Optimal split of orders across liquidity pools: a stochastic algorithm approach Sophie Laruelle , Charles-Albert Lehalle & Gilles Pag` es LPMA & Chevreux Atelier TaMS - Coll` ege de France October 22, 2009 Gilles Pag` es Optimal split of orders across liquidity pools
Outline 1 Introduction A simple model for the execution of orders by dark pools 2 Optimal allocation under constraints Optimization Algorithm The ( IID ) setting: a . s . convergence and CLT The ( ERG ) setting: convergence 3 A reinforcement algorithm Existence of an equilibrium A competitive system 4 Numerical Tests The IID setting The ERG setting The pseudo-real data setting 5 Provisional remarks Gilles Pag` es Optimal split of orders across liquidity pools
Outline 1 Introduction A simple model for the execution of orders by dark pools 2 Optimal allocation under constraints Optimization Algorithm The ( IID ) setting: a . s . convergence and CLT The ( ERG ) setting: convergence 3 A reinforcement algorithm Existence of an equilibrium A competitive system 4 Numerical Tests The IID setting The ERG setting The pseudo-real data setting 5 Provisional remarks Gilles Pag` es Optimal split of orders across liquidity pools
Static modelling The principle of a Dark pool is the following: It proposes a bid price with no guarantee of executed quantity at the occasion of an OTC transaction. Usually this price is lower than the bid price offered on the regular market. So one can model the impact of the existence of N dark pools ( N ≥ 2) on a given transaction as follows: Let V > 0 be the random volume to be executed, Let θ i ∈ (0 , 1) be the discount factor proposed by the dark pool i . Let r i denote the percentage of V sent to the dark pool i for execution. Let D i ≥ 0 be the quantity of securities that can be delivered (or mase available) by the dark pool i at price θ i S . Gilles Pag` es Optimal split of orders across liquidity pools
Cost of the executed order The rest of the order is to be executed on the regular market, at price S . Then the cost C of the whole executed order is given by � � � N � N V − C = S θ i min ( r i V , D i ) + S min ( r i V , D i ) i =1 i =1 � � N � = S V − ρ i min ( r i V , D i ) i =1 where ρ i = 1 − θ i ∈ (0 , 1) , i = 1 , . . . , N . Gilles Pag` es Optimal split of orders across liquidity pools
Mean Execution Cost Minimizing the mean execution cost, given the price S , amounts to solving the following maximization problem � N � � ρ i E ( S min ( r i V , D i )) , r ∈ P N max (1) i =1 � � + | � N r = ( r i ) 1 ≤ i ≤ N ∈ R N where P N := i =1 r i = 1 . It is then convenient to include the price S into both random variables V and D i by considering � � V := V S and D i := D i S instead of V and D i . Gilles Pag` es Optimal split of orders across liquidity pools
The dynamical aspect We consider the sequence Y n := ( V n , D n 1 , . . . , D n N ) n ≥ 1 . We will take two types of stationarity assumptions on the sequence The sequence ( Y n ) n ≥ 1 is i.i.d. with ≡ ( IID ) � � R N +1 , B ( R N +1 distribution ν = L ( V , D 1 , . . . , D N ) on ) . + + the sequence ( V n , D n ( i ) i ) n ≥ 1 is a stationary Feller homogeneous Markov chain with distribution L ( V , D i ) , ≡ ( ERG ) i the sequence ( V n , D n ( ii ) i ) n ≥ 1 is ergodic i . e . n � 1 ( R 2 + ) P - a . s . δ ( V k , D k = ⇒ ν i = L ( V , D i ) , i ) n k =1 Gilles Pag` es Optimal split of orders across liquidity pools
Towards some solutions There are two approaches to deal with this problem. A classical maximization (under constraints using a Lagrangian). An approach, somewhat more intuitive, based on a reinforcement principle; the algorithm is devised by R. Berenstein and C.-A. Lehalle in keeping with R. Almgren. We will study both and try comparing their assets and drawbacks. Gilles Pag` es Optimal split of orders across liquidity pools
Outline 1 Introduction A simple model for the execution of orders by dark pools 2 Optimal allocation under constraints Optimization Algorithm The ( IID ) setting: a . s . convergence and CLT The ( ERG ) setting: convergence 3 A reinforcement algorithm Existence of an equilibrium A competitive system 4 Numerical Tests The IID setting The ERG setting The pseudo-real data setting 5 Provisional remarks Gilles Pag` es Optimal split of orders across liquidity pools
The mean execution function of a dark pool Let ϕ : [0 , 1] → R + be the mean execution function of a single dark pool defined by ∀ r ∈ [0 , 1] , ϕ ( r ) = ρ E (min ( rV , D )) (2) where ρ > 0, ( V , D ) is an R 2 + -valued random vector defined on a probability space (Ω , A , P ). To ensure the consistency of the model, we assume that V > 0 P − a . s ., V ∈ L 1 ( P ) and P ( D > 0) > 0 (3) The positivity of V means that we consider only true orders. The fact that D is not identically 0 means that the dark pool exists in practice. Gilles Pag` es Optimal split of orders across liquidity pools
The function ϕ is clearly concave, non-decreasing, bounded and if the distribution function of D V is continuous on R ∗ + , (4) then ϕ is everywhere differentiable on the unit interval [0 , 1] with � � ϕ ′ ( r ) = ρ E 1 { rV < D } V , r ∈ [0 , 1] . (5) So the distribution of D V has no atom except possibly at 0. It can be interpreted as the fact the dark pool has no ”quantized” answer to an order. One extends ϕ on the whole real line into a concave non-decreasing function with lim ±∞ ϕ = ±∞ . Gilles Pag` es Optimal split of orders across liquidity pools
Extension of the mean execution function One extends ϕ in the whole real line into a concave non decreasing function with lim ±∞ ϕ = ±∞ . Extension 8 6 4 2 0 − 2 − 4 − 6 − 1 − 0.5 0 0.5 1 1.5 2 2.5 3 Gilles Pag` es Optimal split of orders across liquidity pools
Optimal allocation of orders among N dark pools Assume that V satisfies (3). We set for every r = ( r 1 , . . . , r N ) ∈ P N , N � Φ( r 1 , . . . , r N ) := ϕ i ( r i ) . i =1 where for every i ∈ I N = { 1 , . . . , N } , ϕ i ( u ) := ρ i E (min ( uV , D i )) , u ∈ [0 , 1] Based on the extension of the functions ϕ i , we can formally extend Φ on the whole affine hyperplan spanned by P N i . e . � � N � r = ( r 1 , . . . , r N ) ∈ R N | H N := r i = 1 i =1 Gilles Pag` es Optimal split of orders across liquidity pools
The Lagrangian Approach We aim at solving the following maximization problem r ∈P N Φ( r ) max (6) The Lagrangian associated to the sole affine constraint is � N � � L ( r , λ ) = Φ( r ) − λ r i − 1 (7) i =1 So, for every ∂ L = ϕ ′ i ∈ I N , i ( r i ) − λ. ∂ r i This suggests that any r ∗ ∈ arg max P N Φ iff ϕ ′ i ( r ∗ i ) is constant when i runs over I N or equivalently if N � i ) = 1 ϕ ′ i ( r ∗ ϕ ′ j ( r ∗ ∀ i ∈ I N , j ) . (8) N j =1 Gilles Pag` es Optimal split of orders across liquidity pools
Existence of maximum To ensure that the candidate provided by the Lagragian approach is the true one, we need an additional assumption on ϕ to take into account the bahaviour of Φ on the boundary of ∂ P N . Proposition 1 Assume that ( V , D i ) satisfies (3) and (4) for every i ∈ I N . Assume that the functions ϕ i satisfy the following assumption � � 1 ϕ ′ ϕ ′ ( C ) ≡ min i (0) > max . (9) i N − 1 i ∈ I N i ∈ I N Then arg max H N Φ = arg max P N Φ ⊂ int ( P N ) where � � r ∈ P N | ϕ ′ i ( r i ) = ϕ ′ arg max P N Φ = 1 ( r 1 ) , i = 1 , . . . , N . Gilles Pag` es Optimal split of orders across liquidity pools
Interpretation and Comments Assumption ( C ) is a kind of homogenity assumption on the rebates made by the involved dark pools. If we assume that i ∈ I N P ( D i = 0) = 0 for every (all dark pools buy or sell at least one security at the announced price!), then � � E V 1 { V N − 1 ≤ D i } ( C ) ≡ min ρ i > max ρ i E V i ∈ I N i ∈ I N since ϕ ′ i (0) = ρ i E V . In particular, it is always satisfied when ρ i = ρ, i ∈ I N . Gilles Pag` es Optimal split of orders across liquidity pools
Interpretation of Condition ( C ) We consider the case where N = 2. We have then the two following derivatives ′ ′ ′ ϕ 1 ( r 1 ) and ϕ 2 ( r 2 ) = ϕ 2 (1 − r 1 ) Condition (C) satisfied Condition (C) not satisfied 4.5 4 φ ’ φ ’ 1 (r) 1 (r) φ ’ φ ’ 2 (1 − r) 2 (1 − r) 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 r r Gilles Pag` es Optimal split of orders across liquidity pools
Design of the stochastic algorithm Remark: a 1 , . . . , a N are equal iff a i = a 1 + ··· + a N , ∀ 1 ≤ i ≤ N . Then using N the representation of the derivatives ϕ ′ i yields that, if Assumption ( C ) is satisfied, then r ∗ ∈ arg max P N Φ ⇔ � � �� � N i V < D i } − 1 ∀ i ∈ { 1 , . . . , N } , E V ρ i 1 { r ∗ j =1 ρ j 1 { r ∗ = 0 . j V < D j } N Consequently, this leads to the following recursive zero search procedure i + γ n +1 H i ( r n , Y n +1 ) , r 0 ∈ P N , i ∈ I N , r n +1 = r n (10) i Gilles Pag` es Optimal split of orders across liquidity pools
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