# «The Navigation of an Iceberg: The Optimal Use of Hidden Orders ∗ Angelika Esser⋆ and Burkart M¨nch⋆⋆ o ⋆ DekaBank, Hahnstrae 55, 60528 ...»

The Navigation of an Iceberg: The Optimal Use of

Hidden Orders ∗

Angelika Esser⋆ and Burkart M¨nch⋆⋆

o

⋆

DekaBank, Hahnstrae 55, 60528 Frankfurt am Main, Germany

e-mail: Angelika.Esser@dekabank.de

⋆⋆

Department of Finance, Faculty of Economics and Business Administration,

Uni-PF 77, Goethe University, 60054 Frankfurt am Main, Germany

e-mail: moench@wiwi.uni-frankfurt.de

January 14, 2005

Abstract. Market participants with large orders to execute are often reluctant to expose these to an open order book in their entirety in order to avoid a potential adverse market impact. Therefore, investors often split large orders into smaller tranches. Iceberg orders facilitate these trading practices by executing such business automatically in the order book. This paper analyzes the rationale for the use of iceberg orders in continuous trading by assessing the costs and beneﬁts of this trading instrument. We present a parsimonious framework that allows the determination of the optimal limit and the optimal peak size of an iceberg order for a static liquidation strategy. Examples with real world order book data demonstrate how the setup can be implemented numerically and provide a deeper insight into relevant properties of the model.

Keywords: optimal liquidation, order book imbalance, limit order, iceberg order, liquidity JEL classiﬁcation: G12

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-based electronic markets. Typically, electronic trading platforms provide market participants with information on an anonymous open order book during continuous trading in real time. Usually the limits, the accumulated order volumes of each limit, and the number of orders This research was partially supported by the Deutsche Forschungsgemeinschaft.

∗ We are grateful to the Trading Surveillance Oﬃce of the Deutsche B¨rse AG for o providing us a limit order book dataset. We are indebted to Nicole Branger and Christian Schlag for inspiration and insightful discussions. Special thanks for helpful comments go to Matthias Birkner, G¨tz Kersting, Michael Melvin, Roderich o Tumulka, Anton Wakolbinger, and our colleagues Christoph Benkert and Micong Klimes.

2 Angelika Esser, Burkart M¨nch o in the book at each limit are displayed, so that traders can assess the altering order ﬂow and the market liquidity.

What does the existence of an open order book imply for investment ﬁrms who want to submit limit orders, the total volume of which is large relative to others in the market? No doubt, exposing large limit orders in an open order book may reveal the investor’s motives for trading and may raise suspicion that the originator of the large order has access to private information about the true value of the security under consideration. Consequently, other market participants change their own order submission strategy, which in turn lowers the probability that the large order will be executed at the prespeciﬁed limit. The investment ﬁrm then has to choose a less favorable limit if it wants to increase the probability of execution and thus suﬀers losses from the 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 the order into several smaller limit orders, which are submitted over time. For this reason many electronic trading platforms introduced so-called iceberg orders. Euronext, the Toronto Stock Exchange, the London Stock Exchange (with its order driven services SETS, SETSmm, and IOB), and XETRA are just some prominent examples. Iceberg orders allow market participants to submit orders with only a certain portion of the order publicly disclosed. The metaphor alludes to the fact that in nature an iceberg’s biggest part ﬂoats unobservable under the water.

Only one-ninth of the mass of ice is seen above the water surface.

An iceberg order is speciﬁed by its mandatory limit, its overall volume, and a peak volume. The peak is the visible part of the iceberg order and is introduced into the order book with the original time stamp of the iceberg order according to price/time priority. As soon as the disclosed volume of an iceberg order has received a complete ﬁll and a hidden volume is still available, a new peak is entered into the book with a new time stamp. The new peak behaves in an identical manner to a conventional limit order. From this point of view a pure limit order is basically a special case of an iceberg order where the peak volume coincides with the total order volume.

However, it is important to note that iceberg orders exhibit a less favorable time priority compared with pure limit orders. After the peak of an iceberg order is completely matched, all visible limit orders at the same limit that were entered before the new peak is displayed take priority, i.e. they must have received a complete ﬁll before the new peak comes to execution.

When submitting an iceberg order to the market, several issues have to be considered. Imagine, for example, that the management of a mutual fund has to close a large position in a single stock within one Optimal Use of Hidden Orders trading day. Using an iceberg order with only a small peak size allows it to minimize the adverse informational impact of disclosing the actual order volume. However, the 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 would make a complete ﬁll unlikely.

Moreover, the fund managers have to choose a reasonable limit for the order. If the limit is too low, one may miss some trade opportunities, i.e.

one would give away the chance to participate in raising stock prices.

Otherwise, if the limit is too ambitious, the order is unlikely to receive a complete ﬁll.

In the present paper this tradeoﬀ is modeled analytically in a continuous time setup where a large position in a single stock is to be liquidated within a ﬁnite trading window.1 We assume that the investor uses an iceberg order and follows a static strategy, i.e. once the limit and the peak size of the iceberg 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 by maximizing the expected payoﬀ of the liquidation strategy under certain assumptions concerning the execution risk of the iceberg order. Note that a 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 low that it is actually a market sell order, the probability of receiving a complete ﬁll within a ﬁnite time horizon is strictly smaller than one. In principle at least 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 remaining shares with a market order if the iceberg order fails to receive complete execution. We call this setup the self-contained approach. Market orders are executed immediately. They use liquidity from the book until they are completely 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 not always be justiﬁed in practice, especially if the remaining order volume under consideration cannot be absorbed by the market without a signiﬁcant price change.

In this case, investors typically follow an adaptive strategy, i.e. they review their orders frequently and adjust them if the market moves away from the prespeciﬁed order limit. For this reason we also propose a diﬀerent approach that considers the execution probability as a boundary condition, i.e. only those combinations of peak size and limit The analysis for a purchasing strategy is symmetric.

4 Angelika Esser, Burkart M¨nch o are admissible that assure a certain execution probability within a prespeciﬁed time horizon. We call this model the open approach. Compared with the ﬁrst one the latter framework is rather ﬂexible and does not require any assumption concerning the liquidation of the unexecuted part of the iceberg order. To get a ﬂavor of the concept, imagine, for example an investor who wants to liquidate a large position, say, within one week. At the end 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.2 The open approach can assist the investor in this procedure. It deals with the optimal combination of order limit and peak size that maximizes the expected liquidation revenues in the case of complete execution, given that the probability to receive a complete ﬁll exceeds a certain level, for example 40% within one trading day.

If the order remains partially or completely unexecuted by the end of the ﬁrst day, the investor may wish to rerun the optimization at the second day and thereby increase the execution probability, let’s say, to 60% and so on. If a substantial part of the order is still unexecuted on the last day of the week, the investor will probably choose a minimum execution probability that is close to one. In principle one can also specify a utility function for the investor to model the trade-oﬀ between expected payoﬀs and execution risk. However, in order to keep the problem tractable for exposition we will not address this issue in this paper.

We present a theoretical framework for both the open and the selfcontained approach. Although the underlying assumptions of the latter model are certainly questionable from an empirical point of view we believe that its basic structure may serve as a guideline to build more sophisticated models, for example by implementing an individual penalty function for the unexecuted part of the iceberg order that meets the speciﬁc requirements of the investor under consideration. The numerical analysis that illustrates the theoretical part will focus on the open approach.

**The technical design of the model can be summarized as follows:**

During continuous trading a transaction takes place if an order becomes executable against orders on the other side of the book. Thus, for an iceberg sell order that is stored on the ask side of the book the dynamics of the best bid price are of special interest. We model the best bid price as a stochastic process in continuous time and assume a constant best bid size. If the stochastic process hits the limit of the iceberg order a Note that at some exchanges unexecuted iceberg orders are deleted automatically by the system on the end of each trading day and must be resubmitted if desired by the investor at the next trading day. In this case a daily adjustment of order limit and peak size seems very plausible.

Optimal Use of Hidden Orders transaction is executed and the stochastic process jumps back to the next lower limit. Whether the peak of the iceberg order or another sell order at the same limit is processed at this event depends on the relative time priority of the orders. If new orders with the same limit as the iceberg order are submitted continuously to the book the time priority of the iceberg order deteriorates compared with a pure limit order. The smaller the peak size of the iceberg order, the more often the limit must be hit such that the iceberg order receives a complete ﬁll.

On the other hand, a smaller peak size lowers the adverse informational impact of showing the actual order volume in an open book. We model the drift 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 order imbalance changes. We deﬁne the order imbalance as the total visible order volume (in number of shares) stored on the bid side of the order book divided by the total visible order volume stored on the bid side and on the ask side of the order book. We exemplify empirically, using order book data, that current variations in the visible order book imbalance are positively correlated with future returns.

Thus, the higher the peak size of an iceberg sell order, the smaller the order imbalance and the smaller the expected returns in the next time intervals. Consequently, a higher peak size results in a smaller 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

**iceberg sell order is reduced in our model:**

− The drift of the stochastic process is reduced to a smaller extent when the order enters the book.

− The number of times the limit must be hit in order to process the iceberg order completely increases.

While the ﬁrst eﬀect is beneﬁcial for the originator of the iceberg order, the latter is not. The proposed framework weights these eﬀects and identiﬁes the optimal combination of peak size and order limit.

The rest of the paper is organized as follows: Section 2 brieﬂy reviews the related literature. The dataset used to exemplify the theoretical ideas throughout this paper is described in Section 3. Section 4 introduces the theoretical setup for both the self-contained and the open approach. In Section 5 we explicitly model the drift as a function of the order imbalance. The open approach to determine the optimal combination of order limit and peak size is calibrated with a clinical order book data sample in Section 6 so that one can get an impression 6 Angelika Esser, Burkart M¨nch o of the optimal strategies for diﬀerent scenarios. The paper concludes in Section 7 with a brief summary and a discussion of issues for further research.

A number of empirical studies shed light upon the use of hidden orders3 and the associated motives of traders.