Liquidation in Limit Order Books with Controlled Intensity Erhan - - PowerPoint PPT Presentation

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions

Liquidation in Limit Order Books with Controlled Intensity

Erhan Bayraktar and Mike Ludkovski

University of Michigan and UCSB Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions

Outline

1

Limit Order Book Model Price Model Inventory Process

2

Power-Law Intensity Law Explicit Solutions Continuous Selling Limit

3

Exponential Decay Order Books Finite Horizon Infinite Horizon

4

Extensions

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Price Model Inventory Process

Liquidation via Limit Orders

An investor wishes to liquidate a large position. To have a guaranteed execution price, use limit orders. No guaranteed execution time: trading frequency depends on the spread between limit price and bid price. Objective: maximize total revenue by a fixed liquidation date T. Use a queue representation of limit order book. Study a stochastic control problem where the investor controls the frequency of trading. Leads to nonlinear first-order ordinary differential equations.

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Price Model Inventory Process

Liquidation via Limit Orders

An investor wishes to liquidate a large position. To have a guaranteed execution price, use limit orders. No guaranteed execution time: trading frequency depends on the spread between limit price and bid price. Objective: maximize total revenue by a fixed liquidation date T. Use a queue representation of limit order book. Study a stochastic control problem where the investor controls the frequency of trading. Leads to nonlinear first-order ordinary differential equations.

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Price Model Inventory Process

Price Model

(Pt): bid price process. Assume e−rtPt is a martingale. (Nt): counting process of order fills and τk the corresponding arrival times, Nt =

k 1{τk≤t}. Each order is unit size.

Λt: (controlled) intensity of order fill. st ≥ 0: spread between the bid price and the limit order of the investor. Nt − t

0 Λs ds is a martingale and expected revenue is

E n

• i=1

e−rτi(Pτi + sτi1{τi≤T})

• .

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Price Model Inventory Process

Background

Unaffected price is a martingale. No price impact, however, trade intensity depends on the strategy. Inspired by the model of Stoikov and Avellaneda (2009). Our view of the LOB as a system of Poisson processes is similar to recent work by Cont and co-authors on multi-queue formulations. In previous work (BL11) considered a similar case but without control

• ver the spreads.

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Price Model Inventory Process

Optimization Problem

Start with n shares to sell. Maximize expected revenue until T: E n

i=1 e−rτi(Pτi + sτi1{τi≤T})

• .

Assume intensity of order fills is Λ(st). Remaining shares at T are liquidated at the bid price. Since e−rtPt is a martingale, ignore the baseline revenue nP0. Maximize expected profit due to limit orders: V(n, T) := sup

(st)∈ST

E n

• i=1

e−rτisτi1{τi≤T}

• .

ST is the collection of F-adapted controls, st ≥ 0 with Ft := σ(Ns : s ≤ t).

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Price Model Inventory Process

Inventory Process

Xt: remaining inventory at time t with X0 = n. Xt := X0 − Nt is a “death" process with intensity Λ(st). Re-write

V(n, T) = sup

(st )∈ST

E T∧τ(X) e−rtst dNt

• =

sup

(st )∈ST

E T∧τ(X) e−rtstΛ(st) dt

• .

τ(X) := inf{t ≥ 0 : Xt = 0} is the time of liquidation. Boundary conditions are V(n, 0) = 0 ∀n (terminal condition in time) V(0, T) = 0 ∀ T (exhaustion).

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Price Model Inventory Process

Nonlinear ODE

Using standard methods, value function is the viscosity solution of − VT + sup

s≥0

• Λ(s) · (V(n − 1, T) − V(n, T) + s)
• − rV(n, T) = 0,

with boundary conditions V(0, T) = V(n, 0) = 0 and VT denoting partial derivative wrt time-to-expiration. Optimal control is of Markov feedback type, s∗

t = s(X ∗ t , T − t).

Focus on:

◮ Special functional forms of Λ(s) that admit closed-form solutions. ◮ The fluid limit where order size ∆ → 0 and trade intensity is Λ(s)/∆ → ∞. Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Explicit Solutions Continuous Selling Limit

Power-Law LOB: Explicit Solution

Proposition

Assume that Λ(s) = λs−α, α > 1. Then V(n, T) = cn

• 1 − e−rαT1/α ,

s∗(n, T) = λ αrcn 1/(α−1) ·

• 1 − e−rαT1/α ,

with cn satisfying the recursion rcn = Aαλ(cn − cn−1)1−α, n ≥ 1, c0 = 0, where Aα := (α−1)α−1

αα

.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Explicit Solutions Continuous Selling Limit

Solution Structure

Empirical tests suggest that α ∈ [1.5, 3]. V(n, T) is concave in n. n → s∗(n, T) is decreasing. Λ(s∗(n, T)) =

C(n) 1−e−rαT so P(σi ≤ T − τi−1) = 1 for all i ≤ n. Liquidate

everything by T. On infinite horizon, limT→∞ V(n, T) = cn. Also have an explicit solution when r = 0.

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Explicit Solutions Continuous Selling Limit

Continuous Selling Limit

Consider the problem where shares are sold at ∆ increments and Λ∆(s) := Λ(s)/∆. The corresponding value function V ∆(x, T) with x ∈ {0, ∆, 2∆, · · · }, T ∈ R+ is the viscosity solution of − V ∆

T + sup s≥0

λ sα∆(V ∆(x − ∆, T) − V ∆(x, T) + s∆) − rV ∆ = 0. (1) As ∆ → 0, the limiting PDE is −vT + sups≥0 λ s−vx

sα

− rv = 0, which has explicit solution v(x, T) = λ rα 1/α x(α−1)/α 1 − e−rαT1/α . The optimizer above is s(0)(x, T) = λ

αr

1/α

1 x1/α

• 1 − e−rαT1/α.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Explicit Solutions Continuous Selling Limit

Relationship with Fluid Limit

Theorem

As ∆ → 0, V ∆ → v uniformly on compact sets. Use viscosity arguments of Barles-Souganides. Idea: the pre-limit is a space-discretization of the pde. Immediately get convergence to the viscosity solution and related convergence of the controls. Note: our control set and payoffs are unbounded. Extends results on fluid limit of queues due to Bäuerle, Piunovskiy, Day, etc.

Proposition

For any sequence (∆k) with ∆k = δ2−k, we have V ∆k ↑ v as k → ∞.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Explicit Solutions Continuous Selling Limit

Relationship with Fluid Limit

Theorem

As ∆ → 0, V ∆ → v uniformly on compact sets. Use viscosity arguments of Barles-Souganides. Idea: the pre-limit is a space-discretization of the pde. Immediately get convergence to the viscosity solution and related convergence of the controls. Note: our control set and payoffs are unbounded. Extends results on fluid limit of queues due to Bäuerle, Piunovskiy, Day, etc.

Proposition

For any sequence (∆k) with ∆k = δ2−k, we have V ∆k ↑ v as k → ∞.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Explicit Solutions Continuous Selling Limit

Implications for the Original Model

Theorem tells us that cn ∼ λ

rα

1/α n(α−1)/α as n → ∞.

Corollary

Let us denote by s(∆) the pointwise optimizer in (1). Then we have that s(∆) → s(0) uniformly on compacts. ⇒ Clearly, the marginal spread will go to zero as n → ∞. The corollary gives the rate of convergence: s∗(n, T) ∼ λ

αr

1/α

1 n1/α

as n → ∞.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Explicit Solutions Continuous Selling Limit

Numerical Illustration of the Convergence

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.96 0.98 1 1.02 1.04 1.06 1.08

x v(x)/V 0.05 (x) v(x)/V 0.01 (x) ˜ V 0.05 (x)/V 0.05 (x)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.9 0.92 0.94 0.96 0.98 1

x

s(0)(x) s(0.05) (x) s(0)(x) s(0.01) (x)

Figure: Convergence to the fluid limit for Λ(s) = s−2. Left panel: the ratio between discrete and continuous V ∆(x)/v(x) for ∆ = 0.05 and ∆ = 0.01. Right panel: ratio of the fluid limit optimal control s(0)(x) to the discrete s(∆)(x) for ∆ ∈ {0.01, 0.05}.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Finite Horizon Infinite Horizon

Exponential Decay Order Books

In power-law LOB, can trade instantaneously at the bid price. Realistically, order fill intensity might be finite even at zero spread. Also, the LOB tail for large spreads might be thinner. ⇒ Consider Λ(s) = λe−κs.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Finite Horizon Infinite Horizon

Explicit Solution II: Finite Horizon

Proposition

Consider order size ∆ ∈ [0, 1] and Λ(s) = λe−κs. Then for x = n∆, the value function V ∆(x, T) with boundary condition V ∆(x, 0) = V ∆(0, T) = 0 is

V ∆(x, T) = ∆ κ log  

n

• j=0

1 j! λT ∆e j   , s∗(n∆, T) = 1 κ + V ∆(n∆) − V ∆((n − 1)∆) ∆ .

As ∆ ց 0, V ∆(n∆, T) → v(x, T) uniformly on compacts, where v(x, T) solves the nonlinear first order PDE vT(x, T) = λ κe−1−κvx(x,T), v(0, T) = v(x, 0) = 0.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Finite Horizon Infinite Horizon

Solution Structure

s∗(n∆, T) ≥ 1

κ. Never trade close to the bid.

Have the bounds x κ log λ x T

• ≤ v(x, T) ≤ λ

κeT. Cannot guarantee liquidation by T. Impossible to do so when x > λe−1T.

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Finite Horizon Infinite Horizon

Explicit Solution III: Infinite Horizon

Proposition

For exponential-decay LOB with T = +∞ and discounting rate r > 0 we have V ∆(x) = ∆ κ W

• λr −1∆−1 exp
• κV ∆(x − ∆)

∆ − 1

• ,

x ∈ {0, ∆, · · · }. where W is the Lambert-W function. As ∆ ↓ 0, V ∆(x) → v(x) uniformly on compacts where li eκrv(x) λ

• := −erx

λ , li(y) := y 1 log t dt.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Finite Horizon Infinite Horizon

Power Law vs. Exponential LOB Fluid Limit Optimizer

Figure: Optimal controls for power-law and exponential-decay order books. We take r = 0.1 and depth functions Λ(s) ∈ {s−2, s−3, e1−s}, which have been normalized such that Λ(1) = 1 in all three cases. The plot shows the resulting fluid limit spreads s(0)(x).

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions

General Limit Order Book Shapes

Our fluid limit Convergence Theorem holds for general shapes of limit

• rder books.

Furthermore if s → Λ(s) is decreasing and Λ(s)Λ′′(s) (Λ′(s))2 < 2, ∀s ∈ R+, then:

◮ both V ∆ and v are concave; ◮ s(∆) and s(0) are decreasing; ◮ s(∆) → s(0) uniformly on compacts. Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions

Other Extensions

Liquidity might be stochastic. As a first approximation, have closed-form solution to a power-law LOB with regime-switching Λ(s, t) = λMt/sα where (Mt) is a two-state

• indep. Markov chain.

Can also consider multiple trading venues. For instance, continuous trading on exchange C, discrete orders of ∆ on exchange L. Obtain a nonlinear delay ODE. Aαλ0v′(x)1−α + Aαλ1(x ∧ δ)α(v(x) − v((x − δ)+))1−α − rv = 0, Can do asymptotic expansions in λ1.

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions

Still To Do

Price impact. Combine market orders + limit orders. THANK YOU!

Bayraktar LOBs with Controlled Intensity

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions

Still To Do

Price impact. Combine market orders + limit orders. THANK YOU!

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Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions

References

• M. AVELLANEDA AND S. STOIKOV, High-frequency trading in a limit order book
• Quant. Finance, 8 (2008), pp. 217–224.
• G. BARLES AND P. E. SOUGANIDIS, Convergence of approximation schemes for fully nonlinear second
• rder equations,

Asymptotic Anal., 4 (1991), pp. 271–283.

• E. BAYRAKTAR AND M. LUDKOVSKI, Optimal trade execution in illiquid financial markets
• Math. Finance, 21(4) (2011), pp. 781–801.
• E. BAYRAKTAR AND M. LUDKOVSKI, Liquidation in Limit Order Books with Controlled Intensity

Available on ArXiv.

• A. CARTEA AND S. JAIMUNGAL, Modeling asset prices for algorithmic and high frequency trading, 2010.

Available at SSRN: http://ssrn.com/abstract=1722202.

• R. CONT, S. STOIKOV, AND R. TALREJA, A stochastic model for order book dynamics

Operations Research, 58 (2010), pp. 549–563.

• M. V. DAY, Weak convergence and fluid limits in optimal time-to-empty queueing control problems.

Available at http://www.math.vt.edu/people/day/research/Publications.htm (2010).

Bayraktar LOBs with Controlled Intensity