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Liquidation in Limit Order Books with Controlled Intensity

Erhan Bayraktar, Michael Ludkovski

arXiv:1105.0247v5q-fin.TRcs.SYeess.SYmath.OC

TL;DR

Liquidating large securities positions is an important financial-mathematics problem, and the paper studies maximizing expected revenue by date T. It starts from a discrete-state investor problem and uses viscosity techniques to show convergence.

  • Problem

    Liquidating large securities positions is an important problem in financial mathematics, with the goal of maximizing expected revenue by date T.

    Source: §1 · p. 1

  • Method

    The paper starts from a discrete-state problem for an investor holding n shares of an illiquid asset.

    Source: §1 · p. 2

  • Results

    The paper uses viscosity techniques to show convergence.

    Source: §1 · p. 2

  • Takeaways & Limitations

    s*(n,T) decreases with n, which the passage links to the concavity of V(n,T).

    Source: §2 · p. 4

    A full treatment of the problem is deferred to a separate paper.

    Source: §1 · p. 2

Abstract

from arXiv · show

We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power-law and exponential-decay order book models and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.

1. Introduction

The paper frames limit-order liquidation as an expected-revenue control problem under fill risk, with the investor choosing an order price that affects execution. It develops discrete-inventory and fluid-limit analyses, including closed-form cases and convergence results.

1. Introduction

  • Limit-order liquidation exposes the investor to fill risk: she chooses the order price, trading off a higher price against a lower chance of execution while maximizing expected revenue by T.The paper focuses on fill risk associated with limit orders being lifted by sufficiently large market orders.Source: §1 · pp. 1–2
  • The model treats fills as discrete point-process events and lets the investor control fill intensity through the offered spread.The investor's inventory decreases as fills occur.Source: §1 · pp. 1–2
  • A full treatment of simultaneous multi-exchange trading is deferred; the ideal broader model would also reconcile market- and limit-order approaches and investor risk preferences.The introduction identifies these as aspects a fully specified model should address.Source: §1 · p. 2
  • For large inventories, the fluid limit supplies asymptotic results for the discrete-state problem; viscosity techniques establish convergence of both value functions and corresponding optimal controls.The paper identifies viscosity methods as a different approach from earlier probabilistic arguments.Source: §1 · p. 2
  • Closed-form solutions in most examples link the limit-order-book model and depth function to liquidation strategy, including power-law and exponential-shape intensities.The introduction also previews extensions and numerical illustrations, including generic intensity functions.Source: §1 · pp. 3–4

2. Power-Law Limit Order Books

For a power-law spread intensity, the section derives the value and optimal spread and shows that the optimal spread decreases with inventory. In the infinite-horizon limit, the strategy is time-homogeneous.

2. Power-Law Limit Order Books

  • Under Λ(s)=λs^-α, with α>1, Proposition 2.1 gives the value function and optimal spread, with c_n determined by a recursion.Source: §2 · p. 4
  • The value function is concave in inventory, so the optimal spread decreases as inventory increases.Source: §2 · p. 4
  • Even without a direct penalty for leftover orders, the full inventory is liquidated with probability one.Source: §2 · p. 5
  • Adding a liquidation penalty −M1{x≥0}, for any M≥0, leaves the Proposition 2.1 solution unchanged.Source: §2 · pp. 4–5

2.1. Infinite Horizon.

  • As T→∞, the strategy becomes time-homogeneous and the value function is V(n)=c_n.Source: §2.1 · p. 5

3. Continuous selling limit

This section studies the continuous-selling limit, establishing convergence of discrete liquidation values and optimal strategies to their limits using viscosity-solution methods, along with monotone convergence on dyadic grids and asymptotics for related quantities. It also describes the spread’s convergence rate as a function of remaining inventory and the infinite-horizon execution of orders in logarithmic time, with the remainder executed arbitrarily slowly as x1 approaches zero.

Continuous selling limit

  • As Δ→0, V^Δ converges to v uniformly on compact sets.Theorem 3.1 states this convergence for the value functions.Source: §3 · p. 7
  • The proof uses analytical approximation and viscosity-solution convergence, viewing the prelimit problem as a space-only discretization of the fluid-limit PDE rather than using probabilistic arguments.Source: §3 · p. 9
  • For Δ_k=δ2^-k, V^Δk increases to v as k→∞.Source: §3 · p. 9
  • For fixed x and grids passing through x, the pointwise optimizer s^(Δ)(x,T) converges to s^(0)(x,T).Source: §3 · p. 10
  • The value and optimizer convergence results also yield asymptotics for c_n and the marginal price.Source: §3 · pp. 7, 10, 11

3. Continuous selling limit

  • The spread goes to zero as n →∞, and its convergence rate is obtained as a function of remaining inventory.Source: §3 · p. 11
  • On an infinite horizon, orders are filled in logarithmic time, and as x1 →0 the remainder is executed arbitrarily slowly.Source: §3 · p. 11

4. Exponential-Decay Order Books

The section studies exponential-decay order books as an alternative for less-liquid markets where power-law books’ execution behavior may be unrealistic. It gives finite- and infinite-horizon results and establishes convergence to continuous-inventory limits.

4. Exponential-Decay Order Books

  • Exponential-decay books keep the optimal spread bounded away from zero, so trades are never placed close to the bid.This contrasts with power-law books, where the optimal spread tends to zero as inventory grows.Source: §4 · p. 11, §4.2 · p. 14

4.1. Finite Horizon.

  • With a finite horizon and no discounting (r=0), the discrete value function V∆(x,T) has a closed form for x=n∆, subject to V∆(0,T)=0 and Λ(s) given in (4.1).Source: §4.1 · p. 11
  • The discrete finite-horizon value function converges uniformly on compacts to a continuous limit solving a nonlinear first-order PDE.The limit has boundary conditions v(0,T)=v(x,0)=0.Source: §4.1 · p. 12
  • For x>λe^-1T, the bounded trading rate prevents full liquidation by horizon T, and the continuous value is independent of x in that region.The upper bound on the value is tight there.Source: §4.1 · pp. 12–13
  • The cited spread rule gives s*=1/κ otherwise; its displayed threshold is λT/x ≥ e, so this value applies when λT/x < e.Source: §4.1 · p. 13

4.2. Infinite Horizon.

  • The infinite-horizon exponential-decay model has a closed-form solution, and its discrete value functions converge uniformly on compacts to the continuous limit.The result is stated for positive discounting in the infinite-horizon case.Source: §4.2 · pp. 13–14
  • Doubling the depth parameter κ halves both value functions and the corresponding optimal spreads.The passage describes this as making the order book more shallow.Source: §4.2 · p. 14
  • As inventory grows, the exponential-book optimal spread stays bounded away from zero, unlike the power-book spread, which tends to zero.For small inventories, the optimal spread diverges in both models, but more slowly for exponential decay.Source: §4.2 · p. 14

5. Discussion and Further Extensions

This section discusses convergence to the fluid limit and execution curves, then extends liquidation models to general depth functions, regime-switching liquidity, and trading across two exchanges with different order scales. For the two-exchange model, it reports a unique value-function solution and a leading-order solution from the one-exchange problem when large-order intensity is small.

5.1. Numerical Example: Convergence to the Fluid Limit.

  • For Δ=0.01 and x>1, the fluid-limit approximation differs from V^Δ by less than 1%; V^Δ decreases with Δ and converges to v, while the controls are also very close.Source: §5.1 · pp. 14, 16

5.2. Execution Curves.

  • Execution curves depend on book depth: large α produces an S-shape with high initial and final trading rates, while small α keeps inventory above the linear baseline.For deep books (small α), the profit effect dominates and slows execution.Source: Figure 3 · p. 16, §5.2 · p. 17
  • In the fluid limit, execution is linear at a constant rate, whereas the discrete stochastic model can leave positive expected inventory at the deadline.With no discounting in the exponential-book case, the trading rate is independent of depth parameter κ.Source: §5.2 · p. 17

5.3. General Order Book Depth Functions.

  • For infinite-horizon optimal liquidation with general intensity depth function Λ, Theorem 5.1 says Theorem 3.1’s statements hold; with an additional assumption, VΔ and v are concave and controls s(Δ) and s(0) decrease.Source: §5.3 · pp. 17–18

5.4. Regime Switching Market Liquidity.

  • Motivated by evidence that liquidity is not constant, the section introduces a two-state active/slow-market Markov model with power-law order-arrival intensities.In regimes 0 and 1, the intensities are Λ0(s)=λ0/s^α and Λ1(s)=λ1/s^α, respectively.Source: §5.4 · p. 19
  • The regime-switching bounds correspond to single-regime solutions, assuming λ0 > λ1 and that market participants observe and know M.Source: §5.4 · p. 19

5.5. Two-Exchange Multi-scale Model.

  • The two-exchange model combines infinitesimal orders on exchange C with orders of size δ on exchange L, with a final trade liquidating any remaining inventory below δ.Source: §5.5 · p. 21
  • The two-exchange value function solving (5.8) is unique.The equation is a first-order nonlinear delay ODE solved by successive patching.Source: §5.5 · p. 21
  • Numerical computation must handle a singularity, and the solution is generally not concave near knots δ, 2δ, …, where v′ may not exist.For sufficiently small x, the benefit of large orders is negligible because the probability of receiving one is very small.Source: §5.5 · p. 21
  • With λ0 fixed at 1 and λ1 small, the leading-order value v0(x) solves the one-exchange problem.Source: §5.5 · pp. 21–22
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