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Optimal Portfolio Liquidation with Limit Orders
Olivier Guéant, Charles-Albert Lehalle, Joaquin Fernandez Tapia
TL;DR
The paper addresses the problem of jointly choosing a liquidation schedule and the prices of passive limit orders, rather than treating scheduling and order placement separately. It models executions and price risk in a Hamilton-Jacobi-Bellman framework, derives tractable optimal quotes, and evaluates them through numerical experiments and backtests. The results are promising, while extreme slow-execution cases indicate that limit orders alone may be unsuitable and require market-order approximations.
Problem
Existing liquidation work emphasizes scheduling or liquidity-consuming orders, while the joint problem of scheduling and posting passive limit orders remains insufficiently addressed.
Method
The paper models limit-order executions with a Poisson process tied to a diffusive fair price and solves the resulting price- and non-execution-risk problem using a Hamilton-Jacobi-Bellman equation.
Results
Numerical experiments and backtests provide promising results for the proposed optimal liquidation and quoting strategy.
Takeaways & Limitations
The framework jointly solves optimal scheduling and limit-order posting and can be used for entire liquidations or short slices intended to follow a trading curve.
Takeaways & Limitations
When execution is too slow or liquidation urgency is high, negative quotes can appear and a model including market orders may be better suited.
Abstract
from arXiv · showhide
This paper addresses the optimal scheduling of the liquidation of a portfolio using a new angle. Instead of focusing only on the scheduling aspect like Almgren and Chriss, or only on the liquidity-consuming orders like Obizhaeva and Wang, we link the optimal trade-schedule to the price of the limit orders that have to be sent to the limit order book to optimally liquidate a portfolio. Most practitioners address these two issues separately: they compute an optimal trading curve and they then send orders to the markets to try to follow it. The results obtained here solve simultaneously the two problems. As in a previous paper that solved the "intra-day market making problem", the interactions of limit orders with the market are modeled via a Poisson process pegged to a diffusive "fair price" and a Hamilton-Jacobi-Bellman equation is used to solve the problem involving both non-execution risk and price risk. Backtests are carried out to exemplify the use of our results, both on long periods of time (for the entire liquidation process) and on slices of 5 minutes (to follow a given trading curve).
Introduction
The paper jointly optimizes liquidation timing and limit-order prices using a liquidity-providing approach that accounts for price and non-execution risk. It extends prior scheduling and market-making models with tractable optimal quotes and applications to full liquidations and short trading slices.
- Prior liquidation models mainly trade off market impact or execution costs against price risk, while few model interactions with the order book.
- The paper solves optimal scheduling and limit-order posting simultaneously rather than computing a trading curve and following it separately.
- Limit orders remove execution costs in the model but introduce non-execution risk because execution timing and probability depend on the posted price.
- Trade arrivals are modeled with a Poisson intensity linked to the distance between the limit order and a Brownian-motion fair price.
- Unlike related market-making models relying on approximations or numerical PDE methods, the paper provides simple, easy-to-compute optimal quotes.
1 Setup of the model
The model describes a trader liquidating inventory through continuously updated ask limit orders whose executions arrive stochastically. The trader chooses quotes to maximize expected CARA utility while accounting for price risk, remaining inventory, and terminal liquidation cost.
- The reference stock price follows a Brownian motion with drift, and the trader must liquidate an initial quantity q0 over horizon T.
- The trader continuously posts an ask quote, while a unit-jump counting process records executions and reduces inventory from q0.
- Execution intensity depends on the quoted price, with lower-priced orders executed faster.
- The objective maximizes expected CARA utility of terminal profit and loss, including cash, remaining inventory, risk aversion, and a cost for shares left at T.
2 Optimal quotes
The optimal-quote problem is formulated through a Hamilton-Jacobi-Bellman equation and transformed into linear ordinary differential equations. Numerical examples show quotes varying with inventory and time, while also revealing cases where limit orders alone are unsuitable.
- The control problem is represented by a Hamilton-Jacobi-Bellman equation whose solution is the value function.
- A CARA-based change of variables and exponential execution intensity reduce the problem to a linear system of ordinary differential equations.
- The verification theorem derives the value function and expresses the optimal ask quote from the ODE solution.
- 2.2 Numerical example: For q = 6 shares over 5 minutes, optimal quotes decrease as inventory increases because larger inventories require faster trading to reduce price risk.
- 2.2 Numerical example: Quotes are lowered as the deadline approaches, except near T, where they converge to a value determined by liquidation cost b.
- 2.2 Numerical example: Negative quotes can arise when inventory is large relative to remaining time, liquidation urgency is high, or risk aversion and volatility are high.
- 2.2 Numerical example: The simulated trading curve can leave inventory at T when b = 3 because the incentive to liquidate strictly before the deadline is weak.
3 Special cases
The paper derives tractable special cases for optimal limit-order liquidation, clarifying how quotes and trading curves respond to time horizon, price risk, liquidation costs, and execution incentives.
- 3.1 Asymptotic behavior as T →+∞: As T tends to infinity, the optimal quotes converge to limiting values under the stated parameter condition.The asymptotic result is established for µ < 2γσ^2; the risk-neutral case does not generally share this convergence.
- 3.1 Asymptotic behavior as T →+∞: The asymptotic quote decreases with inventory and volatility but increases with drift and the arrival rate of liquidity-consuming orders.For positive quotes, it decreases with k; higher volatility speeds liquidation, while positive drift encourages slower execution.
- 3.2 Absence of price risk and risk-neutrality: With no drift and no volatility, the optimal quote increases with liquidity-arrival intensity and decreases with risk aversion and liquidation cost.The liquidation cost b encourages faster execution because each share remaining at time T incurs that cost.
- 3.2 Absence of price risk and risk-neutrality: In the no-drift/no-volatility case, the optimal quote is bounded below by −b because execution is guaranteed at price s −b at time T.This lower bound follows from the terminal liquidation-cost structure.
- 3.3 Limiting behavior as b →+∞: As b tends to infinity, the limiting trading curve becomes independent of the liquidity-arrival parameter A.The limiting solution has the form Aqvq(t), with v independent of A, and the resulting expected inventory curve inherits that independence.
- 3.3 Limiting behavior as b →+∞: In the high-liquidation-incentive limit, greater risk aversion steepens the trading curve, while a sufficiently strong positive trend can make it concave.The trend slows execution to benefit from rising prices, competing with the incentive created by price risk.
4 Comparative statics
The comparative statics show how optimal liquidation quotes respond to drift, volatility, execution intensity, risk aversion, liquidation cost, and the intensity-shape parameter. Most parameters shift quotes monotonically, while k has an ambiguous effect when negative quotes are possible.
- Influence of the drift µ: Optimal quotes increase with expected price drift: anticipated price declines prompt lower prices for faster execution, whereas anticipated rises support deeper quotes.The relationship is reported as increasing in µ and exemplified by Table 1.
- Influence of the volatility σ: Higher volatility lowers optimal quotes because the trader reduces the additional price risk.This monotonic effect is observed numerically in Table 2.
- Influence of the intensity scale parameter A: Higher A raises optimal quotes because greater execution probability allows transactions at higher prices.The relationship is observed numerically in Table 3.
- Influence of the intensity shape parameter k: The effect of k is ambiguous: it lowers quotes when optimal quotes remain positive but reverses when high price risk produces negative quotes.The reversal follows from the exponential execution intensity and the possibility of negative quotes; Tables 4 and 5 illustrate the two cases.
- Influence of the risk aversion γ: Higher risk aversion γ lowers optimal quotes by reducing both price risk and non-execution risk.The reference case γ = 0.05 differs strongly from γ = 0.01 and the risk-neutral case.
- Influence of the liquidation cost b: Higher liquidation cost b lowers optimal quotes because it increases the need to sell before T.The value of remaining shares at T decreases as b increases, matching Table 7.
5 Historical simulations
The historical simulations adapt the continuous model to discrete prices and times, calibrate selected parameters using market data, and test limit-order liquidation on AXA. Examples cover five-minute slices in bullish and bearish periods and a two-hour liquidation of 20 times ATS.
- Model adaptation: The implementation rounds continuous optimal quotes to ticks because real trading is discrete in price and order priority.Changing order position too often can reduce the chance of execution.
- Parameter calibration: A and k are calibrated from trade-by-trade limit-order-book data using the market bid-ask spread, while γ is chosen separately.The spread dependence is an off-model calibration hypothesis because the underlying market is not explicitly modeled.
- Backtest assumptions: The backtests assume complete fills whenever a trade occurs at or above the agent’s quoted ask price.They use trade-by-trade data and are intended as examples across several situations.
- Five-minute simulations: Five-minute AXA examples liquidate 3 times ATS in both bullish and bearish periods; the bullish example executes its first order after 50 seconds and outperforms a market order in that example.The strategy raises quotes after early execution and lowers them successively when an order is not filled.
- Five-minute simulations: In the bearish AXA example, early execution is followed by substantially lower final-order prices as urgency to sell increases.The authors raise the practical question of linking a trend detector to the liquidation algorithm.
- Two-hour simulation: A longer simulation liquidates 20 times ATS over two hours, representing about 5% of the period’s volume.The example is shown in Figure 6.
Conclusion
The paper formulates optimal liquidation with passive orders as a joint scheduling-and-posting problem that accounts for price risk and non-execution risk. It derives an analytically tractable solution, studies limiting cases, and reports promising numerical experiments and backtests while identifying important extensions.
- The framework jointly optimizes liquidation scheduling and limit-order posting while accounting for price risk and non-execution risk.
- An innovative change of variables reduces the four-variable Hamilton-Jacobi-Bellman equation to a system of linear ordinary differential equations.The optimal quote is then obtained by solving the ODE system and deducing the order price.
- Limiting-case analyses characterize the optimal strategy's asymptotic behavior and recover the risk-neutral result obtained in parallel by Bayraktar and Ludkovski.
- Numerical experiments and backtests produce promising results.
- The model does not explicitly represent passive market impact from liquidity provision, which the authors propose to address in future versions.
- The separation of variables relies on CARA utility and Brownian price dynamics, while generalizing the order-arrival function remains ongoing work.
Appendix
The appendix verifies the proposed value function and optimal control, reduces the system to linear dynamics, and derives asymptotic and limiting-case results for quotes and trading intensity.
- The proof constructs a candidate value function from a system whose components are strictly positive, then verifies its value-function and optimal-control properties.
- The optimal-control derivation uses a first-order condition for a maximum and treats β = 0 by continuity or an analogous argument.
- The non-local equation is represented through a vector system w′(t) = Mw(t), with eigenvectors associated with eigenvalues αj^2 − βj.
- Under α > β, the long-horizon limit satisfies w∞ := limT→+∞ w(0) = c0f0 and characterizes the asymptotic optimal ask quote.
- The limiting trading intensity is independent of A, so the resulting trading curve is also independent of A.
- In the risk-neutral limiting case, the trading intensity is proportional to q and the trading curve is characterized by an ordinary differential equation.