Optimal execution in limit order books with stochastic liquidity Antje Fruth Joint work with Torsten Sch¨ oneborn and Mikhail Urusov Technische Universit¨ at Berlin Deutsche Bank Quantitative Products Laboratory
Bachelier meeting, Toronto, June 2010
Antje Fruth (QP Lab)
Optimal execution
June 2010
1 / 16
Outline
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Problem: Minimize impact on execution prices (as in Predoiu, Shaikhet, Shreve)
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Limit order book model with stochastic liquidity
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Structure of optimal strategies
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Examples and numerical implementation
Antje Fruth (QP Lab)
Optimal execution
June 2010
2 / 16
Block order book model ◮
Market buy order of x0 shares at t = 0 has linear price impact Number of shares
q x0 shares
0 ◮
◮
B0
A0
A0+D0+
Price per share
Ask price At martingale and bid Bt < At effect of A can be neglected for risk neutral investor Dynamic of price displacement D with resilience speed ρ > 0 1 dΘt − ρDt dt qt Impact cost at t: Dt + 2q1 t xt xt dDt =
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Antje Fruth (QP Lab)
Optimal execution
June 2010
3 / 16
Model with stochastic liquidity ◮ ◮
Dynamic order book height: Kt := q1t e.g. positive diffusion Risk-neutral investor wants to purchase x shares on [t, T ]
Singular control problem in continuous time U(t, δ, x, κ) :=
inf
Θ∈A(x)
J(t, δ, Θ, κ)
Admissible strategies A(x) Θ : Ω × [t, T ] → [0, x] adapted,increasing,c` agl` ad, Θt = 0, ΘT + = x a.s. Trading costs (∆Θs := Θs+ − Θs ) hZ i Ks ∆Θs dΘs Dt = δ, Kt = κ J(Θ) := J(t, δ, Θ, κ) := E Ds + 2 [t,T ] Antje Fruth (QP Lab)
Optimal execution
June 2010
4 / 16
Intuition: Wait and Buy region ◮
Scaling property of value function reduces dimension: U(t, aδ, ax, κ) = a2 U(t, δ, x, κ) for a ∈ R≥0 a= δ1 x ⇒ U(t, δ, x, κ) = δ2 U(t, 1, , κ) δ
Antje Fruth (QP Lab)
Optimal execution
June 2010
5 / 16
Intuition: Wait and Buy region ◮
Scaling property of value function reduces dimension: U(t, aδ, ax, κ) = a2 U(t, δ, x, κ) for a ∈ R≥0 a= δ1 x ⇒ U(t, δ, x, κ) = δ2 U(t, 1, , κ) δ
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How could optimal strategy look like for fixed t and κ? x δ
is small, say
x δ
≤ c ∈ (0, ∞]
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Wait if
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Otherwise buy ξ > 0 shares s.t.
x−ξ δ+ ξq
!
=c
At time t
x/d
Barrier c(t,k) Buy (BR) Buy x
Wait (WR) k=1/q
0 Antje Fruth (QP Lab)
Optimal execution
June 2010
5 / 16
WR-BR-WR example Binomial model and resilience=2
kt 3
k0=2.1
p=1/2
1
Scenario B Scenario A t
t0=0 t1=0.0001
Antje Fruth (QP Lab)
T=1
Optimal execution
June 2010
6 / 16
Unique optimal strategies Theorem (F./Sch¨oneborn/Urusov) dKs = µ(s, Ks )ds + σ(s, Ks )dWs Let K be a positive, continuous diffusion satisfying 2
2ρ s) s) + µ(s,K i) ηs := K − σ (s,K > 0 for all s ∈ [t, T ] Ks2 Ks3 s sup K2 ii) E inf s∈[t,T ] Kss < ∞ s∈[t,T ] i hR T 2 0 for all s ∈ [t, T ] Ks2 Ks3 s sup K2 ii) E inf s∈[t,T ] Kss < ∞ s∈[t,T ] i hR T 2
