Liquidation in Limit Order Books. LOBs with Controlled Intensity

Transcription

1 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Liquidation in Limit Order Books with Controlled Intensity Erhan and Mike Ludkovski University of Michigan and UCSB 1 / 23

2 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 2 / 23

3 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books PriceExtensions 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. 3 / 23

4 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books PriceExtensions 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. 3 / 23

5 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books PriceExtensions Model 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. N t t 0 Λ s ds is a martingale and expected revenue is [ n ] E e rτ i (P τi + s τi 1 {τi T }). i=1 4 / 23

6 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books PriceExtensions 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 over the spreads. 5 / 23

7 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books PriceExtensions Model Inventory Process Optimization Problem Start with n shares to sell. Maximize expected revenue until T : E [ n i=1 e rτ i (Pτi + s τi 1 {τi T }) ]. 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 ] V (n, T ) := sup E e rτ i s τi 1 {τi T }. (s t ) S T S T is the collection of F-adapted controls, s t 0 with F t := σ(n s : s t). i=1 6 / 23

8 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books PriceExtensions Model 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) V (n, T ) = sup E e rt s t dn t = sup E e rt s tλ(s t) dt. (s t ) S T (s t ) S T 0 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

9 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books PriceExtensions Model 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, st = s(xt, 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

10 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Extensions Solutions Continuous Selling Limit Power-Law LOB: Explicit Solution Proposition Assume that Λ(s) = λs α, α > 1. Then ( V (n, T ) = c n 1 e rαt ) ( ) 1/(α 1) 1/α λ, s (n, T ) = (1 e rαt ) 1/α, αrc n with c n satisfying the recursion where A α := (α 1)α 1 α α. rc n = A α λ(c n c n 1 ) 1 α, n 1, c 0 = 0, 9 / 23

11 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Extensions 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 )) = everything by T. C(n) 1 e rαt On infinite horizon, lim T V (n, T ) = c n. Also have an explicit solution when r = 0. so P(σ i T τ i 1 ) = 1 for all i n. Liquidate 10 / 23

12 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Extensions 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 λ + sup s 0 s α (V (x, T ) V (x, T ) + s ) rv = 0. (1) As 0, the limiting PDE is v T + sup s 0 λ s vx s α explicit solution rv = 0, which has v(x, T ) = ( ) 1/α λ x (α 1)/α ( 1 e rαt ) 1/α. rα The optimizer above is s (0) (x, T ) = ( ) λ 1/α 1 ( ) αr x 1 e rαt 1/α. 1/α 11 / 23

13 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Extensions 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. 12 / 23

14 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Extensions 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. 12 / 23

15 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Extensions Solutions Continuous Selling Limit Implications for the Original Model Theorem tells us that c n ( 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 1/α 1 αr as n. n 1/α 13 / 23

16 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Explicit Extensions Solutions Continuous Selling Limit Numerical Illustration of the Convergence 1.08 v(x)/v 0.05 (x) v(x)/v 0.01 (x) Ṽ 0.05 (x)/v 0.05 (x) x 0.92 s (0) (x) s (0.05) (x) s (0) (x) s (0.01) (x) 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 = Right panel: ratio of the fluid limit optimal control s (0) (x) to the discrete s ( ) (x) for {0.01, 0.05}. 14 / 23

17 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books FiniteExtensions 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. 15 / 23

18 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books FiniteExtensions 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 ) = n ( ) j κ log 1 λt, s (n, T ) = 1 j! e κ + V (n ) V ((n 1) ). j=0 As 0, V (n, T ) v(x, T ) uniformly on compacts, where v(x, T ) solves the nonlinear first order PDE v T (x, T ) = λ κ e 1 κvx (x,t ), v(0, T ) = v(x, 0) = / 23

19 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books FiniteExtensions Horizon Infinite Horizon Solution Structure s (n, T ) 1 κ. Never trade close to the bid. Have the bounds ( ) x λ κ log x T v(x, T ) λ κe T. Cannot guarantee liquidation by T. Impossible to do so when x > λe 1 T. 17 / 23

20 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books FiniteExtensions Horizon Infinite Horizon Explicit Solution III: Infinite Horizon Proposition For exponential-decay LOB with T = + and discounting rate r > 0 we have V (x) = ( (λr κ W 1 1 exp κ V (x ) 1) ), x {0,, }. where W is the Lambert-W function. As 0, V (x) v(x) uniformly on compacts where ( ) eκrv(x) li λ := erx y, li(y) := λ 0 1 log t dt. 18 / 23

21 Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books FiniteExtensions 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, e 1 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). 19 / 23

22 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 order books. Furthermore if s Λ(s) is decreasing and then: both V and v are concave; s ( ) and s (0) are decreasing; s ( ) s (0) uniformly on compacts. Λ(s)Λ (s) (Λ (s)) 2 < 2, s R +, 20 / 23

23 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) = λ Mt /s α where (M t ) 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. A α λ 0 v (x) 1 α + A α λ 1 (x δ) α (v(x) v((x δ) + )) 1 α rv = 0, Can do asymptotic expansions in λ / 23

24 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! 22 / 23

25 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! 22 / 23

26 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 G. BARLES AND P. E. SOUGANIDIS, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), pp E. BAYRAKTAR AND M. LUDKOVSKI, Optimal trade execution in illiquid financial markets Math. Finance, 21(4) (2011), pp 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, Available at SSRN: R. CONT, S. STOIKOV, AND R. TALREJA, A stochastic model for order book dynamics Operations Research, 58 (2010), pp M. V. DAY, Weak convergence and fluid limits in optimal time-to-empty queueing control problems. Available at (2010). 23 / 23