Optimal execution strategies in limit order books Antje Fruth J oint work with Aur´ elien Alfonsi and Alexander Schied www.math.tu-berlin.de/˜fruth Technische Universit¨ at Berlin Deutsche Bank Quantitative Products Laboratory IRTG Summer School, Disentis, Switzerland, July 2008 Antje Fruth (TU Berlin) Optimal execution July 2008 1 / 1
Outline ◮ Problem ◮ Limit order book model ◮ Optimal execution strategy ◮ Examples ◮ Sketch of the proof ◮ Model ramifications Antje Fruth (TU Berlin) Optimal execution July 2008 2 / 1
Problem ◮ Trade a big position of a single asset in fixed time � Price impact! ◮ More precisely: Buy X ∈ N shares over [0 , T ] at equidistant trading times ( t n ) n =0 ,..., N ◮ Find optimal strategy ξ 0 , ..., ξ N with � N n =0 ξ n = X such that expected costs are minimized � risk neutral investor � � N � min ξ E π t n ( ξ n ) n =0 ◮ We need a market model for the transaction cost π ! Antje Fruth (TU Berlin) Optimal execution July 2008 3 / 1
Market: Limit order book (LOB) ◮ Snapshot of a LOB in t = 0: ◮ LOB form: f : R → ]0 , ∞ [ continuous ◮ Unaffected best ask A t is a martingale and the best bid satisfies B t ≤ A t Antje Fruth (TU Berlin) Optimal execution July 2008 4 / 1
Market: Limit order book (LOB) ◮ Price impact of a market buy order x 0 Antje Fruth (TU Berlin) Optimal execution July 2008 5 / 1
Market: Limit order book (LOB) ◮ Resilience of the LOB ◮ Exponential resilience with resilience speed ρ Model E Model D E t 1 = e − ρτ E t 0 + D t 1 = e − ρτ D t 0 + ◮ Our model is a generalization of Obizhaeva, Wang (2005) Antje Fruth (TU Berlin) Optimal execution July 2008 6 / 1
Model Cost of transaction of size x t at time t � D A t + A t x t + xf ( x ) dx buy order D A π t ( x t ) := t � D B t + B t x t + xf ( x ) dx sell order D B t Stochastic optimization problem (risk neutral investor) � N � � min π t n ( ξ n ) ξ E n =0 for all adapted strategies ξ = ( ξ 0 , ..., ξ N ) such that ξ n is bounded from below and � N n =0 ξ n = X Antje Fruth (TU Berlin) Optimal execution July 2008 7 / 1
Optimal strategy Theorem Under some technical assumptions, there exists a unique optimal strategy ξ in both models. It is deterministic , consists only of buy orders and is determined by: Model E Model D � � ξ 0 h E ( ξ 0 ) = 0 h D ( ξ 0 ) = 0 ξ 0 (1 − e − ρτ ) ξ 0 − F ( e − ρτ F − 1 ( ξ 0 )) ξ 1 = ... = ξ N − 1 ξ N X − ξ 0 − ( N − 1) ξ 1 ◮ Interpretation: E t 1 = ... = E t N � ”Optimal level of E ”, trade-off between price impact and attracting new limit sell orders Antje Fruth (TU Berlin) Optimal execution July 2008 8 / 1
Example 1 ◮ Constant limit order book form: f(x)=5,000 x -4 -2 0 2 4 0 X ◮ Same optimal strategy for Model E and D: ξ 0 = ξ N = ( N − 1)(1 − e − ρτ )+2 Number�of�shares 10,223 n 0 10 Antje Fruth (TU Berlin) Optimal execution July 2008 9 / 1
Example 2 ◮ Limit order book form: ◮ Optimal strategy for Model E and D: Antje Fruth (TU Berlin) Optimal execution July 2008 10 / 1
Proof for Model E ! � D A t + A t x t + xf ( x ) dx buy order D A π t ( x t ) := t � D B t + B t x t + xf ( x ) dx sell order D B t � � N � min π t n ( ξ n ) ξ E n =0 1. Reduction to deterministic strategies 2. Lagrange method to determine optimal strategy 3. Uniqueness and positivity of the strategy Antje Fruth (TU Berlin) Optimal execution July 2008 11 / 1
Proof: 1. Reduction to deterministic strategies ◮ W.l.o.g consider only buy orders ◮ Martingale property of A and integrating by parts yields: � D A � � � N � N � � tn + E π t n ( ξ n ) = XA 0 + E xf ( x ) dx D A n =0 n =0 tn � �� � =: C ( ξ 0 ,...,ξ N ) > 0 | � N ◮ Show C has unique minimum in { ( x 0 , ..., x N ) ∈ R N +1 n =0 x n = X } Antje Fruth (TU Berlin) Optimal execution July 2008 12 / 1
Proof: 2. Lagrange method | x |→∞ ◮ Show C ( x ) − → ∞ to guarantee the existence of a Lagrange multiplier ν ∈ R with ∂ C ( x ∗ 0 , ..., x ∗ ν = N ) ∂ x n � �� C − F − 1 � ∂ a ( a n x ∗ 0 + ... + x ∗ + F − 1 ( a n x ∗ 0 + ... + x ∗ = a n ) n ) ∂ x n +1 with resilience coefficient a := e − ρτ ◮ This leads to the system h E ( a n x ∗ 0 + ... + x ∗ n ) = ν (1 − a ) for n = 0 , ..., N − 1 which is explicitly solved by h − 1 x ∗ = E ( ν (1 − a )) 0 x ∗ x ∗ = 0 (1 − a ) for n = 1 , ..., N − 1 n x ∗ X − x ∗ 0 − ( N − 1) x ∗ = N n ∂ x C ( x ) = � ◮ Find x ∗ 0 : C ( x ∗ 0 , ..., x ∗ N ) = C ( x ∗ ∂ 0 ) with h E ( x ) Antje Fruth (TU Berlin) Optimal execution July 2008 13 / 1
Ramifications ◮ Inhomogeneous trading times ( t n ) n =0 ,..., N and time varying resilience ( ρ t ) t ∈ [0 , T ] − � tn tn − 1 ρ t dt a n := e ◮ If f ( x ) ≡ const., then the optimization can be reduced to a quadratic form min x 1 2 � x , Mx � with 1 a 1 a 1 a 2 · · · a 1 ... a N . . a 1 1 a 2 . . ... ∈ ]0 , 1] N +1 , N +1 . M := a 1 a 2 a 2 1 . . ... ... . . a N a 1 ... a N · · · · · · a N 1 Antje Fruth (TU Berlin) Optimal execution July 2008 14 / 1
Ramifications Optimal strategy without constraints There is a unique, deterministic, positive optimal strategy: � � c 1 a n +1 c ξ 0 = , ξ n = c − for n = 1 , ..., N − 1, ξ N = 1 + a 1 1 + a n 1 + a n +1 1 + a N Optimal strategy with constraints x ∈ R N +1 � � � � � N � � � v j , x Linear constraints n =0 x n = X , ≥ 0 Then the optimal strategy is � x = cM − 1 1 + c j M − 1 v j j for constants c , c j uniquely determined by a system of linear equations. Antje Fruth (TU Berlin) Optimal execution July 2008 15 / 1
Conclusion ◮ Market microstructure model for LOB ◮ Improvements compared to Obizhaeva, Wang: ◮ LOB form not necessarily constant � nonlinear price impact ◮ Explicit optimal strategies with similar qualities (”Optimal level of E ”) ◮ More general unaffected best ask, bid Antje Fruth (TU Berlin) Optimal execution July 2008 16 / 1
Thank you for your attention! [1] Alfonsi, A., Fruth, A., Schied, A. Optimal execution strategies in limit order books with general shape functions. Preprint, TU Berlin (2007) [2] Alfonsi, A., Fruth, A., Schied, A. Constrained portfolio liquidation in a limit order book model. Preprint, forthcoming in Banach Center Publications, TU Berlin (2007) [3] Obizhaeva, A., Wang, J. Optimal trading strategy and supply/demand dynamics. Preprint, forthcoming in Journal of Financial Markets Antje Fruth (TU Berlin) Optimal execution July 2008 17 / 1
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