# Optimal Order Placement in Limit Order Markets, from Alex Castaldo

May 2, 2013 |

I attended a presentation yesterday on "Optimal Order Placement in Limit Order Markets" given by Arseniy Kukanov.

It was an interesting presentation but was perhaps best summarized by a well dressed white haired gentleman in the audience who asked at the end: "given all the simplifications and assumptions you had to make, of what use is this 'solution' other than to enable you to get a PhD".

In choosing between limit orders and market orders you have to trade off cost savings provided by limit orders against "non-execution risk" that accompanies limit orders (risk that the limit order will not execute).

The problem is: you must buy S shares of stock within a time horizon T.

There are N markets (exchanges) available, for each market you know: the bid ask spread the bid queue lengths at time zero the maker/taker fees that each market charges.

In each market you can at time zero place a market order (which has a 100% probability immediate of execution) and/or or a limit order at the best bid price (other price choices are not included in this version of the model).

There are two parameters lo and lu which are the penalties (in dollars per share) that you charge yourself for buying fewer than S or more that S shares in the available time.

The model minimizes the sum of the trading costs plus the penalty.

The simplest solution is in the case of only one exchange (ex: the minis are only traded on the CME).

In this case there is an algebraic expression for the optimal strategy, and what to do depends on the value of lo:

If lo is below a certain value lbar1, it is optimal to enter a limit order.

If lo is above a certain value lbar2, meaning there is a high urgency to buying the stock, it is optimal to enter a market order.

For lo intermediate between these two values it is optimal to enter both a limit order and a market order at time zero that add up to the desired quantity.

Certain theorems about the solution can be derived, for example as the amount of stock S to be bought increases there is an increased reliance on market orders.

From my point of view one of the most unrealistic assumptions is that the probability distribution of future order flows and cancellations does not depend on the size of the order you enter. In my experience on the contrary if you enter a big limit order it can 'scare' the market and move it away from you.

All in all it was very good for an academic presentation.

Thanks to Anatoly for the invitation.

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