Smart TWAP trading in continuous-time equilibria

Transcription

1 Smart TWAP trading in continuous-time equilibria Jin Hyuk Choi, Kasper Larsen, and Duane Seppi Ulsan National Institute of Science and Technology Rutgers University Carnegie Mellon University IAQF/Thalesians seminar June 20, / 25

2 2 / 25 TWAP trading (time-weighted average price) How to optimally buy (sell) ã i shares dynamically over t [0,1]? New feature is the use of equilibrium theory (i.e., demand = supply) Trader i is given a target curve for his trades. This means, at time t [0,1], he is incentivized to hold γ(t)ã i, i = 1,...,M Examples of γ(t) with [0,1] split into 100 trading rounds: 1.0 γ n affine ( ), piecewise affine (- - -), square root (- - -)

3 3 / 25 TWAP trading (2) Trader i has the terminal target ã i for i {1,...,M} with M < ãi is private/non-public information (model input) There is a given execution path γ(t) with γ(1) = 1 γ(t) is non-random (deterministic and model input) When γ(t) is smooth, γ (t) is the target execution rate Simple TWAP: at time t [0,1], the trader has reached γ(t)% of his terminal target ã i Smart TWAP: the trader is allowed to deviate from the trading target path γ(t)ã i Deviations from γ(t)ã i are penalized but could still be profitable to make In our model, the optimal smart TWAP strategy γ(t)ãi

4 4 / 25 TWAP trading (3) The deviation penalty is defined by (quadratic costs) L i,t := t 0 κ(s) ( γ(s)ã i θ i,s ) 2ds, t [0,1] Trader i s initial stock position is zero (model output) θ i,t denotes trader i s stock position at time t [0,1] (model output) κ(t) controls the penalty severity (model input) κ n κ 1(t) := 1 ( ), κ 2(t) := 1 + t (- - -), κ 3(t) := (1 t) 1/4 (- - -)

5 5 / 25 TWAP trading (4) Trader i solves the optimization problem max E [U(X i,1 L i,1 )] (1) θ i A i U(x) := x or U(x) := e x/τ for a common risk-aversion τ > 0 (model input) Ai is the set of admissible controls (model input) X i,t is trader i s wealth at time t [0,1] (more about X i,t later - model output) In the above objective (1), we have two competing drivers: (i) Wealth X i,1 : Concavity of U gives incentive to use Merton s utility-maximizing strategy (ii) Penalty L i,1 : The terminal penalty gives incentive to use the simple TWAP strategy γ(t)ã i

6 6 / 25 Questions (qualitative) How does the widespread presence of smart-twap traders affect equilibrium price dynamics? How does the equilibrium return change over the trading day in order to clear the markets (stock and bank)? What is the resulting equilibrium price-impact of orders? How do traders HFT market makers (ã i := 0) and smart-twap traders (ã i 0) supply liquidity and optimally trade? How do traders absorb the inelastic noise-trader orders (they must do this for markets to clear)? How sensitive are the traders optimal positions ˆθ i,t relative to their own trading targets ã i? How sensitive are the traders objective value (i.e., certainty equivalent CE i ) relative to their own trading targets ã i?

7 7 / 25 Questions (quantitative) As with any other model in applied math we need (Hadamard 1902) Existence Uniqueness (led us to a new problem in calculus of variations) Stability Issues related to numerical computation Speed (low dimensional linear or quadratic ODEs) Discretization (convergence of discrete-time pre-limits works numerically but is not developed analytically) Online algorithm (filtering components are not developed) Big data issues (how to handle various forms of data irregularities is not developed)

8 8 / 25 Literature Equilibrium with traders with heterogeneous dynamic random endowments (no asymmetric info related to cash-flows): Discrete-time incomplete models: Vayanos (1999) and Calvet (2001) Vayanos (1999) has both the competitive (Radner) equilibrium and a Nash price-impact equilibrium non-uniqueness Continuous-time incomplete models: Christensen, Larsen, and Munk (2012), Žitković (2012), Christensen and Larsen (2014), Choi and Larsen (2015), Larsen and Sae-Sue (2016), Weston (2018), and Xing and Žitković (2018) Continuous-time equilibrium models with targets, linear utilities, and absolutely continuous optimal controls (rates) Brunnermeier and Pedersen (2005): elastic noise-trader supply and hard targets Gârleanu and Pedersen (2016): one agent, inelastic noise-trader supply, and competitive equilibrium with quadratic costs Bouchard, Fukasawa, Herdegen, and Muhle-Karbe (2018): many agents, inelastic noise-trader supply, and competitive equilibrium with quadratic costs

9 9 / 25 Literature (2) Optimal order-splitting when the price-impact of order flow is model input Find an optimal trading strategy to minimize E[cost] of trading a fixed number of shares Bertsimas and Lo (1998), Almgren and Chriss (1999, 2000), Gatheral and Scheid (2011), and Predoiu, Shaikhet and Shreve (2011) Equilibrium with long-lived asymmetric info Kyle (1985), Holden and Subrahmanyam (1992), Foster and Viswanathan (1994, 1996), and Back, Cao, and Willard (2000) Equilibrium with asymmetric cash-flow info and trading targets Degryse, de Jong, and van Kervel (2014): short-lived non-public info Choi, Larsen, and Seppi (2018): long-lived non-public info

10 10 / 25 Model details Continuous-time with t [0,1]. Stock and bank account with r = 0. Non-public variables (ã 1,...,ã M ) and Brownian motions (W,D) on Ω: (ã 1,...,ã M ) W = (W t ) t [0,1] D = (D t ) t [0,1] The stock price S = (S t ) t [0,1] is to be determined. S must satisfy the terminal condition S 1 = D 1 Inelastic supply of shares w t from noise traders (OU-process) dw t := (α πw t )dt + σ w dw t, w 0 := 0 M traders with heterogenous terminal targets (ã i ) M i=1 Zero initial stock positions Smart-TWAP traders have ã i 0 HFT traders (liquidity providers/market makers) have ã i := 0 κ(t), γ(t), and U(x) are the same for all traders The only(!) heterogeneity is in the realization ã1 (ω),...,ã M (ω), ω Ω

11 11 / 25 Model details (2) Trader i solves the optimization problem max E [U(X i,1 L i,1 )] θ i A i U(x) := x or U(x) := e x/τ for a common risk-aversion τ > 0 Penalty process L i,t := t 0 κ(s) ( γ(s)ã i θ i,s ) 2ds, t [0,1] θ i,t = cumulative stock position for trader i at time t (model output) γ(t) = target path such as γ(t) := t for simple TWAP (model input) κ(t) = penalty severity such as (model input) κ 1 (t) := 1, κ 2 (t) := 1 + t, κ 3 (t) := (1 t) 0.25

12 12 / 25 Model details (3) The aggregate target is ã Σ := M i=1 ãi turns out to be public Wealth process for trader i is dx i,t := θ i,t ds θ i t θi,t σ(ã i, ã Σ, w u,d u ) u [0,t] is trader i s stock position at t [0,1] The stock price has the dynamics In (2) the drift is defined by ds θi t := µ i,t dt + σ SW (t)σ w dw t + dd t. (2) µ i,t := µ 0 (t)ã Σ + µ 1 (t)θ i,t + µ 2 (t)ã i + µ 3 (t)w t where µ 0 (t),...,µ 3 (t), and σ SW (t) are deterministic (model output) µ1 (t) controls the drift-impact of trader i s position The equilibrium stock price Ŝ = (Ŝ t ) t [0,1] satisfies σ(ã i, ã Σ, w u,d u ) u [0,t] = σ(ã i, Ŝ u,d u ) u [0,t] There is also an integrability condition placed on θ i,t for θ i A i (needed to rule out doubling strategies)

13 13 / 25 Equilibrium definition (reduced-form notion) The deterministic functions (µ 0,...,µ 3,σ SW ) form an equilibrium if the associated optimal stock positions (ˆθ i,t ) M i=1 satisfy (i) Market-clearing condition w t = M ˆθ i=1 i,t (ii) Terminal stock-price condition Ŝ 1 = D 1 holds at time t = 1 (iii) Equilibrium price drift ˆµ t does not depend on trader-specific variables (ã i, θ i,t ) when θ i,t is set as the maximizer ˆθ i,t Equilibrium is not unique: Uncountably many equilibria One degree of freedom: each equilibrium is pinned down by a different choice of the function µ 1 (t) Market-clearing and equilibrium condition (iii) give one too few restrictions

14 14 / 25 What makes the analysis hard? For market-clearing, the traders must absorb the inelastic noise-trader orders in aggregate Requires an equilibrium risk premium (or discount) to induce this behavior As t 1, the terminal price condition S 1 = D 1 constrains prices. The wiggle room for prices to induce market-clearing inventory shrinks To induce the terminal trading constraint (i.e., forcing θ i,1 to be close to ã i ), the penalty severity needs to explode lim κ(t) = + t 1 Turns out square-integrablity of κ(t) is sufficient whereas integrability is insufficient

15 15 / 25 Existence theorem for U(x) := x Let γ : [0,1] [0, ) and µ 1,κ : [0,1) (0, ) be continuous functions with (µ 1,κ) square integrable such that µ 1(t) < κ(t). Then a unique Nash equilibrium exists in which (i) Trader optimal holdings ˆθ i in equilibrium are given by (ii) The equilibrium stock price is ˆθ i,t = wt M + 2κ(t)γ(t) ( ) ã i ãσ 2κ(t) µ 1(t) M Ŝ t = g 0(t) + g(t)ã Σ + σ SW(t)w t + D t, where the deterministic functions g 0, g, and σ SW : [0,1] R are unique solutions of three linear ODEs: g 2κ(t)(γ(t) 1) 0(t) = w 0 ασ SW(t), g 0(1) = 0, M g (t) = 2γ(t)κ(t), g(1) = 0, M σ SW(t) 2κ(t) µ1(t) = + πσ SW(t), σ SW(1) = 0, M (iii) The pricing functions µ 0,µ 2,µ 3 are explicitly available in terms of µ 1

16 16 / 25 Contributions First equilibrium model with smart TWAP targets and price impact TWAP targets and penalties are model input Derive price and optimal holding processes Endogenous intraday liquidity premium in prices Continuous-time model for rebalancing (closed-form solutions) Model flexibility (calibration) As usual, model input parameters can be calibrated New degree of freedom: Infinitely many equilibria Distinguish between risk-aversion over wealth lotteries vs. trading Risk-neutral or exponential utilities Quadratic costs with bounded or potentially exploding penalty severities κ(t) near maturity

17 17 / 25 Contributions (2) Analytical results (proofs) Existence of equilibria (infinitely many) Each equilibrium is pinned down by a different pricing function µ 1(t) Closed-form solutions up to solution of ODEs (linear or quadratic) Existence of a welfare-maximizing equilibrium Pinned down by a function µ 1 (t) attaining sup µ 1 (t) M E[CE i] i=1 where the sup is taken over all continuous functions µ 1 : [0,1) R satisfying a 2nd-order condition Welfare-maximizing competitive (Radner) with µ 1(t) := 0 Tractable; the numerics are fast and stable (linear or quadratic ODEs) Numerical results Hump-shaped intraday patterns in liquidity premium SD in prices Price impact of noise-trader orders shrinks over day

18 18 / 25 Different price-impact functions µ 1 (t) Recall our drift-impact relation ds θ i t := ( µ 0(t)ã Σ + µ 1(t)θ i,t + µ 2(t)ã i + µ 3(t)w t ) dt + σsw(t)σ wdw t + dd t Competitive equilibrium (Radner) sets µ 1 (t) := 0 Welfare-maximizing equilibrium sets µ 1 (t) := µ 1 (t) where µ 1(t) argmax µ1 (t) M E[CE i ] i=1 For risk-neutral utilities U(x) := x we have sufficient conditions guaranteeing the existence of µ 1 (t) ( 0,κ(t) ) Finding µ 1 (t) requires us to solve a new problem in calculus of variations Vayanos (1999) adds the restriction µ 1 (t) = µ 3 (t) Produces a negative drift-impact µ 1 (t) < 0 Many refinements of Nash equilibria exist (not developed) Empirical calibration (not developed because we don t have data)

19 Key mathematical elements No filtering needed because the aggregate target ã Σ is inferred from Ŝ 0 = g 0 (0) + g(0)ã Σ + σ SW (0)w 0 + D 0 For w0 random, filtering is needed (not developed) The functions (model input) U(x), γ(t), κ(t) are homogeneous whereas the targets (ã i ) M i=1 are heterogeneous The extension to heterogeneous functions is not developed The following maps are affine (drift and Sharpe ratio/market price of risk process are both model output) θ i,t µ i,t, θ i,t µ i,t σsw (t) 2 + 1, with coefficients controlled by µ 1 (t) Non-linear price-impacts are empirically relevant (not developed) 19 / 25

20 20 / 25 Comments The equilibrium stock price dynamics are dŝ t := ˆµ t dt + σ SW (t)σ w dw t + dd t where ˆµ t := 2κ(t) µ 1(t) M w t 2κ(t)γ(t) M ãσ Shocks to price level ds t due to Dividend shocks dd t. One-to-one Supply shocks dw t from noise trader orders. Scaled by σ SW (t)σ w Shocks to equilibrium price drift ˆµ t For the traders to absorb the inelastic noise orders wt 0, they require a higher expected return. When ã Σ := M i=1 ãi > 0, expected price drift needs to be depressed to deter the traders from buying stock

21 Comments (2) Intuition for trader i s optimal holding process ˆθ i,t = w t M + 2κ(t)γ(t) ( 2κ(t) µ 1 (t) ã i ãσ M ) Split noise trader supply w t equally This implies that no optimal execution rate can exist (i.e., ˆθ i,t is not absolutely continuous with respect to time) Traders with larger targets ã i than the average target ã Σ /M take larger positions 21 / 25

22 Numerics: welfare-maximizing µ 1 (t) * μ n -0.1 κ 1 ( ), κ 2 ( ), κ 3 ( ). κ 1 (t) := 1, κ 2 (t) := 1 + t, κ 3 (t) := (1 t) 0.25 Welfare-maximizing µ 1 (t) is larger when penalty severity is larger 22 / 25

23 23 / 25 Numerics: price impact of noise trader orders σsw 0.1 σsw n n A: [Welfare] κ 1 ( ), B: [Radner] κ 1 ( ), κ 2 ( ), κ 3 ( ). κ 2 ( ), κ 3 ( ) κ 1 (t) := 1, κ 2 (t) := 1 + t, κ 3 (t) := (1 t) 0.25 Price-level loading σ SW (t) on w t is unique The terminal condition Ŝ 1 = D 1 forces σ SW (t) 0 as t 1

24 Numerics: liquidity premium standard deviation n n A: [Welfare] κ 1 ( ), B: [Radner] κ 1 ( ), κ 2 ( ), κ 3 ( ). κ 2 ( ), κ 3 ( ) κ 1 (t) := 1, κ 2 (t) := 1 + t, κ 3 (t) := (1 t) 0.25 Initially SD[Ŝ t D t ] grows with the variance of the noise-trader supply but eventually converges to 0 because Ŝ 1 = D 1 24 / 25

25 25 / 25 Next steps Find µ 1 (t) based on empirical data (calibration) Find µ 1 (t) based on Kyle estimates of λ(n) where S n = λ(n)( W n + θ i,n) The following plot is by curtesy of Yashar Barardehi (Chapman University) Endogenies ã Σ (e.g., by making w 0 random) Convert the model into an online algorithm