Smart TWAP and VWAP trading in continuous-timeequilibria

Jin Hyuk Choi†, Kasper Larsen‡, and Duane J. Seppi∗

†Ulsan National Institute of Science and Technology (South Korea)‡Rutgers University

∗Carnegie Mellon University

Michigan seminar

Jan 16, 2019

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TWAP trading (time-weighted average price) How to optimally buy (sell) ai shares dynamically over t ∈ [0,1]?

I New feature is the use of equilibrium theory (i.e., demand = supply) Trader i is given a trading target curve so that at time t ∈ [0,1], the

trader is incentivized to hold

γ(t)ai, i = 1,...,M

Examples of γ(t) with [0,1] split into 100 trading rounds:

20 40 60 80 100n

0.2

0.4

0.6

0.8

1.0

γ

affine (—–), piecewise affine (- - -), square root (- · - · -)

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TWAP trading (2)

Trader i has the terminal target ai for i ∈ 1,...,M with M <∞I ai is private/non-public information (model input)

There is a given monotone path γ(t) with γ(0) ∈ [0,1] and γ(1) = 1I γ(t) is non-random (model input — stochastic γ for VWAP comes

later)I When γ(t) is smooth, γ′(t) is the target execution trading rate

Simple TWAP: At time t ∈ [0,1], the trader has reached γ(t)% of histerminal target ai

Smart TWAP: The trader is allowed to deviate from the tradingtarget path γ(t)ai

I Deviations from γ(t)ai are penalized but could be profitable to makeI In our equilibrium model, the optimal smart TWAP strategy 6= γ(t)ai

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TWAP trading (3) The deviation penalty is defined by (quadratic costs)

Li,t :=

∫ t

0κ(s)

(γ(s)(ai − θi,−)− (θi,s − θi,−)

)2ds, t ∈ [0,1]

I Trader i’s initial stock position is θi,− (model output)I θi,t denotes trader i’s stock position at time t ∈ [0,1] (model output)I κ(t) controls the penalty severity (model input)

20 40 60 80 100n

0.5

1.0

1.5

2.0

2.5

3.0

κ

κ1(t) := 1 (——), κ2(t) := 1 + t (- - -), κ3(t) := (1− t)−1/4 (- · - · -)

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TWAP trading (4)

Trader i solves the optimization problem

maxθi∈Ai

E [U(Xi,1 − Li,1)] (1)

I U(x) := x (model input - exponential utilities are also possible)I Ai is the set of admissible controls (model input)I Xi,t is trader i’s wealth at time t ∈ [0,1] (more about Xi,t later - model

output) In the above objective (1), we have two competing drivers:

(i) Wealth Xi,1: Concavity of U gives incentive to use Merton’sutility-maximizing strategy

(ii) Penalty Li,1: The terminal penalty gives incentive to use the simpleTWAP strategy θi,− + γ(t)

(ai − θi,−

)

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Questions (qualitative)

How does the widespread presence of smart-TWAP traders affectequilibrium price dynamics?I How does the equilibrium return change over the trading day in order

to clear the markets (stock and bank)?I What is the resulting equilibrium price-impact of orders?

How do traders — high frequency (HF) traders with ai := 0 andsmart-TWAP traders with ai 6= 0 supply liquidity and optimallytrade?I How do traders absorb the inelastic noise-trader orders (they must do

this for markets to clear)?I How sensitive are the traders’ optimal positions θi,t relative to their

own trading targets ai?I How sensitive are the traders’ objective value (i.e., certainty

equivalent CEi) relative to their own trading targets ai?

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Questions (quantitative)

As with any other model in applied math we need (Hadamard 1902)I ExistenceI Uniqueness (led us to a new calculus of variations problem)I Stability

Issues related to numerical computationI Speed (low dimensional linear or quadratic ODEs — this is both fast

and stable)I Discretization (convergence of discrete-time pre-limits works

numerically but is not developed analytically)I Online algorithm (filtering components are not developed)I Big data issues (how to handle various forms of data irregularities is

not developed)

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Literature Equilibrium with traders with heterogeneous dynamic random

endowments (no asymmetric info related to cash-flows):I Discrete-time incomplete models: Vayanos (1999) and Calvet (2001)

I Vayanos (1999) has both the competitive (Radner) equilibrium and aNash price-impact equilibrium — non-uniqueness

I Continuous-time incomplete models: Christensen, Larsen, and Munk(2012), Žitkovic (2012), Christensen and Larsen (2014), Choi andLarsen (2015), Larsen and Sae-Sue (2016), Weston (2018), Xing andŽitkovic (2018), and Weston and Žitkovic (2018)

I Continuous-time equilibrium models with targets, linear utilities, andabsolutely continuous optimal controls (rates)I Brunnermeier and Pedersen (2005): elastic noise-trader supply and

hard public targetsI Gârleanu and Pedersen (2016): one agent, inelastic noise-trader

supply, and competitive equilibrium with quadratic costs (zero targets)I Bouchard, Fukasawa, Herdegen, and Muhle-Karbe (2018): many

agents, inelastic noise-trader supply, and competitive equilibrium withquadratic costs (zero targets)

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Literature (2)

Optimal order-splitting when the price-impact of order flow is modelinputI Find an optimal trading strategy to minimize E[cost] of trading a

fixed number of sharesI Bertsimas and Lo (1998), Almgren and Chriss (1999, 2000), Gatheral

and Scheid (2011), and Predoiu, Shaikhet and Shreve (2011)

Equilibrium with long-lived asymmetric infoI 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 targetsI Degryse, de Jong, and van Kervel (2014): short-lived non-public infoI Choi, Larsen, and Seppi (2018): long-lived non-public info

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TWAP model details Continuous-time with t ∈ [0,1]. Stock and bank account with r := 0. Non-public variables (a1,...,aM) and Brownian motions (W,D) on Ω:

(a1,...,aM) ⊥ W = (Wt)t∈[0,1] ⊥ D = (Dt)t∈[0,1]

The equilibrium stock price S = (St)t∈[0,1] is to be determined. Smust satisfy the terminal condition

S1 = D1

Inelastic supply of shares wt from noise traders (OU-process)

dwt := (α− πwt)dt + ηdWt, w0 := 0

M traders with heterogenous terminal targets (ai)Mi=1

I Zero initial stock positionsI Smart-TWAP traders have ai 6= 0I HF traders (liquidity providers/market makers) have ai := 0I The functions κ(t), γ(t), and U(x) are the same for all tradersI The only(!) heterogeneity is in the realization a1(ω),...,aM(ω), ω ∈ Ω

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TWAP model details (2)

Trader i solves the optimization problem (linear utilities U(x) := x)

maxθi∈Ai

E [Xi,1 − Li,1]

Penalty process

Li,t :=

∫ t

0κ(s)

(γ(s)(ai − θi,−)− (θi,s − θi,−)

)2ds, t ∈ [0,1]

I θi,− = initial stock position for trader i (model input)I θi,t = cumulative stock position for trader i at time t (model output)I γ(t) = target path such as γ(t) := t for simple TWAP (model input)I κ(t) = penalty severity such as (model input)

κ1(t) := 1, κ2(t) := 1 + t, κ3(t) := (1− t)−0.25

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Model details (3) The aggregate target is aΣ :=

∑Mi=1 ai turns out to be public info

Wealth process for trader i is

dXi,t := θi,tdSθit

I θi,t ∈ σ(ai, aΣ,wu,Du)u∈[0,t] is trader i’s stock position at t ∈ [0,1]I The stock price dynamics faced by investor i are

dSθit := µi,t dt + σw(t)ηdWt + dDt. (2)

In (2), the drift is defined by

µi,t := µ0(t)aΣ + µ1(t)θi,t + µ2(t)ai + µ3(t)wt

where µ0(t),...,µ3(t), and σw(t) are deterministic (model output)I µ1(t) controls the drift-impact of trader i’s position

I The equilibrium stock price S = (St)t∈[0,1] satisfies

σ(ai, aΣ,wu,Du)u∈[0,t] = σ(ai, Su,Du)u∈[0,t]

There is also an integrability condition placed on θi,t for θi ∈ Ai

(needed to rule out doubling strategies)12 / 25

Equilibrium definition (reduced-form notion)

The deterministic functions (µ0,...,µ3,σw) form an equilibrium if theassociated optimal stock positions (θi,t)

Mi=1 satisfy

(i) The market-clearing condition wt =∑M

i=1 θi,t holds(ii) The terminal stock-price condition S1 = D1 holds at time t = 1

(iii) The equilibrium price drift µt does not depend on any trader-specificvariables (ai, θi,t)

Our equilibrium is not unique:I Uncountably many equilibriaI One degree of freedom: Each equilibrium is pinned down by a

different choice of the function µ1(t)I The market-clearing and equilibrium conditions (i)-(iii) give one too

few restrictions

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What makes the equilibrium model non-trivial?

For market-clearing, the traders must absorb the inelasticnoise-trader orders (driven by the OU process wt)I Requires an equilibrium risk premium (or discount) to induce this

behavior

As t ↑ 1, the terminal price condition S1 = D1 constrains prices. The“wiggle room” for prices to induce market-clearing inventory shrinks

To induce the terminal trading constraint (i.e., forcing the optimizersθi,1 to be close to ai), the penalty severity needs to explode

limt↑1

κ(t) = +∞

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Existence theoremLet γ : [0,1]→ [0,∞) and µ1,κ : [0,1)→ (0,∞) be continuous functions with (µ1,κ)integrable such that µ1(t) < κ(t). Then a unique equilibrium exists in which

(i) Trader optimal holdings θi in equilibrium are given by

θi,t =wt

M+

2κ(t)γ(t)2κ(t)− µ1(t)

(ai −

aΣ

M

)(ii) The equilibrium stock price is

St = g0(t) + g(t)aΣ + σw(t)wt + Dt,

where the deterministic functions g0, g, and σw : [0,1]→ R are the unique solutionsof the three linear ODEs:

g′0(t) = −ασw(t), g0(1) = 0,

g′(t) = −2γ(t)κ(t)M

, g(1) = 0,

σ′w(t) =2κ(t)− µ1(t)

M+ πσw(t), σw(1) = 0,

(iii) The pricing functions µ0,µ2,µ3 are explicitly available in terms of µ1

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Different price-impact functions µ1(t)

Recall our drift-impact relation

dSθit :=

(µ0(t)aΣ + µ1(t)θi,t + µ2(t)ai + µ3(t)wt

)dt + σw(t)ηdWt + dDt

Competitive equilibrium (Radner) sets µ1(t) := 0

Collusive equilibrium sets µ1(t) := µ∗1(t) where

µ∗1(t) ∈ argmaxµ1(t)

M∑i=1

E[CEi]

I For risk-neutral utilities U(x) := x we have sufficient conditionsguaranteeing the existence of µ∗1(t) ∈

(0,κ(t)

)I Finding µ∗1(t) requires us to solve a new problem in calculus of

variations

Many refinements of Nash equilibria exist (not developed)

Empirical calibration (not developed because we don’t have data)

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TWAP numerics: γ(t) and κ(t) functions

The target-ratio function is

γ(t) := 0.1 + 0.9t, t ∈ [0,1] (3)

The penalty functions are

κ1(t) := 1, t ∈ [0,1],

κ2(t) := 1 + t, t ∈ [0,1],

κ3(t) :=98

(1− t)−0.25, t ∈ [0,1),

κ4(t) :=

.0002 for t ∈ [0,.75],

2.3791 + 11.8954(t − 0.95) for t ∈ [.75,.95],98(1− t)−0.25 for t ∈ [.95,1)

(4)

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TWAP numerics: Collusive equilibrium

0.2 0.4 0.6 0.8 1.0t

-1.0

-0.5

0.5

1.0μ1*(t)

κ1 (———), κ2 (−−−), κ3 (− · − · −), κ4 (− · ·−).

µ∗1(t) is larger when penalty severity is larger

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TWAP numerics: Price impact of noise trader orders

0.2 0.4 0.6 0.8 1.0t

-0.4

-0.3

-0.2

-0.1

σw(t)

0.2 0.4 0.6 0.8 1.0t

-0.4

-0.3

-0.2

-0.1

σw(t)

A: [Collusive] κ1 (—-), B: [Radner] κ1 (—-),κ2 (−−), κ3 (− · −), κ4 (− · ·−). κ2 (−−), κ3 (− · −), κ4 (− · ·−).

Price-level loading σw(t) on wt is unique

The terminal condition S1 = D1 forces σw(t)→ 0 as t ↑ 1

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TWAP numerics: Liquidity premium standard deviation

0.2 0.4 0.6 0.8 1.0t

0.05

0.10

0.15SD[St-Dt|σ(aΣ)]

0.2 0.4 0.6 0.8 1.0t

0.05

0.10

0.15SD[St-Dt|σ(aΣ)]

A: [Collusive] κ1 (—), B: [Radner] κ1 (—),κ2 (−−), κ3 (− · −), κ4 (− · ·−). κ2 (−−), κ3 (− · −), κ4 (− · ·−).

Initially, SD[St − Dt] grows with the variance of the noise-tradersupply but eventually converges to 0 because S1 = D1

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VWAP model with multiple sets of different investors Stochastic targets: Replace γ(t) with a stochastic process γt

I Gamma bridge model used in Frei and Westray (2015)I Volume weighted average price (VWAP) benchmarking

VWAP penalty process for i ∈ 1,...,M

Li,t :=

∫ t

0κ(s)

(γs(ai − θi,−)− (θi,s − θi,−)

)2ds, t ∈ [0,1]

I θi,− = initial stock position for trader i (model input)I θi,t = cumulative stock position for trader i at time t (model output)I κ(t) = penalty severity such as (model input)

Liquidity trader penalty process for i ∈ M + 1,...,M + M

Li,t :=

∫ t

0κ(s)

(εBs − θi,s

)2ds, t ∈ [0,1]

I θi,t = cumulative stock position for trader i at time t (model output)I κ(t) = penalty severity such as (model input)I ε is a constant and B is a Brownian motion (both aΣ and B turn out to

become public in equilibrium)21 / 25

VWAP model with multiple sets of different investors (2) We have multiple sets of investors

I Target traders i ∈ 1,...,M with private targets aiI When ai 6= 0, these are called smart VWAP tradersI When ai = 0, these are called HF traders

I Target traders i ∈ 1,...,M with common public target εBsI When ε 6= 0, these traders are similar to those in Sannikov and

Skrzypacz (2016)I When ε = 0, these are HF traders as above (but κ could differ from κ)

I Noise traders suppling wt + ργγt− + ρBBt shares of stock with wt

being the OU process from before and ργ ,ρB are constants

The market clearing condition becomes

wt + ργγt− + ρBBt =

M∑i=1

θi,t +

M+M∑i=M+1

θi,t

There is an existence theorem with infinitely many equilibria(parameterized by functions µ3(t) for t ∈ [0,1])

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VWAP numerics: Penalty severity functions The penalty functions are

κ(t) :=(1− p)

(1− t)p , t ∈ [0,1), p ∈ [0,1),

κ(t) :=(1− p)

(1− t)p , t ∈ [0,1), p ∈ [0,1).

(5)

0.0 0.2 0.4 0.6 0.8 1.0t

0.5

1.0

1.5

2.0

2.5κ(t)

p = 0 (——), p = 0.1 (−−), p = 0.5 (− · −), p = 0.99 (− · ·−).

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VWAP numerics: Collusive equilibrium

Figur: The collusive equilibrium µ3(t) . The parameters are wt := w0 (constant),ργ := −1, ρB := ε := 0 (second group consists of HF traders), and the tradingday is split into 1000 trading rounds.

0.0 0.2 0.4 0.6 0.8 1.0t

0.1

0.2

0.3

0.4

0.5μ3(t)

p = p = 0 (——), p = 0.5, p = 0 (−−),p = 0, p = 0.5 (− · −), p = p = 0.5 (− · ·−).

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