Optimal Placement of a Small Order Under a Diffusive Limit Order Book (LOB) Model

Transcription

1 Optimal Placement of a Small Order Under a Diffusive Limit Order Book (LOB) Model José E. Figueroa-López Department of Mathematics Washington University in St. Louis INFORMS National Meeting Houston, TX Oct. 23, 2017 (Joint work with Hyoeun Lee and Raghu Pasupathy from Purdue U.) J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

2 Outline 1 Introduction 2 Model and Assumptions 3 Main Results 4 Concluding Remarks and Future Work J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

3 LOB based Market: Market Order vs. Limit Order Market Order (MO) pros: immediate execution (i.e., no execution risk), cons: pay spread and an additional fee (f ). Limit Order (LO) pros: get discount" or better price (when executed) and an additional rebate (r), cons: execution risk (the order may never be executed), discount and execution risk are related to the book depth and flow. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

4 Optimal Placement Problem Find the level in the LOB (say, buy side) to place a limit order for one share so that to minimizing the "expected cost" during a fixed time horizon t: Cost is understood as the price paid for the share (taking into account rebate or fee) minus the initial ask price. The optimal placement is the best tradeoff between execution risk and discount. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

5 Guo, de Larrard, and Ruan 2016 Discrete-time model for the best ask price as a symmetric correlated random walk (CRW): [t/δ] S t = S 0 + X i, X i { ε, ε}, δ > 0, i=1 P(X 1 = ε) = 1 P(X 1 = ε) = p, P(X i = ε X i 1 = ε) = P(X i = ε X i 1 = ε) = p < 1/2. Spread is always constant to one-tick ε Constant probability q (0, 1] that a limit order (of size 1) is executed when sitting at the best bid. Order placement (either market or limit order) takes place at time 0 and there is no intermediate cancellation. If the order is not executed by a specified time t, this is cancelled and changed to a market order at time t. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

6 Guo, de Larrard, and Ruan Key Result: The optimal placement strategy is one of the following: (i) placement at the best bid S 0 ε, (ii) placement at the second best bid S 0 2ε, (iii) placement of a market order at time 0. 2 Drawbacks: No consideration of the initial state of the LOB at time 0; however, everything else the same, placement at a large queue should be less desirable; Model does not incorporate any local drift" or momentum (e.g., say, P(X i = ε X i 1 = ε) < P(X i = ε X i 1 = ε)); J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

7 Our work We consider a variation of the problem which incorporates the initial state of the LOB and some assumptions about the order flow during the specified time horizon [0, t]; We analyze the problem when the time changes are frequent enough so that the dynamics of the best ask price can be well approximated by a diffusive process. Find condition under which a nontrivial solution to the problem exists. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

8 Notation S (δ,ε) u 1 S u := denotes the best ask price at time u 0, when tick size is ε and time step is δ (or another parameter such that average time between price changes goes to 0 when δ 0); 2 C δ,ε (x, t): expected cost when the time horizon is t and the limit order is placed at x lower than the best ask price; 3 Ȳt = inf{ S u : u t}: 4 ρ = ρ(t, x): the probability that a limit order placed at level S 0 x is executed, before time t, during the first time when this is possible (i.e., when S u = S 0 x + ε), conditional on the latter event to occur. In general, ρ(t, x) would depend on the initial queue size Q x (0) at the level S 0 x; we expect that ρ(t, x) when Q x (0). J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

9 Expected Cost when order is placed at S 0 x Case 1: Ȳt > S 0 x + ε Order not executed, buy MO at S t = cost = S t S 0 + f. Case 2: The first time S u reaches S 0 x + ε, τ, happens before t: If order is executed before or at τ + δ t: = cost = x r. This happens with probability ρ (0, 1] If order is not executed before τ + δ t: Cancel LO and buy with MO at S0 x + 2ε = cost = x + f + 2ε. Expected Cost and Optimal Placement C δ,ε (x, t) = E [ St S 0 Ȳt > S 0 x + ε ] P ( Ȳ t > S 0 x + ε ) + P(Ȳt S 0 x + ε) [ x (r + f )ρ + 2ε(1 ρ)] + f. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

10 Expected Cost in Continuous Case Motivation: Several LOB models (including, symmetric and some asymmetric CRW) have been shown to admit a diffusive limit {S u } u 0 when δ 0 and ε 0; Hence, it is natural to consider the following analog continuous time problem: Expected Cost, Continuous Case. C(x, t) = E [S t S 0 Y t > S 0 x] P (Y t > S 0 x) + P(Y t S 0 x) ( x ρ(x, t)(r + f )) + f. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

11 Price Models for S Brownian Motion with Drift (Bachelier Model): ds u = µdu + σdw u, (reasonable approximation for intermediate intraday time horizons) Geometric Brownian Motion (Black Scholes Model) ds u = µs u du + σs u dw u, (better model for asset price movement at longer time periods) J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

12 Expected Cost Lemma (F-L, Lee, & Pasupathy, 2017) For the BM model and placement at level S 0 x, ( x + µt C(x, t) =µtn σ t { ( x µt + N σ t ) + e 2xµ σ 2 (2x µt)n ) + e 2xµ σ 2 N ( x + µt σ t ( x + µt σ t ) + f )} ( x ρ(x, t)(r + f )). For the GBM model and placement at level S 0 e y (y > 0) (i.e., x = S 0 S 0 e y ), C(y, t) = S 0 e µt[ ( y + α+ t N σ t { ( y α t + N σ t where α ± := µ ± σ 2 /2. ) e 2yα + σ 2 N ) + e 2yα σ 2 N ( y + α+ t σ t ( y + α t σ t )] + f S 0 )} (S 0 e y ρ(y, t)(r + f )), J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

13 Expected Cost for GBM: µ > 0 vs. µ < 0 C(y,t), mu=0.1 C(y,t), mu= y y Figure 2: C(y, t) against y with ρ(r + f ) = 0.01, σ = 0.2, S 0 = 10, µ = 0.1, t = 0.1 Figure 3: C(y, t) against y with ρ(r + f ) = 0.01, σ = 0.2, S 0 = 10, µ = 0.1, t = 0.1 J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

14 Expected Cost for GBM with µ < 0: large vs. small t C(y,t), mu= 0.05,t=0.02 C(y,t), mu= 0.05, t= y y Figure 4: C(y, t) against y with ρ(r + f ) = 0.01, σ = 0.2, S 0 = 10, µ = 0.05, t = 0.02 Figure 5: C(y, t) against y with ρ(r + f ) = 0.01, σ = 0.2, S 0 = 10, µ = 0.05, t = J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

15 Optimal Placement Solution Simplifying Assumption ρ(x, t) is independent of t; Theorem (F-L, Lee, & Pasupathy, 2017) Let x (t) [0, ] be such that x (t) = arg inf C(x, t), x>0 where x (t) = 0 (resp., x (t) = ) means that C(0 +, t) < C(x, t) (resp., C(, t) < C(x, t)), for all x > 0. 1 Suppose that µ 0 and x ρ(x) is decreasing. Then, x C(x, t) is strictly increasing and, thus, x (t) = 0; 2 Suppose that µ < 0 and ρ (0 + ) > 0. Then, there exists a t 0 > 0 such that x (t) (0, ), for all t t 0. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

16 Comments 0 + in C(0 +, t) can be interpreted as an order placed at the best bid; The case µ = 0 is the analog of Guo et al. (2016) result. As it turns out, in the case of µ < 0, 2 C t x (0+, t) < 0 and t 0 is such that C x (0+, t 0 ) = 0, C x (0+, t) > 0, t < t 0, C x (0+, t) < 0, t > t 0. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

17 Asymptotic Behavior of the Optimal Placement Problem Goals: Approximation of critical horizon (t 0 ) as r + f 0, 1 optimal placement (x (t) or y (t)) behavior when t t 0, optimal placement behavior in the low volatility regime (σ 0), optimal placement behavior when t is large enough. 1 e.g., the fee and rebate for NYSE are and , respectively. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

18 Critical Horizon (t 0 ) Behavior Theorem (F-L, Lee, & Pasupathy, 2017) Let µ < 0. Then, as r + f 0, BM : t 0 ρ(0+ )(r + f ), GBM : t 0 ρ(0+ )(r + f ). 2 µ 2 µ S 0 t0 t S_0 Figure 6: t 0 (black) for GBM vs. ˆt 0 := ρ(0 + )(r + f )/(2 µ S 0 ) (red) against S 0 with ρ(r + f ) = 0.01, σ = 0.2, µ = S_0 Figure 7: t 0 (black) for GBM vs. ˆt 0 := ρ(0 + )(r + f )/(2 µ S 0 ) (red) against S 0 near t=1 day (same parameters). J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

19 Optimal Placement Behavior Near t 0 Theorem (F-L, Lee, & Pasupathy, 2017) Suppose that µ < 0 and let t 0 be defined as above. Then, as t t 0, where κ 1 := 2 C y (t) = κ 1 (t t 0 ) + κ 2 (t t 0 ) 2 + o((t t 0 ) 2 ), t y (0, t 0) 2 C y 2 (0, t 0 ), κ 2 := 1 3 C 2 (0, t y 3 0 )κ C 3 C (0, t t y 2 0 )κ (0, t y t 2 0 ). 2 C (0, t y 2 0 ) J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

20 Optimal Placement behavior near t 0 : example y*(t) y*(1 day) time Figure 8: y (t) (black) vs. κ 1 (ˆt 0 )(t ˆt 0 ) (red) against t with ρ(0 + )(r + f ) = 0.01, σ = 0.2, µ = 0.1, S 0 = 50. Here, ˆt 0 := ρ(0 + )(r + f )/2 µ S S_0 Figure 9: y (1day) (black) and κ 1 (ˆt 0 )(1day ˆt 0 ) (red) against S 0 with ρ(0 + )(r + f ) = 0.01, σ = 0.2, µ = 0.1, S 0 = 50. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

21 Optimal Placement behavior near t 0 : example Optimal Placement solution and approximations Optimal Placement S0*exp(y*(t)) S0*exp( κ 1(t0)*(t t0)) S0*exp( κ 1(t0^ )*(t t0^ )) S0*exp( κ 1(t0)*(t t0) κ 2(t0)*(t t0) 2 ) S0*exp( κ 1(t0^ )*(t t0^ ) κ 2(t0^ )*(t t0^ ) 2 ) t(days) Figure 10: Approximations against t(days) when ρ(0 + )(r + f ) = 0.006, σ = 0.2, µ = 0.1, S 0 = 50, ˆt 0 = ρ(r + f )/(2 µ S 0 ). J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

22 Optimal placement behavior in the low volatility regime Lemma (F-L, Lee, & Pasupathy, 2017) When σ 0, there is no local minimum of C(y, t), when y ( µt, ). In other words, σ 0 such that 0 < σ < σ 0 C y > 0 for all y > µt. Lemma (F-L, Lee, & Pasupathy, 2017) Let us denote the Expected Cost as C(y, σ), the function of y and σ, for fixed t. Let us denote y (σ) [0, ] be such that y (σ) = arg inf C(y, σ), y>0 When σ 0, y (σ) exists in (0, µt) and y (σ) µt. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

23 Optimal placement behavior in the low volatility regime Theorem (F-L, Lee, & Pasupathy, 2017) Let y (σ) be arg inf C(y, σ). Then, as σ 0, Furthermore, y (σ) µt lim σ 0 y (σ) + µt 2σ 2 t ln(1/σ) = 1. 2σ 2 c t ln(1/σ) + 2σ 2 ln(1/σ) 2 t ln(1/σ), where c = ln S 0 + µt ln t ln(r + f ) ln 2π. J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

24 Optimal placement behavior in the low volatility regime y*(sigma) C(y*(sigma),sigma) sigma sigma Figure 11: y (σ)(black), 1 st order approximation(red) against σ when S 0 = 50, ρ(r + f ) = 0.01, µ = 0.1, t = 0.01 Figure 12: C(y (σ), σ)(black), C(1 st order approximation)(red) against σ when S 0 = 50, ρ(r + f ) = 0.01, µ = 0.1, t = 0.01 J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

25 Conclusions BM and GBM Model nontrivial optimal placement solution exists when t is larger than a critical value t 0 accurate and simple estimation of threshold horizon, t 0, behavior of optimal placement solution when: time horizon is near the threshold, volatility is low, time horizon is long enough. Future and ongoing work: introduce robust optimization (unknonw µ and σ), Optimal placement problem when ρ(x, t) depends on both x and t introduce randomness of parameters in the cost function. consider other price dynamics (e.g., stochastic volatility model, Lévy process) J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30

26 References Xin Guo, Adrien De Larrard, and Zhao Ruan. Optimal placement in a limit order book: an analytical approach. Mathematics and Financial Economics, J.E. Figueroa-López, Hyoeun Lee, and Raghu Pasupathy. Optimal Placement of a Small Order Under a Diffusive Limit Order Book (LOB) Model. Available at J.E. Figueroa-López (WUSTL) Optimal Placement of a Small Order INFORMS / 30