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 1 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Outline Limit Order Book Model 1 Price Model Inventory Process Power-Law Intensity Law 2 Explicit Solutions Continuous Selling Limit Exponential Decay Order Books 3 Finite Horizon Infinite Horizon Extensions 4 2 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Price Model Extensions 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. 3 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Price Model Extensions 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. 3 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Price Model Extensions Inventory Process Price Model ( P t ) : bid price process. Assume e − rt P t is a martingale. ( N t ) : counting process of order fills and τ k the corresponding arrival times, N t = � k 1 { τ k ≤ t } . Each order is unit size. Λ t : (controlled) intensity of order fill. s t ≥ 0: spread between the bid price and the limit order of the investor. � t N t − 0 Λ s ds is a martingale and expected revenue is � n � � e − r τ i ( P τ i + s τ i 1 { τ i ≤ T } ) E . i = 1 4 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Price Model Extensions 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 over the spreads. 5 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Price Model Extensions Inventory Process Optimization Problem Start with n shares to sell. �� n i = 1 e − r τ i ( P τ i + s τ i 1 { τ i ≤ T } ) � Maximize expected revenue until T : E . Assume intensity of order fills is Λ( s t ) . Remaining shares at T are liquidated at the bid price. Since e − rt P t is a martingale, ignore the baseline revenue nP 0 . Maximize expected profit due to limit orders: � n � � e − r τ i s τ i 1 { τ i ≤ T } V ( n , T ) := sup . E ( s t ) ∈S T i = 1 S T is the collection of F -adapted controls, s t ≥ 0 with F t := σ ( N s : s ≤ t ) . 6 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Price Model Extensions Inventory Process Inventory Process X t : remaining inventory at time t with X 0 = n . X t := X 0 − N t is a “death" process with intensity Λ( s t ) . Re-write �� T ∧ τ ( X ) �� T ∧ τ ( X ) � � e − rt s t dN t e − rt s t Λ( s t ) dt V ( n , T ) = sup E = sup E . ( s t ) ∈S T 0 ( s t ) ∈S T 0 τ ( X ) := inf { t ≥ 0 : X t = 0 } is the time of liquidation. Boundary conditions are V ( n , 0 ) = 0 ∀ n (terminal condition in time) V ( 0 , T ) = 0 ∀ T (exhaustion). 7 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Price Model Extensions Inventory Process Nonlinear ODE Using standard methods, value function is the viscosity solution of � � − V T + sup Λ( s ) · ( V ( n − 1 , T ) − V ( n , T ) + s ) − rV ( n , T ) = 0 , s ≥ 0 with boundary conditions V ( 0 , T ) = V ( n , 0 ) = 0 and V T 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 ) / ∆ → ∞ . 8 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Solutions Extensions Continuous Selling Limit Power-Law LOB: Explicit Solution Proposition Assume that Λ( s ) = λ s − α , α > 1 . Then � λ � 1 / ( α − 1 ) 1 − e − r α T � 1 /α , 1 − e − r α T � 1 /α , � s ∗ ( n , T ) = � V ( n , T ) = c n · α rc n with c n satisfying the recursion rc n = A α λ ( c n − c n − 1 ) 1 − α , n ≥ 1 , c 0 = 0 , where A α := ( α − 1 ) α − 1 . α α 9 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Solutions Extensions 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. C ( n ) Λ( s ∗ ( n , T )) = 1 − e − r α T so P ( σ i ≤ T − τ i − 1 ) = 1 for all i ≤ n . Liquidate everything by T . On infinite horizon, lim T →∞ V ( n , T ) = c n . Also have an explicit solution when r = 0. 10 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Solutions Extensions 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 λ s α ∆( V ∆ ( x − ∆ , T ) − V ∆ ( x , T ) + s ∆) − rV ∆ = 0 . − V ∆ T + sup (1) s ≥ 0 As ∆ → 0, the limiting PDE is − v T + sup s ≥ 0 λ s − v x − rv = 0, which has s α explicit solution � λ � 1 /α 1 − e − r α T � 1 /α . x ( α − 1 ) /α � v ( x , T ) = r α � λ � 1 /α 1 − e − r α T � 1 /α . The optimizer above is s ( 0 ) ( x , T ) = 1 � α r x 1 /α 11 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Solutions Extensions 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 = δ 2 − k , we have V ∆ k ↑ v as k → ∞ . 12 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Solutions Extensions 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 = δ 2 − k , we have V ∆ k ↑ v as k → ∞ . 12 / 23 Bayraktar LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Solutions Extensions Continuous Selling Limit Implications for the Original Model � λ � 1 /α n ( α − 1 ) /α as n → ∞ . Theorem tells us that c n ∼ r α 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 � λ � 1 /α gives the rate of convergence: s ∗ ( n , T ) ∼ 1 as n → ∞ . α r n 1 /α 13 / 23 Bayraktar LOBs with Controlled Intensity
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