithms orMonte Carlo methods to compute high-dimensional integrals is very timeconsuming and thus not suitable for the dimensions under consideration in
3.6 Numerical Results
107
our framework. Therefore, we use interpolating cubic splines 5o,n for n > 2to approximate the convolutions in the following way:/o,n (t) ^ «5o,n (t) ,
such thatSo, n ( 4 ) =
/
5 o , n - l ( 4 - U)fn-l,n{u)
Jo
dw « / o , n ( 4 ) ,
where tk denotes the equally spaced spline knots.Alternatively, one can also invert the Laplace transform of the density function.^ The Laplace transform for the (defective) density function of the firsthitting time ri is well known (see, e.g., Karlin and Taylor, p. 362), and isgiven byE exp (—ATI )
=
exp
r i Inn (5/5o)^
M o - y j +2a2A- (/^o - yAs the following sequences of hitting times r^ for i > 2 are identically distributed their Laplace transform is given byEexp(—Ar^)
=
exp
{-
ln[5/5(l-e)]Mi-i - Y I + 2<7^-^
( " ' - - T ) }•
The Laplace transform of the sum of the independent hitting timestn
=
YA=I ^i is equal to the product of the corresponding exponential
functions:Eexp(—A^n)
=
exp
{-
ln(5/5o)
^Numerical routines that do this job pretty fast are, for example,imsl_d_inverse_laplace from the IMSL C-Library or C06LAF/C06LBF from theNag Fortran-Library.
The Navigation of an Iceberg
108
y ^ r ^ Y 2 f ^ ^ 2 ^ - (MO - ^2/2)^
/-ln(l-£)
i=2
l^i^i - G-^jif + 2a2A - (//,_i - ^ 7 2 )
}
3.6.3 Numerical ExamplesFor the first example, suppose that the investor wants to liquidate 10,000MAN shares (approximately 1-2% of daily turnover) within 10 hours. Assume, furthermore, that (/)p has to be a multiple of 1,000 shares.Euro
1.0
Fig. 3.6: Optimal limit 5 as a function of the probability P* that the iceberg orderreceives a complete fill. Other parameters: T = 10 hours, 0o = 10,000MAN shares.Figure 3.6 represents the optimal hmit 5 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, for
3.6 Numerical Results
109
S in Euro29.50
29.25
29.00
28.75So=28.550
10
20
30
40
50
60
70
80
90
T in h100
Fig. 3.7: Optimal limit 5 as a function of the final time horizon T. Other parameters: P* = 50%, 00 = 10,000 MAN shares.
example by approx. 1% if P* = 30% and (j)p = 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 observed empirically in Section 3.3. Two reasons may explain the difference.First, one may argue that the model systematically underestimates the negative price impact of displaying a large order volume in the book. There aregood reasons to believe that a variation of the order imbalance within entries 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 order 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.
The Navigation of an Iceberg
no
8,0007,0006,0005,000 \4,0003,0002,0001,0000
T in h10
20
30
40
50
60
70
90
100
Fig. 3.8: Optimal peak size (/)p as a function of the final time horizon T. Otherparameters: P* = 50%, (t)o = 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.P*
=
We set
50% and (/)o = 10,000 shares. Figures 3.7 and 3.8 report the re-
sults, If T < 6 hours, then the probabihty of receiving a complete fill is lessthan 50%, no matter which hmit 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 hmit increases aswell. Second, a longer time horizon allows for a reduction of the peak size.However, (j)p 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 0o and the optimal pairs of 0^ and S. We setp* = 25% and T = 100 hours.
Figure 3.9 corroborates the hypothesis
that if more shares have to be hquidated within the same period of timethe limit has to be lowered to keep the execution probability at the samelevel Furthermore, an increase in (/)o tends to result in higher peak sizes,
3.6 Numerical Results
111
Euro
000
20,000
30,000
40,000
50,000
60,000
Fig. 3.9; Optimal limit 5 as a function of (pQ. Other parameters: P* = 25%,T = 100 hours.
11, 00010, 0009, 0008, 000
f t
• •
•
t
7, 0006, 0005, 0004, 000
t f 11
•
3,0002, 0001,000010,000
20,000
30,000
40,000
50,000
Fig. 3.10: Optimal peak size (t)p as a function ofP* = 25%, T = 100 hours.
60,000
Other parameters:
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 mayseem somehow counterintuitive. The main reason for this phenomenon can
112
The Navigation of an Iceberg
be found in the discrete setup of the order execution process. Wheneverthe hmit S is hit, a fixed transaction size ^s is processed. At the n*-th hitthe last part of the iceberg order, which is given by (/)o — h{n* — 1), becomesexecutable. However, if 0o — h{rC — 1) <^ 0^ a small reduction in 0^ would notchange n* but would increase the drift component /Xt and thus the probabilitythat in < T.
3.7 Summary and ConclusionThis 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 hquidated 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 approach
assumes that the unexecuted part is liquidated by a market order.
The
open 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 probabihty of completeorder execution. Using real-world order book data we illustrate how the openapproach can be implemented and explore major properties of the model bymodifying input parameters.To our knowledge, this framework is the first analytical approach that investigates 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 interaction between observable market variables and order specific parameters thatare 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 assumption 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 simulationbased 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 dynamicinstead of static limit-setting strategies as shown in Table 3.1 in Section 3.3.
3.8 Appendix: Conditional Density g of In {ST/StJThe conditional density g of In {Sr/Str,) is given byg(s\\
max \n{SJ St J
<
ln{S/StJ,n)
tn<U<T
/
^(Jt {s, InP (In (^ma^^SJ
{S/St^))
St„)<
In {S/St„)
whereiPt (x, y)
=
(1/a) exp [ {n* - a''ji)
x/a^
- (M* - a 7 2 ) ' (T - t„)/2a^ 1 <5 {x/a,
5{x,y)
y/a).
v{x{T-tnr"^)
-^{{x-2y){T-U)-"^)]andp(^ln(^^max^5„/5,„^
<
In (5/5t„)
[T-tn) - 1 / 2
114
The Navigation of an Iceberg=
^An(^/gO-(/i*-aV2)(T-^0a^{T - tn)J- exp [2 (/x* - ay2) In {S/St^)/a^] x$
a^{T - tn)
where (p (z) denotes the standard normal density function and $ {z) the standard normal cumulative distribution function. If the drift is a constant, asassumed in Section 3.4, set //* = fj,. If the drift is modeled as a timedependent variable, as proposed in Section 3.5, replace /i* by fitn- For thederivation of the respective formulas, see, e.g., Harrison (1990), pp. 1-16.
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