ive 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 partof the iceberg order. To get a flavor of the concept, imagine, for example aninvestor who wants to liquidate a large position, say, within one week. At theend of each trading day the investor inspects the state of the iceberg order
and, if necessary, adjusts the limit or the peak size to reach the target.”
The open approach can assist the investor in this procedure. It deals withthe optimal combination of order limit and peak size that maximizes theexpected liquidation revenues in the case of complete execution, given thatthe probability to receive a complete fill exceeds a certain level, for example40% within one trading day. If the order remains partially or completelyunexecuted by the end of the first day, the investor may wish to rerun theoptimization at the second day and thereby increase the execution proba-bility, let’s say, to 60% and so on. If a substantial part of the order is stillunexecuted on the last day of the week, the investor will probably choosea minimum execution probability that is close to one. In principle one canalso specify a utility function for the investor to model the trade-off between
2Note that at some exchanges unexecuted iceberg orders are deleted automatically bythe system at the end of each trading day and must be resubmitted if desired by theinvestor on the next trading day. In this case a daily adjustment of order limit and peak
size seems very plausible.3.1 Introduction 83
 
expected payoffs and execution risk. However, in order to keep the problemtractable for exposition we will not address this issue in this chapter.
We present a theoretical framework for both the open and the self-containedapproach. Although the underlying assumptions of the latter model are cer-tainly questionable from an empirical point of view we believe that its basicstructure may serve as a guideline to build more sophisticated models, forexample by implementing an individual penalty function for the unexecutedpart of the iceberg order that meets the specific requirements of the investorunder consideration. The numerical analysis that illustrates the theoreticalpart will focus on the open approach.
The technical design of the model can be summarized as follows: Duringcontinuous trading a transaction takes place if an order becomes executableagainst orders on the other side of the book. Thus, for an iceberg sell orderthat is stored on the ask side of the book the dynamics of the best bid priceare of special interest. We model the best bid price as a stochastic processin continuous time and assume a constant best bid size. If the stochasticprocess hits the limit of the iceberg order a transaction is executed and thestochastic process jumps back to the next lower limit. Whether the peakof the iceberg order or another sell order at the same limit is processed atthis event depends on the relative time priority of the orders. If new orderswith the same limit as the iceberg order are submitted continuously to thebook the time priority of the iceberg order deteriorates compared with a purelimit order. The smaller the peak size of the iceberg order, the more oftenthe limit must be hit such that the iceberg order receives a complete fill.
On the other hand, a smaller peak size lowers the adverse informationalimpact of showing the actual order volume in an open book. We model thedrift of the stock price process as a function of the visible order imbalance.When the peak size of an iceberg order enters the book the visible orderimbalance changes. We define the order imbalance as the total visible ordervolume (in number of shares) stored on the bid side of the order book dividedby the total visible order volume stored on the bid side and on the askside of the order book. We exemplify empirically, using order book data,84 The Navigation of an Iceberg
 
that current variations in the visible order book imbalance are positivelycorrelated with future returns. Thus, the higher the peak size of an icebergsell order, the smaller the order imbalance and the smaller the expectedreturns in the next time intervals. Consequently, a higher peak size resultsin asmaller probability that the stock price process will reach the prespecified
limit within the given time horizon.
In total, one can observe two opposite effects if the peak size of an icebergsell order is reduced in our model:
e The drift of the stochastic process is reduced to a smaller extent whenthe order enters the book.
e The number of times the limit must be hit in order to process theiceberg order completely increases.
While the first effect is beneficial for the originator of the iceberg order, thelatter is not. The proposed framework weights these effects and identifies theoptimal combination of peak size and order limit.
The rest of the chapter is organized as follows: Section 3.2 briefly reviewsthe related literature. The dataset used to exemplify the theoretical ideasthroughout this chapter is described in Section 3.3. Section 3.4 introducesthe theoretical setup for both the self-contained and the open approach. InSection 3.5 we explicitly model the drift as a function of the order imbalance.The open approach to determine the optimal combination of order limit andpeak size is calibrated with a clinical order book data sample in Section 3.6 sothat one can get an impression of the optimal strategies for different scenarios.The chapter concludes in Section 3.7 with a brief summary and a discussionof issues for further research.3.2 Related Literature 85
 
3.2 Related Literature
A number of empirical studies shed light upon the use of hidden orders? and
the associated motives of traders.
Aitken, Brown, and Walter (1996) state that approximately 6% of ordersaccounting for 28% of shares traded at the Australian Stock AutomatedTrading System (SEATS) were undisclosed in 1993. Aitken, Berkman, andMak (2001) find that undisclosed limit orders are used to reduce the optionvalue of limit orders. This follows the intuition that limit orders can becomemispriced when new public information arrives. Some authors, for exampleCopeland and Galai (1983), therefore characterize limit buy (sell) orders asfree put (call) options to other market participants. Pardo and Pascual (2003)use six months of limit order book and transaction data on 36 stocks fromthe Spanish Stock Exchange (SSE) and report that 26% of all trades (20%of all non-aggressive trades and 42% of all aggressive trades)* involve hiddenvolume. They provide evidence that liquidity suppliers use iceberg orders tomitigate adverse selection costs if new information is released to the market,and that hidden orders temporarily increase the aggressiveness of traderswhen revealed to the public. D’Hondt, De Winne, and Francois-Heude (2003)investigate data for six CAC40 stocks traded at Euronext and show that 30%of the depth is hidden in the whole book. The authors highlight that hiddenorders are more frequently canceled than usual orders, that iceberg orders areless likely to be totally filled and that the limit of hidden orders is modifiedmore often than that of pure limit orders.
The modeling of optimal liquidation strategies attracts more and more atten-tion by researchers. Bertsimas and Lo (1998), Almgren and Chriss (2000),Hisata and Yamai (2000), Dubil (2002), and Ménch (2003) investigate liqui-
’Note that at some exchanges the expression “hidden order” is reserved exclusivelyfor orders that are completely invisible to other market participants. However, as it iscommon practice in the literature that is related to this chapter we use it as a synonym
for an iceberg order.4A buyer- or seller-initiated trade is defined as aggressive if it consumes, at least, the
best quote on the opposite side of the book.86 The Navigation of an Iceberg
 
dation strategies for large security positions if market orders are employed astrading instruments. The papers differ from each other mainly in the defini-tion of the stock price dynamics, the modeling of the price impact function,and whether the final time horizon is given exogenously or modeled endoge-
nously.
Compared with market orders, the analysis of optimal liquidation strategiesfor limit orders is more complex. While the former are matched immediately(provided a sufficient market depth) the execution probability of the latter or-der type depends critically on the respective limit. Wald and Horrigan (2001)estimate a probit model that characterizes the execution probability depend-ing on a number of variables as the order limit, subsequent realized returns,the bid~ask spread and so on. Lo, MacKinlay, and Zhang (2002) compareempirically three different approaches to determine the execution probabilityof limit orders using order book data for the 100 largest stocks in the S&P500 from August 1994 to August 1995. First, they model the execution of alimit order as the first passage time of a geometric Brownian motion to thelimit price and find that the predictive power of this setup is only moder-ate. The first passage time model suffers from important shortcomings. Itneither considers the time priority, the order size, a potential adverse impactof revealing large limit orders in the book nor does it distinguish betweentime-to-first-fill and time-to-completion. As mentioned above, these limi-tations are eased in the framework proposed in this chapter. Second, theauthors consider first-passage times determined by historical time series oftransaction data. As this approach also ignores the time priority and cur-rent market conditions it is not able to represent actual limit order executiontimes adequately. Finally, the article proposes an econometric model of limitorder execution times based on survival analysis and actual limit order data.This empirical approach is a reduced form model as it leaves open whichmechanism actually causes the execution of a limit order. The model useseight explanatory variables that are updated in real time to capture currentmarket conditions and three explanatory variables that are updated monthly
to model differences across stocks. The authors make some assumptions that3.2 Related Literature 87
 
may not always be justified empirically to keep the framework tractable.For example the authors assume time-independent covariates. Nevertheless,the empirical framework is able to predict actual limit order executions verywell. However, the model requires a continuous update with order book data.
Thus, it may be only of limited use if such data are not available in real time.
The intention of our framework is closely related to that of Lo, MacKinlay,and Zhang (2002) although we use a structural approach to model explicitlythe functionality of the order book. Both approaches focus on the modelingof the execution probability of limit or, in our case, more generally, icebergorders. In our approach this probability is obtained endogenously and inthe parametric model of Lo, MacKinlay, and Zhang (2002) the form of theprobability density function is specified exogenously. While Lo, MacKinlay,and Zhang (2002) just present a framework to estimate the time-to-first-filland the time-to-complete execution of a limit order, our approach is moreflexible as it is able to capture every state of partial execution. Furthermore,we extend the analysis to the identification of optimal liquidation strategies
for different scenarios.
Cho and Nelling (2000) estimate the execution probability of a limit orderconditional on order-specific variables and other variables that capture gen-eral market conditions using quote data for 144 NYSE stocks from November1990 to January 1991. The authors observe that the longer a limit order isoutstanding, the less likely it is to receive a complete fill. Furthermore, theyfind that the execution probability is low when the limit price is far awayfrom the current quote, when the order volume is high, when spreads are
narrow, and when volatility is low.
The hypothesis that the order imbalance is a proxy for the execution proba-bility of limit orders and influences the order submission strategy of investorsis supported by many authors. Chordia and Subrahmanyam (2002) analyzetime series of daily order imbalances and individual stock returns for theperiod 1988--98 using a comprehensive sample of NYSE stocks. They findthat lagged imbalances bear a positive predictive relation to current dayreturns. Furthermore, they observe that daily imbalances are positively au-88 The Navigation of an Iceberg
 
tocorrelated. Ranaldo (2004) uses data of 15 stocks quoted on the SwissStock Exchange. He finds that orders are submitted more (less) aggressivelywhen the outstanding order volume on the same (opposite) side of the bookis large. Furthermore, he observes that buyers are more concerned aboutthe opposite side of the book, while sellers are more concerned about theirown side. Parlour (1998) presents a dynamic setup of an order book, whereinvestors anticipate that the own order placement strategy influences thefollowing order flow and where the execution probability of limit orders is
modeled endogenously.
3.3 Description of the Market and Dataset
The functionality of the liquidation model proposed in this chapter will beillustrated by using the order book data sample for the company MAN that
was introduced in Section 2.3.
In this dataset we counted 786 iceberg sell orders, 140,948 pure limit sellorders, and 4,130 market sell orders. At first sight the hidden part of theorder book seems tiny. However, analyzing the average volume of each trad-ing instrument more deeply changes this impression slightly. Iceberg ordersexhibit an average volume of 16,037 shares, whereas pure limit and marketorders just have an average order volume of 964 and 1,069 shares. Due to thisfact, hidden orders represent a remarkable proportion of 8.24% of the overallvolume on the ask side of our sample order book. Pure limit sell orders andmarket sell orders provide 88.87% and 2.89% of the liquidity on the ask side.
Figure 3.1 shows the spectrum of the observed initial volumes of all hidden sellorders. In Figures 3.2 and 3.3 we address the issue of how market participantschoose the limit and the peak size in practice. Obviously there exists a strongpreference to specify a peak size that corresponds to a tenth of the overallorder volume, as one can see in Figure 3.2. Roughly 37% of all marketparticipants follow such a strategy. To investigate the difference betweenthe chosen limit and the current best bid price at the time of submission3.3 Description of the Market and Dataset 89
 
Absolute frequencyof iceberg sell orders
462400 |30019620053ar Meet? ttn“ep eeeeeeeeésoEEE ESSESSE ES cxsrMi SMeSHS LES LSS SS whumsLESETSTS EG? — omeso of oh ot ot ot ot 2 of shares)YP PP Pres
Fig. 3.1: Absolute frequencies of order volumes of all observed iceberg sell ordersin the sample.
Absolutefrequency
   
200
100
Fig, 3.2: Ratio of initial order volume to peak size of all different iceberg sellorders in the sample.90 The Navigation of an Iceberg
 
we consider only the 702 iceberg sell orders that entered the book duringcontinuous trading. Figure 3.3 shows that the majority of market participantsset the limit between 5 and 15 cents above the best bid price. Note that theaverage bid-ask spread in our sample is 7 cents and the average midprice€ 26.91.
Absolute frequencyof iceberg sell orders
Best-Bid-Price
 
Fig. 3.3: Absolute price difference between the chosen limit and the current best.bid price at time of entry of all observed iceberg sell orders in the sampleentered during continuous trading.
With respect to the success of the observed trading strategies, Table 3.1delivers an insight into the empirical execution probability of hidden orders.Less than 18% of all iceberg sell orders were executed completely. Almost30% of all iceberg orders received a partial fill before expiry or cancelation bythe investor. The majority (52%) of all hidden sell orders were canceled orexpired completely unexecuted. Looking at the median of the observed timebetween entry and complete execution or deletion one can state that marketparticipants check the state of their orders frequently and cancel them ifprices move away from the limit.3.4 The Basic Setup
91
Table 3.1: Execution or deletion of iceberg sell orders in the order book sample.
 
 
Number |Average ratio of} Average time betweenof iceberg}executed to initialfentry and completesell orders | volume; median in|execution or deletionparentheses (hh:mm:ss); median inparenthesesCompletely 139] 100.00% (100.00%) 00:40:48 (00:10:46)executedPartially 231} 32.44% (28.00%) 00:50:04 (00:09:29)executedCompletely 416 0.00% (0.00%)} 03:32:43 (00:09:45)unexecuted
 
 
 
 
3.4 The Basic Setup
3.4.1 General Idea and Dynamics
This section introduces the basic concepts and provides the motivation forthe assumptions that have been made. Assume that the large investor holdsoo shares that should be liquidated before time T. For this purpose the tradersubmits an iceberg sell order that is stored on the ask side of the order book.The investor assigns a peak size ¢, and a limit S to the iceberg order. Thelatter is strictly higher than the initial best bid price Sp such that the firstproportion of the order is not immediately executable.
The best bid price S; is modeled by a kind of jump-diffusion process. ForS, < § it follows a geometric Brownian motion:
dS, = S,dt+o8,dW, — with Sy < &. (3.1)
Throughout this section the drift jz is assumed to be a constant. In Section 3.5we ease this restriction and model the drift as a function of the chosen peak
size.92 The Navigation of an Iceberg
 
When the process hits the limit of the iceberg order, ie. S,. = 5, a smalldownward jump to the next order book entry on the bid side occurs suchthat S, = (1—<¢)S. For sake of simplicity the jump size is modeled as aconstant throughout this chapter. Figure 3.4 illustrates the general settingdescribed so far.
 
 
 
Fig. 3.4: If the best bid price 5; hits the limit of the iceberg order S, a small
downward jump to the next limit on the bid side, ie. to (1-e«)S,occurs.
Each time the limit 5 is hit by the best bid price a transaction is executed.The transaction size ¢, is assumed to be constant over time. Furthermore,we assume that whenever a new tranche of the iceberg order enters the booka fixed volume ¢, of other sell orders that exhibit a better time priorityis already stored at the same limit. These orders must be matched beforethe current peak of the iceberg order becomes executable. Thus, one canobserve the following sequence of newly displayed order quantities over time:bas Pp, bar Pp, and so on. Table 3.2 summarizes the notation that is usedthroughout the rest of this chapter.3.4 The Basic Setup 93
 
Table 3.2: Notation used in the proposed liquidation model.
 
Variable | Meaningdo Total number of shares that have to be liq-uidated by the large investor before time T
 
dp(< 0) | Assigned peak size to the iceberg order
Da Volume of other sell orders at limit 5 that isalready stored on the ask side when a freshpeak enters the order book
dbs Total transaction volume that is processed
each time the limit is hit
 
 
 
 
The objective of the investor is the identification of the optimal combinationof peak size ¢, and limit S. The time horizon is T, and the investor isinterested in the expected payoff of his or her liquidation strategy. To dealwith execution risk, we consider two alternative approaches. In the openapproach, the investor chooses a lower bound P”* for the probability that thesubmitted iceberg order receives a complete fill up to time T. For the self-contained approach we assume that if the iceberg order is not completelyexecuted before time T, the large investor submits a market order to sellthe remaining part of the shares. Consequently, the trader has to bear asignificant liquidity discount, denoted by Y. Sr, where UV >> ¢. Thus, theinvestor will receive (1 — V) Sr for each of the remaining shares. It seemsreasonable to model W as a function of the number of shares that are sold bysubmitting a market order at T.
The remaining part of this section is dedicated to the derivation of the for-mulas necessary to implement the open and the self-contained approach. Ob-viously, the liquidation value depends on the actual number of times thebest bid price hits the prespecified limit of the iceberg order, which in turnis a random variable in the proposed setup. Thus, in Subsection 3.4.2 weintroduce formulas to compute the executed volume of the iceberg order con-94 The Navigation of an Iceberg
 
ditional on the event that the limit is hit a certain number of times. InSubsection 3.4.3 we calculate the probability that the limit is hit a certainnumber of times. Furthermore, in this subsection we present the objectivefunction of the investor for both the open and the self-contained approach.
3.4.2 Execution of the Iceberg Order
The number of times n* the limit 5 must be hit by the best bid price suchthat the iceberg order is completely satisfied is given by
we foe +4] _ Ee + eu)
ds bs
The brackets [.] are called upper Gaussian brackets, with[a] = min{z €Z: z ><a} and Z as the set of integers. Note that n* - ¢,corresponds to the total order volume that is matched after n* transactions,whereas ¢o shares originate from the iceberg order and (¢9/¢p)¢a shares fromother orders.After the limit is hit n times the number of executed shares of the icebergorder is given bynds |
nosnos —au(1+ [st#]) Joao}
where |] = max{z€Z: z<-}. As long as n < n* the first element of the
 
max-expression is larger than the second if other sell orders at the same limitexhibit a better time priority than the current peak of the iceberg order. Ifthe current peak of the iceberg order takes time priority over other sell ordersat the same limit, the second term in the max-expression is larger than thefirst.
The outstanding order volume of the iceberg order after the n-th hit of thelimit is then equal to
go — h(n).3.4 The Basic Setup : 95
 
3.4.3 Liquidation Value
Armed with the results of the previous section one can now calculate theliquidation value conditional on the event that the limit is hit a certainnumber of times. Let M denote the number of times the limit is hit beforetime T. If M > n*, the liquidation value G is given by G = ¢o5, since theiceberg order is completely executed at time T. The open approach simplymaximizes this expression by solving the following optimization problem:
max ¢5
{4ns5}
st. PY < P(M2n‘)So < §dp S 0.
In contrast to the open approach, which focuses on the case of full execution,the self-contained approach considers also those states of the world whereM <n. In this setup, if M = 7, the trader receives h(n) 5 for the executedpart of the iceberg order and [do — h(n)] [1 — UV (¢9 — h(n))] Sp for the re-maining part that is liquidated using a market order at time T. Given therealizations of M and Sr the liquidation value can be calculated by
Guen = h(n) 5 + [bo ~ h(n)] [1 ~ © (do — h(n) Sr.
However, at time tj, both M and Sy are random variables. Thus, in order toderive the expected liquidation value one has to weight all possible realiza-tions of the liquidation value Gy by their probabilities, whereas Sy dependson the realization of M. Thus, the expected liquidation value can be written
as
n) § + [bo — h(n)] [1 — U (bo ~ h (n))] x
—_—
E(Sp|M =n) \96 The Navigation of an Iceberg
 
= P(M=0)o- [1 — W(4)] E (Sr M = 0)ntl
+ > PUM =n) { A(n) $+ [bo — h(n)] x
[1 — ¥ (dy — A (n))| B (Sr| M =n) \
+P(M > n*) (do 8).
 
For the self-contained approach we need to solve the following optimization
problem:max EG{p53}8.t. S < §oe < bo.
It remains to calculate the following quantities:e P(M = 0)e P(M =n), forn=1,...,n*-1¢ P(M >n*)e E(S7| M = 0)e E(S7|M =n), forn=1,...,n*-1.
For this purpose we must compute the distributions of the hitting times,denoted by t;, i =1,...,n. The time periods between two successive hittingtimes will be denoted by 7; := t; — tii, i = 2,...,m and we will let 7 =).
The distribution of 7 can be calculated as follows: Since the process forthe best bid price S; up to 7 follows a geometric Brownian motion, see
equation (3.1), the logarithm of the process is an arithmetic Brownian motiond(In S;) = (u - oa? /2) dt + odW,.
Note that if an arithmetic Brownian motion has a negative drift, i.e. whenpt < 07/2 in our model, then 7; has a defective density function whose integral3.4 The Basic Setup 97
 
over {0, oo) is less than one, see, e.g., Karlin and Taylor (1975), p. 362. Thus,the probability that the limit will never(!) be reached is positive. Specifically,the probability for the event {71 < co} is given by
1 for 4 > 0/2exp [~—21n (8/5) |u— 0?/2|/o?| for p< 0?/2.The distribution of the first hitting time of In$, starting at InSy < InS,conditional on the event {7 < co}, is given by
. In (S/S) oof [In (8/54) — pee) 83
tfa) ov 2nt3 207t
P(n <o)= { (3.2)
 
Taking the product of (3.2) and (3.3) we can write the unconditional (defec-tive) density of 7 as
foa(t) = for (t)P(n <0)In (S/S0) exp { [in (8/50) — (u — 02/2) rl, (3.4)
ov ant3 20°t
At 7 the process independently restarts at 5(1— e), following again a geo-metric Brownian motion. To derive the (defective) density of the first hittingtime after the restart (denoted by T2 = tz — t,) we just need to replace In Soby In [S(1 — €)] in equation (3.4) if we assume a constant drift 4. Thus, forn* > 2 one can write
-In(1~<) w= [-In(1—«)— poy, (3.5)
ov 2nt3 20°t
fain (t)
Since tz can be decomposed into the sum of the two independent random
variables 7, and 72, ie. te = aH 7;, the (defective) density fo,2 of tz is
i=]
simply the convolution of the corresponding (defective) densities, given by
fo2(t) = (for * fi2)(t) = [ foalt ~ w)fi2(u)du.
One can proceed by iterating this methodology: At 7%, 1 > 2 the processindependently restarts at S(1 — <) following a geometric Brownian motion.Thus, the distribution of t, = Sy, 7: is given by
fonlt) = (fon * fr-an)(t) = (for * fa *--- * fan) (f): (3.6)98 The Navigation of an Iceberg
 
Now, we are able to derive the corresponding probabilities by rewriting the
number of hits in terms of hitting times. Since the events{M =n} and {(t, <T)A (tn41 > T)}
are identical, the desired probability for (M > 1) can be written as
uvsq=tl
P((tn <T)A (tna > T))P(t, <T)—P (trai <T)
n ntlP (x <r) —P (Ss <r)isl t=]T T| fan(t) dt ~ | fomax(t) at.
The probability for the event that the limit is not hit before T is given by
P(M = 0)
P(t; >T)
{|
1- ff (t) dt.
The probability for a complete fill of the iceberg order before T can be com-puted via
P(M > n*)
P(tn» <T)T
J baw (t) dé.
0
HW
Now the expected liquidation value, conditional on the event that the limitis hit n times before time T, can be calculated. To simplify the explana-tion, assume for a moment that the hitting times are deterministic. This
assumption will be relaxed later. In this case the expression
E(Sp|M =n)3.4 The Basic Setup 99
 
is equal to
E,, (se
 
max_(S,) < s)
tn<u<T
4
E,, (ew [In(Sr/St,.)] StaIn(5/Stp)/ exp(s) 5:9 (s
-oo
 
max, [In(S../S:,)] < in(S/S,) J
4
 
pmax, [In(Su/5t4)] < In(S/S;,), n) ds,
since the n-th hit occurs at t, and the process independently restarts at tyfollowing a geometric Brownian motion conditional on the event that thethreshold $ is not hit within the time interval from t, to T.
One may notice that this scenario is similar to the evaluation of knock-outbarrier options (see, e.g., Zhang (1998), pp. 203-259). The formula for theconditional density g of In(.S7/S;,) is given in Appendix 3.8.
However, for M = n > 1, t, is in fact a random variable. Thus, weneed to consider the distribution of ¢,, conditional on the event{tn < T A tna: > Th}. Due to the independence and identical
distributions of 7; for i > 2 this conditional density is given by
fomt(t\dt = P(tr € (t,t +dt)|ma > T 2)
P((tn €(t,t+dt)) A (ma >T— ‘))P(t > T — t)
P((tn <T)A (try > T))JF fon(u) du =f? fonsi(u) du
Armed with this result we are able to write the conditional expectation of
 
 
Sp as100 The Navigation of an Iceberg
 
pt fam(t) (1 ~ J Famen(s)48)do Jo font) du — fO fonsa (tu) du
In(S/Stn)/ exp(s) S; x
~0o
9 («I max [In(S,/5,)] < In(S/S)), n) ds dt. (3.7)
Note that the integral with respect to ¢ 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 Sp simplifies to
E(Sr|M = 0)In($/So) _= /