Optimal posting price of limit orders: learning by trading
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Optimal posting price of limit orders: learning by trading Sophie Laruelle · Charles-Albert Lehalle · Gilles Pagès
Received: 9 January 2012 / Accepted: 10 January 2013 / Published online: 23 March 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract We model a trader interacting with a continuous market as an iterative algorithm that adjusts limit prices at a given rhythm and propose a procedure to minimize trading costs. We prove the a.s. convergence of the algorithm under assumptions on the cost function and give some practical criteria on model parameters to ensure that the conditions to use the algorithm are met (notably, using the co-monotony principle). We illustrate our results with numerical experiments on both simulated and market data. Keywords Stochastic approximation · Order book · Limit order · Market impact · Statistical learning · High-frequency optimal liquidation · Poisson process · Co-monotony principle Mathematics Subject Classification (2000)
62L20 · 62P05 · 60G55 · 65C05
1 Introduction With the growth of electronic trading in recent years, most of the transactions in the markets occur in Limit Order Books. During the matching of electronic orders, traders send orders of two kinds to the market: passive and aggressive; passive orders (i.e. limit or patient orders)
S. Laruelle (B) Laboratoire de Mathématiques Appliquées aux Systèmes, École Centrale Paris, Grande Voie des Vignes, 92290 Châtenay-Malabry, France e-mail: [email protected] C.-A. Lehalle Crédit Agricole Cheuvreux, CALYON Group, 9 quai Paul Doumer, 92920 Paris La Défense, France e-mail: [email protected] G. Pagès Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, UPMC, case 188, 4, pl. Jussieu, Paris Cedex 5, France e-mail: [email protected]
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will not give rise to a trade but will stay in the order book (sell orders at a higher price than the higher bid price or buy orders at a lower price than the lower ask price are passive orders); aggressive orders (i.e. market or impatient orders) will generate a trade (sell orders at a lower price than the higher passive buy price or buy orders at a higher price than the lowest passive price). When a trader has to buy or sell a large number of shares, it’s not optimal for him to just send his large order at once because it would consume all of the available liquidity in the order book, impacting the price at his disadvantage; instead, he has to schedule his trading rate to strike a balance between market risk and market impact (too many orders exhaust the order book and makes the price move). Several theoretical frameworks have been proposed for optimal scheduling of large orders see [2,3,7,24]). Once the optimal trading rate is known, the trader has to send smaller orders in the (electronic) limit order book by alternating limit (i.e. patient or passive) orders and market (i.e. urgent or aggressive) orders. The optimal mix of limit and market orders for a trader has not been investigated in the quantitative literature neither h
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