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Optimal execution strategies in limit order books with general shape functions
Aurélien Alfonsi, Antje Fruth, Alexander Schied
TL;DR
The paper addresses optimal execution for large orders in limit order books with nonuniform shapes and nonlinear market impact. It generalizes a resilience model using two recovery mechanisms, derives explicit discrete-time strategies, and obtains a block-shaped closed form that solves the earlier recursive scheme. The authors also report robustness of optimal strategies across shape functions and resilience types.
Problem
Large orders create significant price impact, while existing resilience-based models do not accommodate general, empirically observed limit-order-book shapes and nonlinear impact.
Method
The paper introduces a density-based generalized limit order book with exponential recovery of either volume or bid-ask spread, then derives explicit optimal strategies for adapted discrete-time execution.
Results
The paper obtains explicit optimal strategies in both models and a block-shaped closed form that solves Obizhaeva and Wang’s recursive scheme.
Takeaways & Limitations
Optimal strategies exhibit a qualitative pattern that is independent of the limit-order-book shape, with evidence of robustness across shape functions and resilience types.
Takeaways & Limitations
The model assumes constant resilience and equally spaced trading times, while time-varying limit-order-book shapes remain future work.
Abstract
from arXiv · showhide
We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modeling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e., of the volume of the LOB, or the exponential recovery of the bid-ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy, which explicitly solves the recursive scheme given in Obizhaeva and Wang (2005). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.
1 Introduction.
The paper generalizes resilience-based optimal execution in limit order books by allowing nonuniform shapes and two resilience modes. It derives explicit strategies, a block-shaped closed form, and evidence of robustness across modeling choices.
- Large block orders can represent up to twenty percent of daily trading volume, motivating their division into smaller orders over time.
- The model generalizes Obizhaeva and Wang’s limit order book by using a nonuniform shape function, producing nonlinear market-order price impact.
- The paper models resilience through exponential recovery of either limit-order volume or the bid-ask spread, while assuming the shape is constant over time.
- Explicit optimal strategies are derived for adapted strategies, including possible intermediate sells, in both resilience models.
- For block-shaped books, the results provide a closed-form strategy that explicitly solves Obizhaeva and Wang’s recursive scheme.
- The model is time homogeneous, with constant resilience and equally spaced trading times; extensions to time-inhomogeneous settings are left to subsequent work.
2 Two market impact models with resilience.
The generalized limit order book uses a positive density shape function to describe available shares and models resilience through exponential recovery of volume or extra spread. Market orders shift prices nonlinearly when the shape is nonconstant.
- A continuous positive density f defines the number of ask-side shares available at each price above the unaffected best ask; a constant f gives a block-shaped book.
- A buy market order consumes shares across the ask-side book and shifts the actual best ask price upward.
- The resulting price impact is nonlinear in order size unless the shape function is constant over the consumed price interval.
- Model 1: Model 1 specifies exponential recovery of the number of limit orders, representing recovery of order-book volume.
- Model 2: Model 2 specifies exponential decay of the extra bid-ask spread after the trader becomes inactive.
- Both resilience models use a positive constant resilience speed ρ, with dynamics fully specified during inactivity.
3 The cost minimization problem.
The execution problem chooses admissible market-order sizes over equally spaced times to buy a fixed quantity while minimizing expected average cost. Intermediate sells are permitted, and cost depends on the resilience model.
- The trader must buy X0 shares by time T through market orders at N + 1 equally spaced trading times.
- An admissible strategy is a sequence of market-order sizes, each bounded from below.
- Strategies may include intermediate sell orders, provided sell sizes satisfy the lower-bound admissibility condition.
- The objective is to minimize the expected total cost, expressed as the average cost C(ξ) of consecutive market orders.
- The quantitative features of optimal strategies depend somewhat on whether Model 1 or Model 2 is used, although their qualitative features are similar.
- Unlimited order-book depth is assumed for convenience, with practical relevance only when the real book can accommodate every optimal-strategy order.
4 Main theorem for Model 1.
For Model 1, exponential recovery of LOB volume yields a unique deterministic optimal strategy under a one-to-one condition on h1. The strategy uses only buy orders, follows the LOB’s recovery pattern, and has a continuous-time limit with an initial block and constant-rate buying.
- Existence and representation: Theorem 4.1 gives a unique optimal strategy when h1 is one-to-one.The strategy is represented by an initial order, intermediate orders, and a final order.
- Existence and representation: The initial market order is determined as the unique solution of the theorem’s equation, while intermediate and final orders follow the stated recursive formulas.The one-to-one property of h1 ensures uniqueness of the initial order and the full strategy.
- Strategy structure: The optimal strategy is deterministic and consists only of nontrivial buy orders.Thus, no intermediate sell orders occur in Model 1’s optimal strategy.
- Strategy structure: Each market order consumes the shares accumulated through LOB recovery since the preceding order, with all remaining shares bought at the terminal time.This recovery-based execution pattern is independent of the LOB shape.
- Conditions: The function h1 is one-to-one when it is strictly increasing; a sufficient shape condition is that f decreases for y > 0 and increases for y < 0.Empirical order-book shapes with maxima near the best quotes are described as conforming to this assumption.
- Continuous-time limit: In the continuous-time limit, execution consists of an initial block order, continuous buying at rate ρξ(1),∞, and a terminal purchase of the remaining shares.This limiting description follows as the number of trades tends to infinity under the stated convergence conditions.
5 Main theorem for Model 2.
Model 2 assumes exponential recovery of the extra bid-ask spread and derives a unique optimal strategy under conditions on h2 and the shape function. The strategy is deterministic, uses only buy orders, and has a continuous-time limit with initial, continuous, and terminal components.
- Model 2 setup: Model 2 uses exponential recovery of the extra spread, with resilience speed ρ governing the recovery dynamics.
- Optimal strategy: If h2 is one-to-one and the theorem’s shape-function conditions hold, Model 2 has a unique optimal strategy determined by an initial, intermediate, and final order scheme.
- Optimal strategy: The optimal Model 2 strategy is deterministic and consists only of nontrivial buy orders.
- Optimal strategy: Its intermediate trades buy the amount of liquidity recovered since the preceding trade, while the remaining shares are purchased at the terminal time.
- Comparison with Model 1: The strategy is qualitatively similar to Model 1, differing mainly in the initial order size and the resilience mode, supporting robustness across the two models.
- Continuous-time limit: In the continuous-time limit, the strategy consists of an initial block, continuous buying at a constant rate, and a final block at time T.
7 Examples.
The examples examine power-law LOB shapes and find that optimal strategies vary only slightly across shape exponents and resilience modes, supporting quantitative robustness.
- For the power-law family, the model assumptions underlying both optimal-strategy theorems hold for α ≤ 1.
- Optimal strategies vary only slightly when the exponent α or resilience mode changes, indicating robustness beyond qualitative patterns.
- Model 1: In Model 1, the extra-spread dynamics depend on the strategy but not the LOB shape, so shape changes produce few quantitative differences.
- Model 2: For Model 2, increasing shape functions strengthen volume resilience relative to Model 1, while decreasing functions weaken it; steeper shapes amplify these effects.
- Model 1: The Model 1 optimal initial trade increases with the parameter μ controlling the LOB slope.
A Reduction to the case of deterministic strategies.
The paper reduces the adapted execution problem to deterministic cost minimization by collapsing the bid-ask spread and exploiting deterministic spread dynamics.
- The full LOB model is simplified by collapsing bid-ask spreads into a single value while preserving an equivalent optimization problem.
- The simplified processes D and E update after trades and then recover exponentially according to the resilience mode.
- The reduction uses the martingale property of the unaffected best ask to separate the unaffected-price contribution from the optimization.
- Given the trading values, the simplified extra-spread process D evolves deterministically, yielding deterministic cost functions C(i) for Models 1 and 2.
- Each cost function has a unique minimum over deterministic strategies, coinciding with the corresponding optimal strategy, whose trades are buy orders.
B The optimal strategy in Model 1.
For Model 1, the cost function is coercive and has a unique optimal strategy with strictly positive trades, characterized through relations linking every trade to the initial trade.
- The Model 1 cost function tends to +∞ as the strategy norm grows, ensuring existence of a local minimum.
- The initial trade is determined by an equation whose solution is unique and, when it exists, positive.
- The optimal strategy ξ(1) is the unique minimizer of C(1), and all its components are strictly positive.
- The first-order conditions link all trades to the initial trade through recursive relations involving the resilience factor a.
- The resulting one-dimensional minimization establishes both uniqueness of the optimal strategy and its representation.
C The optimal strategy in Model 2.
For Model 2, technical lemmas establish existence and uniqueness of a strictly positive optimal strategy by reducing the first-order conditions to a scalar equation.
- The Model 2 cost function diverges to +∞ as the strategy norm grows, guaranteeing existence of a local minimum.
- Equation (21) has at most one positive solution, supported by positivity of the function g(x) := f(x) − af(ax).
- The unique minimizer of C(2) is the strategy ξ(2), whose components are all strictly positive.
- Lagrange multiplier conditions imply that transformed post-trade quantities are constant across trading times, linking all orders to the initial trade.
- The scalar optimality equation has at most one solution, completing the existence and representation argument for ξ(2).
D Optimal strategy for block-shaped LOB.
This section derives explicit formulas for the backward-scheme coefficients and uses them to verify that the proposed block-shaped LOB strategy satisfies the recursive optimality scheme.
- The proof’s aim is to show that strategy (27) satisfies recursion (31), using explicit formulas for the backward-scheme coefficients.
- Lemma D.1 gives explicit expressions for the coefficient sequences α_n, β_n, and γ_n from (33).
- The sequences δ_n, ϵ_n, and φ_n from (32) are also obtained explicitly by combining the coefficient formulas with the backward schemes.
- The resulting process formulas are checked at n = 0 and then established for n ≥ 1 by induction.
- For n ∈ {1, . . . , N − 1}, the explicit formulas verify the recursion, showing that the optimal strategy in (27) solves (31).