Optimal trading strategies Optimal trading strategies with limit orders: with limit orders: a stochastic stochastic programming programming approach approach a Rossella Agliardi Agliardi Rossella University of of Bologna (Italy) Bologna (Italy) University Chemnitz, 27 , 27- -29 March 2019 29 March 2019 Chemnitz
Trading in limit limit order order books books Trading in Consider a trader who has a block of shares to sell (or buy) in Consider a trader who has a block of shares to sell (or buy) in � � an illiquid market. an illiquid market. � The trader's problem is to design an optimal order execution The trader's problem is to design an optimal order execution � strategy such that he/she can unwind a portfolio position strategy such that he/she can unwind a portfolio position - within a fixed time constraint within a fixed time constraint - - subject to the optimization of certain criteria subject to the optimization of certain criteria - (Cost minimization? Mean- -variance optimization? Maximize expected variance optimization? Maximize expected (Cost minimization? Mean utility? etc.) utility? etc.)
Limit orders Limit orders � Limit orders are ex ante commitment to trade. They specify the Limit orders are ex ante commitment to trade. They specify the � security, the amount and a (limit) price. security, the amount and a (limit) price. They are order to trade a certain amount of a security at a given given They are order to trade a certain amount of a security at a price (or a better one). price (or a better one). � Traders post their supply/demand in the form of limit orders to Traders post their supply/demand in the form of limit orders to � an electronic trading system. an electronic trading system. � Accumulation of posted limit orders is known as the limit Accumulation of posted limit orders is known as the limit � order book. order book. � Orders leave the order book either they get executed or Orders leave the order book either they get executed or � cancelled. cancelled. � An active order at time t is an order that has been submitted at An active order at time t is an order that has been submitted at � t ′≤ t, but has not been matched or cancelled by time t. , but has not been matched or cancelled by time t. t ′≤ t � A LOB is the set of all active orders in the market at time t. A LOB is the set of all active orders in the market at time t. �
Example of an Order Book Buy Sell Best Ask Order Order Best Bid 10 10 20 10 20 30 10 10 Spread 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Price
Market orders orders vs vs limit limit orders orders Market ◯ Market order Market order A buy/sell order that is to be traded A buy/sell order that is to be traded ◯ immediately at the best available price. Always executed immediately at the best available price. Always executed if there is sufficient quantity available ⇒ PRICE RISK PRICE RISK if there is sufficient quantity available ⇒ ◯ Limit order Limit order A buy/sell order that is to be executed at ◯ A buy/sell order that is to be executed at its specified limit or at a better price. its specified limit or at a better price. If a trader files a limit order to buy, the limit order sets a a If a trader files a limit order to buy, the limit order sets maximum price that will be paid. In case of a limit order maximum price that will be paid. In case of a limit order to sell the limit sets the minimum price that will be to sell the limit sets the minimum price that will be accepted. accepted. Executed only if the limit price is reached ⇒ Executed only if the limit price is reached ⇒ EXECUTION RISK EXECUTION RISK
A key difference difference A key Market orders have virtually zero execution have virtually zero execution- -risk. Uncertainty of risk. Uncertainty of Market orders � � the execution price, price impact (especially of large orders). the execution price, price impact (especially of large orders). A trader’s main concern is to avoid a high execution cost (e.g. by .g. by A trader’s main concern is to avoid a high execution cost (e splitting a large orders into several small orders). splitting a large orders into several small orders). Limit orders incur non incur non- -execution (or partial execution) risk. Orders execution (or partial execution) risk. Orders Limit orders � � are sorted in price- -time priority: higher bids and lower asks have time priority: higher bids and lower asks have are sorted in price higher priority. If a buy (sell) limit price is too low (high), the order higher priority. If a buy (sell) limit price is too low (high), the order will not trade. will not trade. Buy (sell) limit orders with high (low) prices are aggressively aggressively Buy (sell) limit orders with high (low) prices are � � priced . . priced The limit price of a `marketable’ limit buy (sell) order is at or above r above The limit price of a `marketable’ limit buy (sell) order is at o � � (below) the best ask (bid). They are the most aggressive limit (below) the best ask (bid). They are the most aggressive limit orders. orders.
Optimal trade trade execution execution Optimal � Statement of the optimal scheduling problem: Statement of the optimal scheduling problem: � Given a model for the evolution of the stock price, Given a model for the evolution of the stock price, find an optimal strategy for trading shares . . find an optimal strategy for trading shares � Mathematical tool: Mathematical tool: Stochastic dynamic programming Stochastic dynamic programming � � Market orders: trade Market orders: trade- -off between price risk and trade off between price risk and trade � cost cost � Limit orders: trade Limit orders: trade- -off between limit price and off between limit price and � execution risk execution risk
Literature Literature � Market Market orders orders � D. Bertsimas D. Bertsimas & A. Lo (1998), R. & A. Lo (1998), R. Almgren Almgren & N. & N. Chriss Chriss (1999, 2000), R. (1999, 2000), R. Almgren (2001, 2003), A. Almgren (2001, 2003), A. Alfonsi Alfonsi, , A.Fruth A.Fruth, A. , A. Schied Schied (2010), J. (2010), J. Gatheral Gatheral & A. Schied Schied (2011), (2011), J.Gatheral J.Gatheral, A. , A. Schied Schied, A. , A. Slynko Slynko (2011), A. (2011), A. & A. Obizhaeva & J. & J. Wang Wang (2005, 2013),T. (2005, 2013),T. Schöneborn Schöneborn (2008), R. (2008), R. Agliardi Agliardi & & Obizhaeva R. Gençay Gençay (2014), (2014),………… …………. . R. � Limit Limit orders orders � Bayraktar & & Ludkovski Ludkovski (2011), (2011), O. O. Guéant Guéant, C. , C. Lehalle Lehalle, J. , J. Fernandez Fernandez- -Tapia Tapia Bayraktar (2012), R. Cont Cont & A. & A. Kukanov Kukanov (2012), O. (2012), O. Guéant Guéant, C. , C. Lehalle Lehalle (2015), F. (2015), F. (2012), R. Guilbaud & H. Guilbaud & H. Pham Pham (2013), …..R. (2013), …..R. Agliardi Agliardi (2016), R. (2016), R. Agliardi Agliardi & R. & R. Gençay (2017) (2017) Gençay Market and Limit Limit orders orders Market and Huitema (2011) Huitema (2011)
Our problem problem Our � A trader has a sell order to liquidate before T. A trader has a sell order to liquidate before T. � � She submits sell She submits sell limit orders limit orders by specifying a by specifying a limit limit � price and an and an order size order size at any point in time. at any point in time. price � Quotes and Quotes and sizes sizes are chosen in order to optimize the are chosen in order to optimize the � trader’s profit, while keeping control of her risk trader’s profit, while keeping control of her risk inventory and of the cost of getting rid of the inventory and of the cost of getting rid of the remaining shares within T. remaining shares within T. � The use of limit orders exposes the trader to non The use of limit orders exposes the trader to non- - � execution risk. execution risk. � Traders are allowed to control for order size, and not Traders are allowed to control for order size, and not � just for the posted prices. just for the posted prices.
Nomenclature for for sell sell orders orders Nomenclature (L. Harris, Trading & Exchanges Exchanges (2003), Oxford (2003), Oxford U.P. U.P.) ) (L. Harris, Trading & LIMIT PRICE PLACEMENT The sell order order is is …. …. LIMIT PRICE PLACEMENT The sell above the best the best ask ask ‘behind behind the market’ the market’ above ‘ at the best ask ask ‘at the market’ at the market’ at the best ‘ between the best the best bid bid and and ‘in the market’ in the market’ between ‘ the best ask ask the best at the best bid bid ‘marketable marketable’ ’ at the best ‘ below the best the best bid bid ‘marketable marketable’ ’ below ‘
Recommend
More recommend
