Equilibrium Effects of Intraday Order-Splitting Benchmarks
SSmart TWAP trading in continuous-time equilibria ∗ Jin Hyuk ChoiUlsan National Institute of Science and Technology (UNIST)Kasper LarsenRutgers UniversityDuane J. SeppiCarnegie Mellon UniversitySeptember 17, 2018
Abstract : This paper presents a continuous-time equilibrium model of TWAPtrading and liquidity provision in a market with multiple strategic investorswith heterogeneous intraday trading targets. We solve the model in closed-form and show there are infinitely many equilibria. We compare the compet-itive equilibrium with different non-price-taking equilibria. In addition, weshow intraday TWAP benchmarking reduces market liquidity relative to justterminal trading targets alone. The model is computationally tractable, andwe provide a number of numerical illustrations. An extension to stochasticVWAP targets is also provided.
Keywords : Dynamic trading, TWAP, VWAP, rebalancing, liquidity, market-maker inventory, equilibria, market microstructure ∗ The authors benefited from helpful comments from Yashar Barardehi, Umut Cetin, Chris Frei, Steve Karolyi,Ajitesh Mehta, Johannes Muhle-Karbe, Dan Ocone, Tom Ruchti, Chester Spatt, Harvey Stein, Kim We-ston, Ariel Zetlin-Jones, and seminar participants at Carnegie Mellon University, Columbia University,IAQF/Thalesians, and Intech. The second author is partly supported by the National Science Founda-tion (NSF) under Grant No. DMS 1812679 (2018 - 2021). Any opinions, findings, and conclusions orrecommendations expressed in this material are those of the authors and do not necessarily reflect theviews of the National Science Foundation (NSF). Corresponding author is Kasper Larsen and has email:[email protected]. Jin Hyuk Choi has email: [email protected] and Duane J. Seppi has email:[email protected]. a r X i v : . [ q -f i n . M F ] S e p ntraday trading targets play an important role in the dynamics of liquidity provisionand demand over the trading day in financial markets. For example, intraday liquidityproviders usually target a neutral (e.g., zero) inventory level. In addition, trade executionby large institutional investors is widely benchmarked relative to a time-weighted averageprice (TWAP) or volume-weighted average price (VWAP) reference price. Compensationschemes tied to such benchmarks can lead to intraday target trajectories to trade a constantamount per unit time (TWAP) or an amount indexed to track the daily volume curve(VWAP). Deviations from the intraday target trajectories are then penalized. As a result,investors care, not only about their terminal trading target, but also about the intradaytrajectory of their trading relative to a TWAP or VWAP benchmark trajectory. However,traders sometimes intentionally deviate from their trajectories in order to achieve tradingprofits (such as by market makers) and price improvement (such as by large institutions).We call optimized trading strategies that trade off trading profits and target-deviationpenalties smart TWAP or smart VWAP strategies. Our paper is the first to model the equilibrium impact of TWAP and VWAP bench-marking on intraday trading and market liquidity. We view intraday inventory and tradingtarget trajectories and penalties as solutions to agency and risk-management problems be-tween brokers and portfolio managers (for institutions) and between trading desks andbrokerage owners (for market makers). We take intraday trading target trajectories andpenalties as inputs to our model, and then show how smart TWAP and VWAP tradingstrategies affect price dynamics and liquidity over the trading day. In particular, we modela market with multiple strategic investors with different trading targets who follow optimalcontinuous-time dynamic trading strategies. Our paper solves for equilibria in closed-form.Our TWAP results are as follows: • An infinite number of equilibria exist with each equilibrium pinned down by a con-tinuous function giving the price impact of strategic investors’ individual orders. • Intraday TWAP strategies reduce intraday market liquidity and increase price volatil-ity relative to a market in which investors just have terminal end-of-day trading tar- Hagstr¨omer and Nord´en (2013) and Menkveld (2013) show that high-frequency (HFT) market makersare an important source of intraday liquidity. A common feature of HFT market makers is that they have“very short time-frames for establishing and liquidating positions” (SEC 2010), which is consistent with azero target inventory level. Weller (2018) shows further that liquidity over the trading day is provided byan ecosystem of liquidity providers with slower and faster trading latencies who shift inventory betweenthemselves over holding periods of different lengths. This behavior is also consistent with zero intradayinventory targets. Madhavan (2002) discusses price improvement on order execution relative to VWAP. Domowitz andYegerman (2005) estimate empirical execution costs benchmarked relative to VWAP. Baldauf, Frei, and Mollner (2018) show that compensating schemes based on VWAP are optimal incertain principal-agent problems related to optimal portfolio delegation. • Our model predicts price volatility and market illiquidity are greater on days withstronger TWAP penalties. In addition, our model makes empirical predictions aboutintraday patterns in prices and liquidity: First, liquidity effects due to positive im-balances in terminal TWAP trading targets produce intraday predictable trends inwhich prices fall on average. Second, the intraday pattern of the volatility of cumu-lative intraday liquidity effects in prices is hump-shaped conditional on the openingprice. • The welfare-maximizing equilibrium can differ from the competitive equilibrium. Forexample, investors can have different target deviations depending on their private trad-ing targets in the welfare-maximizing equilibrium, whereas investors all have identicaltarget deviations in the competitive equilibrium.Our discussion focuses on preferences that are linear in investor terminal wealth and aTWAP penalty. In addition, we provide two extensions: First, we extend our TWAP modelto stochastic VWAP target trajectories. Second, we extend the analysis to exponentialpreferences. Dynamic order execution strategies are pervasive in financial markets. O’Hara (2015)describes dynamic order execution both by investors with asymmetric information as wellas by pensions, index funds, and other passive investors rebalancing their portfolios. Thepractice of benchmarking order-execution quality with VWAP and other metrics is describedin Berkowitz, Logue, and Noser (1988) and Madhavan (2002). In contrast to the optimalcontrol literature surveyed in Gatheral and Shied (2013), we model dynamic trading withbenchmarking in an equilibrium framework. For an empirical perspective on the extentof order-execution benchmarking in practice, the trading survey Financial Insights (2006)reports that VWAP execution orders represent around 50% of all institutional trading.Our model is most closely related to previous research by Brunnermeier and Pedersen(2005) and Carlin, Lobo, and Viswanathan (2007) on optimal rebalancing and predatory The market in our model is incomplete in that there is only one stock but two Brownian motionsdriving its price. Existence of continuous-time Radner equilibria with exponential utilities in an incompletecompetitive market has been proved in various levels of generality in Christensen, Larsen, and Munk (2012),ˇZitkovi´c (2012), Christensen and Larsen (2014), Choi and Larsen (2015), Larsen and Sae-Sue (2016), Xingand ˇZitkovi´c (2018), and Weston (2018). To the best of our knowledge, there is no extension of thesemodels to an incomplete market equilibrium with continuous trading and price impact (Vayanos (1999)proves existence in a discrete-time model). Appendix C presents such a continuous-time extension in whichinvestors with exponential utilities have TWAP targets. In contrast to TWAP and VWAP, implementation shortfall, an alternative benchmark (Perold 1988),does not control for intraday price trends and thus does not lead to an intraday target trading trajectory. Third, the active agent’s tradingtarget is publicly known in Brunnermeier and Pedersen (2005), whereas trading targets areinitially private knowledge in our model. Information about trading targets is then partiallyrevealed through the trading process. Our paper also builds on the Vayanos (1999) modelof dynamic strategic trading. That model also has multiple equilibria like ours. However,investors in Vayanos (1999) smooth a series of random personal endowment shocks, whereasour strategic investors have heterogeneous trading target trajectories they would like toreach over the day. In addition, our intraday setting lets us ignore endogenous consumptiondecisions. The discrete-time model in Du and Zhu (2017) also has quadratic penalties butwith zero targets and has private dividend information (whereas dividend information ispublic in our model) as well as endowment shocks.Our model is also related to Gˆarleanu and Pedersen (2016) and the multi-agent extensionin Bouchard, Fukasawa, Herdegen, and Muhle-Karbe (2018). In these fully competitiveequilibrium models, traders incur penalties based on their stock holdings as well as theirtrading rates. Our model differs because we allow for price-impact of individual trades, andwe allow for non-zero trading targets that are private information. Furthermore, becauseour traders are penalized based on deviations from intraday targets and not on their rates oftrade, our optimal stock holdings (i.e., inventories) are not given in terms of trading rates.This property allows our investors to absorb noise trades with only finite second variationsuch as the Brownian motion dynamics in, e.g., Kyle (1985).Lastly, there is no asymmetric information about future asset cash-flow fundamentalsin our model. Thus, the analysis here on trading and the non-informational componentof market liquidity is complimentary to Choi, Larsen, and Seppi (2018), which studiesorder-splitting and dynamic rebalancing in a Kyle (1985) style market in which a strategicinformed investor with long-lived private information and a strategic rebalancer with a hardterminal trading target both follow dynamic trading strategies. Instead, our model extendsthe microstructure literature on inventory costs in market making that began with Garman(1976) and Stoll (1978). In addition, trading constraints here are soft rather than hard. In Brunnermeier and Pedersen (2005), ad hoc liquidity providers trade using an exogenous linear schedulethat does not rationally anticipate predictable future price changes later in the day given earlier trading. Model
We develop a continuous-time equilibrium model with a unit horizon in which trade takesplace at each time point t ∈ [0 , S = ( S t ) t ∈ [0 , . The stock pays random dividends generated by a publicly observable andexogenous standard Brownian motion state-process D = ( D t ) t ∈ [0 , with a given constantfixed initial value D ∈ R , zero drift, and volatility normalized to one. We model D t asthe expectation at time t < D is either a liquidating dividendpaid at time t = 1 or simply the time t = 1 expectation of all future dividends.Two types of investors trade in our model: • There are M ∈ N strategic investors who trade over the day to minimize penaltiesrelative to intraday trading targets and to exploit profitable trading opportunities.For a generic investor i , let ˜ a i denote the investor’s terminal end-of-day target stockholdings, and let θ i,t denote the investor’s actual stock holdings at a generic time t .Let θ i, − and θ (0) i, − denote the investor’s initial stock holdings and initial money marketbalances. Thus, ˜ a i − θ i, − is the total amount of stock investor i ideally wants tohave traded by the end of the day, and θ i,t − θ i, − is the cumulative amount investor i has actually traded by time t . The difference ˜ a i − θ i, − can be viewed as a so-called parent order , and the difference θ i,t − θ i, − is the cumulative total executed viaa sequence of so-called child orders up through time t . The initial money-marketbalances ( θ (0)1 , − , ..., θ (0) M, − ), initial stock holdings ( θ , − , ..., θ M, − ), and terminal stock-holding targets (˜ a , ..., ˜ a M ) are private knowledge of the M investors. At this point,we make no distributional assumptions about these variables except for them beingindependent of D .An important feature of our model is that strategic investors have a target trajectoryfor trading defined over the whole day and that they incur inventory penalties tiedto intraday deviations between their actual cumulative stock trading at each point intime and their target trajectory over time. In other words, investors care not onlyabout their target ˜ a i at the end of the day, but also about how quickly their parentorder is executed over the trading day. The intraday target TWAP trajectory forinvestor i is a function defined over the trading day that gives for each time t ∈ [0 , i would ideally like to havecompleted by time t . In particular, the target trading trajectory at a generic time An investor’s stock holding is often referred to as inventory . See Chapter 5 in Johnson (2010) for more about TWAP trading. is γ ( t )(˜ a i − θ i, − ) where γ ( t ) is a nonnegative continuous function for t ∈ [0 , γ ( t ) := t . For generalized TWAP targets with time-varying weights, the γ ( t ) function is non-decreasing and converges to 1 as t ↑
1. Forexample, γ ( t ) might follow the shape of the average cumulative volume curve over thetrading day. A TWAP target trajectory for cumulative trading can also be expressedequivalently in terms of a trajectory γ ( t )(˜ a i − θ i, − ) + θ i, − for the stock holdings ofinvestor i at time t . The TWAP assumption that the target ratio γ ( t ) is deterministicsimplifies our analysis. Section 7 extends the model to allow for stochastically varyingtargets related to VWAP.The penalty process for investor i ∈ { , ..., M } is L i,t := (cid:90) t κ ( s ) (cid:16) γ ( s )(˜ a i − θ i, − ) − ( θ i,s − θ i, − ) (cid:17) ds, t ∈ [0 , . (1.1)The severity of the penalty is controlled by κ ( t ), which is a deterministic strictly posi-tive function. Intuitively, the penalty severity for deviations from the target trajectoryis likely to be increasing over the trading day. Our results below allow for penalty-severity functions κ ( t ) that explode towards the end of the trading day as t ↑ γ ( t ) and κ ( t ).Section 6 considers several specific numerical examples. We differentiate between twotypes of investors based on their realized trading targets ˜ a i . We refer to investorswith targets ˜ a i (cid:54) = θ i, − as institutional investors . Traders with ˜ a i = θ i, − do not needto trade but can provide liquidity. We call these traders market makers or intradaystrategic liquidity providers. Thus, institutional investors and market makers differ inthe target amount they want to trade but face the same penalties for diverging fromtheir trading target. • There are noise traders whose trading motives are exogenous. Let w t denote thestock’s fixed shares outstanding minus the aggregate noise-trader holdings. Thus, w t denotes the stock supply that the strategic investors must absorb at time t ∈ [0 , Quadratic penalization schemes constitute a cornerstone in research related to mean-variance analysisdating back to Markowitz (1952). The penalty (1.1) penalizes holdings θ i,t (i.e., trader i ’s inventory) andnot buying/selling rates. Rates are penalized in Almgren and Chriss (1999, 2000), Almgren (2003), Gatheraland Schied (2011), Almgren (2012), Gˆarleanu and Pedersen (2016), and Bouchard et al. (2018) and forcesthe optimal stock holdings to be given in terms of rates. Optimal buying/selling rates also exist in thecontinuous-time model in Kyle (1985) as well as in its non-Gaussian extension in Back (1992). Predoiu et al.(2011) consider a model where the optimal holding process is of finite variation but includes discrete orderswhich implies that no optimal rate exists. In contrast, optimal holding processes in our model have infinitefirst variation (and only finite second variation), and no optimal rates can exist. This property allows ourstrategic investors to absorb the noise-trader orders from (1.3) below, which would be impossible in modelswhere optimal holdings are given in terms of rates.
5e assume w t is supplied inelastically by the noise traders. Consequently, the stockmarket clears at time t ∈ [0 ,
1] when the strategic investors’ holdings ( θ i ) Mi =1 satisfy w t = M (cid:88) i =1 θ i,t . (1.2)We assume that the stock supply has dynamics (Gaussian Ornstein-Uhlenbeck) dw t := ( α − πw t ) dt + ηdB t , w ∈ R . (1.3)Gaussian noise traders have been widely used; see, e.g., Grossman and Stiglitz (1980)and Kyle (1985). In (1.3), the parameters α , π , and η are constants, and B is anotherstandard Brownian motion that is independent of the dividend Brownian motion D and of the strategic investors’ targets and initial holdings. The specification (1.3)includes cumulative noise-trader supply that follow an arithmetic Brownian motion( π = 0) with a possible predictable trend ( α ) as well as possibly mean-revertingdynamics ( π >
0) or positively autocorrelated changes ( π < S = D + ϕ ˜ a Σ + ϕ w (1.4)where ϕ , ϕ ∈ R and where the total net target imbalance for all M strategic investors isdenoted by ˜ a Σ := M (cid:88) i =1 ˜ a i . (1.5)In other words, the terminal stock price is the end-of-day dividend factor D adjusted upor down depending on the aggregate target of the strategic investors (where ϕ > a Σ raises prices) and on the end-of-day noise-tradersupply (where ϕ < w lowers prices). There are several possible interpretations for this reduced-form. One is for a special case of(1.4) in which D is a liquidating dividend at time t = 1 (as in, e.g., Grossman and Stiglitz We call ˜ a Σ the target imbalance because ˜ a Σ is the positive buying targets less the negative selling targets. If the terminal restriction (1.4) is eliminated, our model becomes simpler because the stock’s volatilitybecomes a free parameter and can, for example, be set to be one. The fact that competitive (Radner)equilibrium models without dividends have free volatilities is well-known; see, e.g., Theorem 4.6.3 in Karatzasand Shreve (1998). S = D . (1.6)It is clear that for S t to converge as in (1.4) or (1.6), the price dynamics are restricted astime approaches maturity (i.e., as t ↑ D , but rather the overnightvaluation attached to future dividends. We offer two overnight interpretations. One isthat (1.4) is the market-clearing valuation at which the M strategic investors are willing tohold w shares over night. An alternate interpretation is that there is a separate group ofovernight liquidity providers who arrive and trade in an additional wrap-up round of tradingright after time t = 1. In this alternative, (1.4) is a reduced-form for the market-clearingvaluation given the overnight liquidity provider demand plus any overnight demand fromthe M strategic investors. For any of these three interpretations of (1.4), our model thenproduces intraday equilibrium prices and trading dynamics given the TWAP preferencesand the terminal (overnight) stock valuation condition.The information structure of our model is as follows: For tractability, we assume thestrategic investors have homogeneous beliefs. They all believe the processes (
D, B ) arethe same independent Brownian motions. Over time, the realized dividend factor D t andthe noise-trader orders w t are publicly observed. At time t ∈ [0 , i chooses acumulative stock-holding position θ i,t that satisfies the measurability requirement θ i,t ∈ F i,t := σ (˜ a i , θ i, − , θ (0) i, − , ˜ a Σ , B u , D u ) u ∈ [0 ,t ] . (1.7)It might seem unclear why ˜ a Σ is included in investor i ’s information set. One possibilityis that ˜ a Σ may be directly observable in the market. However, public observability is not Our overnight liquidity providers are different from the ad hoc intraday residual liquidity providers whotrade continuously during the day as in Brunnermeier and Pedersen (2005). A natural restriction in this interpretation is | ϕ | < | ϕ | . Noise traders trade inelastically, so thefull noise-trader order imbalance w must be held by the overnight liquidity-providers and the strategicinvestors. In contrast, the strategic investors do not demand to achieve their aggregate ideal holdings ˜ a Σ inelastically. Note that in the overnight-liquidity-provider interpretation markets at time 1 are just clearedby the M strategic investors, whereas in the wrap-up round the market is cleared by both the overnightliquidity providers and the M strategic investors together. However, if the price S at time 1 differs fromthe valuation D + ϕ ˜ a Σ + ϕ w of the overnight liquidity providers, then a discrete jump between S to D + ϕ ˜ a Σ + ϕ w would be inconsistent with optimal continuous trading trajectories for the strategicinvestors. Hence, we must have (1.4). The noise-trader orders w t are either directly observed or are inferred from S t and D t . As usual in continuous-time models, we also need to impose an integrability condition to ensure thatcertain stochastic integrals are martingale; see Definition 1.1 below. σ (˜ a i , θ i, − , θ (0) i, − , ˜ a Σ , B u , D u ) u ∈ [0 ,t ] = σ (˜ a i , θ i, − , θ (0) i, − , S u , D u ) u ∈ [0 ,t ] . (1.8)In other words, although investors only know their own target ˜ a i directly, they can inferthe aggregate net target ˜ a Σ in equilibrium from the initial stock price because σ ( S ) = σ (˜ a Σ ) . (1.9)See Remark 2.1.2 below for details. Thus, our model lets us investigate the effects of knownor inferable aggregate trading targets on intraday trading and pricing. Next, we turn to the strategic investors’ individual optimization problems. For a strategy θ i,t ∈ F i,t , let X i,t denote investor i ’s wealth process, which has dynamics dX i,t := θ i,t dS t , X i, := θ i, − S + θ (0) i, − . (1.10)As usual, integrating (1.10) gives the wealth process X i,t = X i, + (cid:90) t θ i,u dS u , t ∈ [0 , . (1.11)The set of admissible strategies A i for investor i is defined as follows: Definition 1.1.
A jointly measurable and F i adapted process θ i = ( θ i,t ) t ∈ [0 , is admissible ,and we write θ i ∈ A i , if the following integrability condition holds E (cid:20)(cid:90) θ i,t dt (cid:12)(cid:12)(cid:12) F i, (cid:21) < ∞ . (1.12) ♦ It is well-known that an integrability condition like (1.12) rules out doubling strategies(see, e.g., Chapters 5 and 6 in Duffie (2001) for a discussion of such conditions). While allbounded processes θ i,t satisfy the integrability condition (1.12), the optimal stock holdingprocess in (2.8) below is not bounded but does still satisfy (1.12).For simplicity, we assume all strategic investors have linear utility functions: U i ( x ) := x for all i ∈ { , ..., M } . For a given stock-price process S , investor i seeks an expected utility-maximizing holding strategy ˆ θ i ∈ A i that attains V ( X i, , w , ˜ a i , ˜ a Σ ) := sup θ i ∈A i E (cid:104) X i, − L i, (cid:12)(cid:12)(cid:12) F i, (cid:105) . (1.13) In contrast, rebalancing shocks are ex ante common knowledge in Brunnermeier and Pedersen (2005).
8n (1.13), the variable L i, is the terminal penalty value from the penalty process in (1.1),and the terminal wealth X i, is from (1.11). Section 8 extends our analysis to homogeneousexponential utilities U i ( x ) := − e − x/τ , x ∈ R , for a common risk-tolerance parameter τ > L i, in (1.13) gives investor i an incentive to use the optimal Merton strategy(with price-impact). On the other hand, neglecting the wealth term X i, in (1.13) impliesthat investor i should follow the TWAP strategy γ ( t )(˜ a i − θ i, − ) + θ i, − . The equilibriumstrategy ˆ θ i,t in (2.8) below strikes an optimal balance between these two competing forceswhere the penalty-severity function κ ( t ) determines the relative importance of these twoforces over the trading interval t ∈ [0 , We start by discussing the market-clearing conditions. First, the initial stock holdings θ , − , ..., θ M, − (also private information variables) satisfy in aggregate w = M (cid:88) i =1 θ i, − . (2.1)This means that the initial amount of the total outstanding shares not held by the noisetraders is held by the strategic investors. Furthermore, at all later times t ∈ (0 , θ ,t , ..., θ M,t ) must also satisfy the intraday clearing condition (1.2). The moneymarket is assumed to be in zero supply, and so, initially, we have (cid:80) Mi =1 θ (0) i, − = 0. Clearing inthe stock market at later time points t ∈ (0 ,
1] spills over to clearing in the money marketbecause the strategic investors must use self-financing strategies.Next, we turn to the stock-price dynamics. In the equilibria we construct, each investor i ∈ { , ..., M } perceives that they face a price process of the form dS t := µ i,t dt + σ w ( t ) ηdB t + dD t , (2.2)where σ w ( t ) is an endogenous price-impact loading and η is the noise-trader supply volatilitycoefficient in (1.3). The drift in (2.2) is given by µ i,t := µ ( t )˜ a Σ + µ ( t ) θ i,t + µ ( t )˜ a i + µ ( t ) w t + µ ( t ) w + µ ( t ) θ i, − , (2.3)and depends on public information variables (˜ a Σ , w t , w ), private information variables9˜ a i , θ i,t , θ i, − ) known to investor i , and a set of endogenous smooth deterministic functions µ , ..., µ , σ w : [0 , → R . (2.4)These price dynamics can be interpreted as follows: First, prices move one-to-one withchanges dD t in the dividend factor. Second, random shocks ηdB t to the noise-tradersupply move prices linearly as given by the deterministic function σ w ( t ). Third, since˜ a Σ , θ i,t , ˜ a i , w t , w , and θ i, − are known to investor i at time t , they are not sources of futureprice randomness. Rather, they appear in the price drift as sources of predictable futureprice movement. For example, for the strategic investors to be willing in aggregate to absorban inelastic positive noise-trader supply w t at time t ∈ [0 ,
1] given the TWAP penalties, weexpect them to require a higher expected return. Thus, we expect the coefficient µ ( t ) on w t in the price drift in (2.3) to be positive. Similarly, when the M strategic investors wantto buy a positive net trading target ˜ a Σ in aggregate, the expected price drift needs to bedepressed (i.e., µ ( t ) <
0) to deter them from trying to buy for markets to clear.We now discuss our equilibrium notion. In multi-agent models such as Foster andViswanathan (1996) and in Vayanos (1999), the price-impact function an investor faces(i.e., how S depends on the investor’s holdings) is derived from a market-clearing conditioncombined with a conjecture about how other investors’ holdings depend on S (i.e., thedemand curves for the other investors). In contrast, the drift in (2.3) is a reduced-formrelation describing how investor i perceives the impact of any position θ i,t she might hold hason prices. In particular, our price impact is expressed as a function of the underlying modelinput variables. Implicit in (2.3) are deeper underlying perceptions for investor i about howthe demands of the other M − θ i,t taken by investor i and also even deeper game-theoretic beliefs that the other M − θ i,t and their other information. In particular, their in-equilibrium price beliefsare consistent with the equilibrium price dynamics. In addition, Example 3.3 below showshow investor demand-curve models are a special case of our price-impact model (2.2)-(2.3). Definition 2.1 (Equilibrium) . The deterministic functions ( µ , ..., µ , σ w ) constitute a equi-librium if, given the stock-price dynamics (2.2)-(2.3), the resulting optimal stock holdingprocesses (ˆ θ i,t ) Mi =1 from (1.13) satisfy the following three conditions:10i) the market-clearing condition w t = M (cid:88) i =1 ˆ θ i,t (2.5)holds at all times t ∈ [0 , t = 1 for constants ϕ , ϕ ∈ R , and(iii) when θ i,t is set to the optimizer ˆ θ i,t in µ i,t in (2.3), the resulting equilibrium pricedrift — which we denote by ˆ µ t — does not depend on the investor-specific privateinformation variables (˜ a i , θ i, − , θ (0) i, − ). In other words, µ i,t = ˆ µ t when θ i,t = ˆ θ i,t . ♦ Requirement (iii) in Definition 2.1 means that equilibrium prices only depend on in-dividual variables ˜ a i , θ i, − , and θ i,t via their impact on the aggregate target imbalance ˜ a Σ and the market-clearing conditions (2.1) and (1.2). Thus, all investors perceive the sameequilibrium price drift. However, following Vayanos (1999), the function µ ( t ) gives theimpact of investor i ’s holdings on i ’s perceived price drift µ i,t both off-equilibrium (when θ i,t (cid:54) = ˆ θ i,t ) as well as in-equilibrium (when θ i,t = ˆ θ i,t ).A major challenge in constructing an equilibrium is the following: Market clearing(1.2) requires at each time t ∈ [0 ,
1] that the strategic investors in aggregate absorb thetotal shares w t inelastically supplied by the noise traders. However, in order to inducethe strategic investors to take a position ˆ θ i,t that deviates from their ideal TWAP position γ ( t )(˜ a i − θ i, − )+ θ i, − , prices S t must adjust to give them a sufficient expected return given theinventory penalties they will incur. This is less problematic at dates t < S t during the day are flexible. However, the fact that prices must converge to either a terminalliquidating dividend (1.6) or to an inventory-adjusted overnight price (1.4) restricts thewiggle room for providing the market-clearing price inducement as time t ↑
1. The analysisbecomes particularly subtle for penalty severities κ ( t ) that explode as t ↑
1. However, weconstruct equilibria for a large class of unbounded functions κ ( t ) as well as for all boundednon-negative continuous penalty functions.Theorem 2.2 below gives restrictions on the pricing functions in (2.4) for existence ofan equilibrium. As we shall see, there is one degree of freedom in the pricing coefficients µ ( t ) , ..., µ ( t ), and so there are multiple (indeed, infinitely many) equilibria. The intu-ition for the multiplicity of equilibria is the following: By design, the drift (2.3) is linearlyimpacted by θ i,t as well as by (˜ a Σ , ˜ a i , w t , w , θ i, − ). As we shall see in Theorem 2.2 below, op-timal holdings are linear in (˜ a Σ , ˜ a i , w t , w , θ i, − ) and inserting such linear holdings into (2.3) Section 8 extends the model to risk-averse utilities so that then dividend risk also matters. a Σ , ˜ a i , w t , w , θ i, − ). Leaving the function µ ( t )as the free parameter leads to intuitive interpretations in terms of different possible priceimpacts of orders. In Theorem 2.2 below, the second-order condition for optimality of theinvestors’ optimal controls is a restriction on the function µ : [0 , → R given by µ ( t ) < κ ( t ) , t ∈ (0 , . (2.6)Our main equilibrium existence result is the following (proof in Appendix A): Theorem 2.2.
Let γ : [0 , → [0 , ∞ ) be a continuous function, and let κ : [0 , → (0 , ∞ ) and µ : [0 , → R be continuous and square integrable functions; i.e., (cid:90) (cid:0) κ ( t ) + µ ( t ) (cid:1) dt < ∞ , (2.7) that satisfy the second-order condition (2.6) . Then an equilibrium exists in which:(i) Investor optimal holdings ˆ θ i in equilibrium are given by ˆ θ i,t = w t M + 2 κ ( t ) γ ( t )2 κ ( t ) − µ ( t ) (cid:16) ˜ a i − ˜ a Σ M (cid:17) + 2 κ ( t ) (cid:0) − γ ( t ) (cid:1) κ ( t ) − µ ( t ) (cid:16) θ i, − − w M (cid:17) . (2.8) (ii) The equilibrium stock price is given by S t = g ( t ) + g ( t )˜ a Σ + σ w ( t ) w t + D t , (2.9) where the deterministic functions g , g , and σ w : [0 , → R are the unique solutionsof the following linear ODEs: g (cid:48) ( t ) = 2 κ ( t )( γ ( t ) − M w − ασ w ( t ) , g (1) = 0 ,g (cid:48) ( t ) = − γ ( t ) κ ( t ) M , g (1) = ϕ ,σ (cid:48) w ( t ) = 2 κ ( t ) − µ ( t ) M + πσ w ( t ) , σ w (1) = ϕ . (2.10) (iii) The pricing functions µ , µ , µ , µ , and µ in (2.3) are given in terms of µ by (A.15) - (A.19) in Appendix A.Remark . We note several properties of this equilibrium here:12. The system of ODEs in (2.10) has the unique solution: g ( t ) = α (cid:90) t σ w ( u ) du + w (cid:90) t κ ( u ) (cid:0) − γ ( u ) (cid:1) M du, (2.11) g ( t ) = ϕ + (cid:90) t γ ( s ) κ ( s ) M ds, (2.12) σ w ( t ) = e π ( t − ϕ − (cid:90) t e π ( t − u ) κ ( u ) − µ ( u ) M du. (2.13)This follows because the ODE for σ w ( t ) is linear and, thus, has a unique solution,which then uniquely determines g ( t ). In addition, the boundary conditions in (2.10)insure the terminal price condition (1.4) is satisfied at t = 1. Hence, the equilibriumprice in (2.9) can be written as S t = (cid:104) α (cid:90) t σ w ( u ) du + w (cid:90) t κ ( u ) (cid:0) − γ ( u ) (cid:1) M du (cid:105) + (cid:104) ϕ + (cid:90) Tt γ ( s ) κ ( s ) M ds (cid:105) ˜ a Σ + (cid:104) e π ( t − ϕ + (cid:90) t e π ( t − u ) µ ( u ) − κ ( u ) M du (cid:105) w t + D t . (2.14)2. From (2.9), the initial stock price at t = 0 is S = g (0) + g (0)˜ a Σ + σ w (0) w + D . (2.15)Therefore, whenever the solution g ( t ) from (2.10) satisfies g (0) (cid:54) = 0, the aggregatetarget ˜ a Σ can be inferred from S because w and D are constants. Thus, we have σ ( S ) = σ (˜ a Σ ) in (1.9), which produces the measurability equivalence in (1.8). From(2.12), a sufficient condition for g (0) (cid:54) = 0 is γ ( t ) > κ ( t ) > t ∈ (0 , ϕ ≥
0. In that case, the optimal control ˆ θ i,t in (2.8) can written as a linearfunction of (˜ a i , S , S t ) where ( S , S t ) are given in (2.15) and (2.9).3. From (2.9), the equilibrium price dynamics are given by dS t = (cid:104) g (cid:48) ( t ) + g (cid:48) ( t )˜ a Σ + σ (cid:48) w ( t ) w t (cid:105) dt + σ w ( t ) dw t + dD t = (cid:104) g (cid:48) ( t ) + g (cid:48) ( t )˜ a Σ + σ (cid:48) w ( t ) w t + σ w ( t )( α − πw t ) (cid:105) dt + σ w ( t ) ηdB t + dD t . (2.16)13e define the equilibrium stock price drift ˆ µ t byˆ µ t := g (cid:48) ( t ) + g (cid:48) ( t )˜ a Σ + σ (cid:48) w ( t ) w t + σ w ( t )( α − πw t )= 2 κ ( t ) − µ ( t ) M w t + 2 κ ( t ) (cid:0) γ ( t ) − (cid:1) M w − κ ( t ) γ ( t ) M ˜ a Σ . (2.17)The second equality in (2.17) follows from substitution of (2.10) into the first line of(2.17). The quadratic variation of S and quadratic cross-variations between S and( D, w ) are given by d (cid:104) S (cid:105) t = (cid:0) σ w ( t ) η + 1 (cid:1) dt, d (cid:104) S, D (cid:105) t = dt, d (cid:104) S, w (cid:105) t = σ w ( t ) η dt. (2.18)The intuition for the price dynamics in (2.16)-(2.17) is as follows: In contrast to w and ˜ a Σ , which only affect prices over time through their predictable impact onthe market-clearing price drift, the noise-trader supply w t evolves stochastically overtime. As a result, w t affects price dynamics via two channels. First, the drift ˆ µ t in(2.17) required for market clearing at time t is increasing in the accumulated noise-trader supply w t that the M strategic investors must hold at time t . Second, theprice dynamics dS t are decreasing in the random noise-trader shock dw t since σ w ( t )in (2.13) is negative whenever ϕ ≤
0. In particular, the integral in (2.13) for σ w ( t )ensures that the contemporaneous fall in prices given dw t > t is sufficientto allow increased expected price drifts in the future.4. From (2.9), we see that the equilibrium stock-price process is Gaussian. More specif-ically, the price process (2.9) is a Bachelier model with time-dependent coefficients.While the equilibrium stock price can be negative with positive probability, such Gaus-sian models have been widely used in the market microstructure literature by, e.g.,Grossman and Stiglitz (1980) and Kyle (1985). Gaussian models are also widely usedin the optimal execution literature including Almgren and Chriss (1999, 2001); see,e.g., the discussion in Section 3.1 in the Gatheral and Schied (2013) survey.5. The difference S t − D t is the price effect of imbalances in liquidity supply and demand.We call this the liquidity premium and note that it can be positive or negative depend-ing on the aggregate target imbalance ˜ a Σ and on whether the noise traders are buyingor selling. From (2.9) and (2.14), the liquidity premium has (i) a deterministic compo-nent g ( t ) (due to any initial supply w and to predictable future noise-trader supplytrends α ), (ii) deterministic effects g ( t )˜ a Σ due to the net strategic-investor target im-balance ˜ a Σ , and (iii) a random component σ w ( t ) w t due to the changing noise-tradersupply w t . 14. Optimal investor holdings (2.8) have an intuitive structure in equilibrium. To see this,we rearrange (2.8) as follows:ˆ θ i,t = w t M + 2 κ ( t )2 κ ( t ) − µ ( t ) (cid:20) γ ( t ) (cid:16) ˜ a i − ˜ a Σ M (cid:17) + (cid:0) − γ ( t ) (cid:1) (cid:16) θ i, − − w M (cid:17)(cid:21) . (2.19)The first term in (2.19) shows that strategic investors share the noise-trader supply w t equally in equilibrium. The second term adjusts these equal positions to take intoaccount differences in different investors’ target trajectories subject to the market-clearing constraint. The term in the square brackets is the difference between investor i ’s personal TWAP target trajectory γ ( t )˜ a i + (cid:0) − γ ( t ) (cid:1) θ i, − and i ’s pro rata share (i.e.,one M th) of the aggregate TWAP target trajectory γ ( t ) ˜ a Σ M + (1 − γ ( t )) w M for all ofthe strategic investors. This difference is scaled by the coefficient κ ( t )2 κ ( t ) − µ ( t ) , whichdepends on the particular equilibrium pinned down by the price-impact function µ ( t ).One immediate implication of (2.19) is that in equilibrium there is no predatorytrading since the signs of the coefficients on w t and ˜ a Σ do not switch over time. Asecond implication is that an investor with an above-average target ˜ a i > ˜ a Σ M holds morestock than an investor with a below-average target keeping θ i, − fixed. This implicationfollows because the second-order condition (2.6) implies that the coefficient κ ( t )2 κ ( t ) − µ ( t ) is positive. A third implication of (2.19) is that the difference ˜ a i − ˜ a Σ M becomes moreimportant than the difference θ i, − − w M in the investor’s holdings ˆ θ i,t as γ ( t ) ↑ t ↑ i places a discrete order at time t = 0 but then trades continuously thereafter.From (2.8), investor i ’s initial trade isˆ θ i, − θ i, − = 2 κ (0) γ (0)2 κ (0) − µ (0) (cid:16) ˜ a i − ˜ a Σ M (cid:17) + (cid:104) κ (0) (cid:0) − γ (0) (cid:1) κ (0) − µ (0) − (cid:105)(cid:16) θ i, − − w M (cid:17) , (2.20)which is generically non-zero when ˜ a i (cid:54) = ˜ a Σ /M and/or θ i, − (cid:54) = w /M .8. The presence of w t in (2.8) implies that the ˆ θ i,t holding paths are non-differentiable.Consequently, there is no dt -rate at which buying and selling occur. However, when κ ( t ) , γ ( t ), and µ ( t ) are smooth functions, the equilibrium holding paths have Itˆodynamics d ˆ θ i,t = 1 M ( α − πw t ) dt + 1 M ηdB t + (cid:34)(cid:18) κ ( t ) γ ( t )2 κ ( t ) − µ ( t ) (cid:19) (cid:48) (cid:16) ˜ a i − ˜ a Σ M (cid:17) + (cid:32) κ ( t ) (cid:0) − γ ( t ) (cid:1) κ ( t ) − µ ( t ) (cid:33) (cid:48) (cid:16) θ i, − − w M (cid:17)(cid:35) dt. (2.21)15. The holdings ˆ θ i,t of investor i differ generically from i ’s TWAP target trajectory. First,this is because the strategic investors follow “smart” (i.e., optimized) strategies ratherthan mechanical TWAP strategies with hard constraints. In particular, the squaredintegrability condition (2.7) means that, even if the penalty severity κ ( t ) is unboundedas t ↑
1, the TWAP penalties still lead to a soft, rather than a hard, constraint relativeto the investor’s target trajectory. Second, market clearing leads to prices such thatthe M strategic investors, in aggregate, take the other side of the noise-trader ordersthroughout the day. From (2.19) and Remark 2.1.6 above, the equilibrium deviationsof investor holdings ˆ θ i,t from their TWAP trajectory at time t areˆ θ i,t − (cid:2) θ i, − + γ ( t )(˜ a i − θ i, − ) (cid:3) = w t M + µ ( t )2 κ ( t ) − µ ( t ) (cid:104) γ ( t )˜ a i + (cid:0) − γ ( t ) (cid:1) θ i, − (cid:105) − κ ( t )2 κ ( t ) − µ ( t ) (cid:104) γ ( t ) ˜ a Σ M + (cid:0) − γ ( t ) (cid:1) w M (cid:105) . (2.22)In other words, the TWAP deviation for investor i consists of their share of the noise-trader order plus additional components tied to investor i ’s personal TWAP trajectoryand the aggregate target trajectory in ways that depend on µ ( t ).The average target error in any equilibrium is always the pro rata share of the noise-trader supply less the pro rata share of the aggregate TWAP target trajectory (cid:80) Mi =1 ˆ θ i,t M − (cid:80) Mi =1 (cid:2) θ i, − + γ ( t )(˜ a i − θ i, − ) (cid:3) M = w t M − (cid:104) w M + γ ( t ) (cid:16) ˜ a Σ M − w M (cid:17)(cid:105) . (2.23)However, in all equilibria with µ ( t ) (cid:54) = 0, there is generically cross-section dispersionin individual investor target deviations around the average target deviationˆ θ i,t − (cid:2) θ i, − + γ ( t )(˜ a i − θ i, − ) (cid:3) − w t M + (cid:104) w M + γ ( t ) (cid:16) ˜ a Σ M − w M (cid:17)(cid:105) = µ ( t )2 κ ( t ) − µ ( t ) (cid:104) γ ( t ) (cid:16) ˜ a i − ˜ a Σ M (cid:17) + (cid:0) − γ ( t ) (cid:1)(cid:16) θ i, − − w M (cid:17)(cid:105) . (2.24)10. A special case of interest is an arithmetic Brownian motion noise-trader imbalanceprocess ( π := 0) and equal initial sharing ( θ i, − := w M ). From (2.8), the conditionalexpected investor holdings at time t in this case are E [ˆ θ i,t | σ (˜ a i , ˜ a Σ )] = w + αtM + 2 κ ( t ) γ ( t )2 κ ( t ) − µ ( t ) (cid:16) ˜ a i − ˜ a Σ M (cid:17) . (2.25)16rom (2.21) the resulting conditional variance of investor holdings is given by V [ˆ θ i,t | σ (˜ a i , ˜ a Σ )] = η tM , (2.26)which is independent of the private target ˜ a i .The equilibrium in Theorem 2.2 has the following comparative statistics: Corollary 2.3.
In the setting of Theorem 2.2, we have:(i) The function σ w ( t ) is increasing in µ ( t ) and M , decreasing in κ ( t ) , and independentof γ ( t ) , α , and η .(ii) When ϕ ≤ in (1.4) , σ w ( t ) is non-positive and increasing in π . Consequently,liquidity | σ w ( t ) | is decreasing in µ ( t ) , π , and M and increasing in κ ( t ) .(iii) The conditional variance of the liquidity premium is V [ S t − D t | σ (˜ a Σ )] = σ w ( t ) V [ w t ] . (2.27) Proof.
The linear ODE (2.10) for σ w ( t ) has the unique solution (2.13). This produces thefirst two claims in the corollary. The third claim follows from the representation of S t − D t using (2.9). ♦ Empirical prediction 1:
Corollary 2.3 leads to a set of empirical predictions about chang-ing market conditions across different days. In particular, suppose that on different daysthere are different numbers of strategic investors M , penalty severities κ ( t ), and speeds ofnoise-trader order-flow mean-reversion π . Corollary 2.3 predicts that price volatility andmarket illiquidity, as measured by | σ w ( t ) η | , should be higher on days on which there arefewer strategic investors (to share the noise-trader order flow), stronger TWAP penalties,less noise-trader order-flow mean reversion, and greater noise-trader supply volatility η . Empirical prediction 2:
Our model predicts two types of intraday patterns in pricing.The first prediction is that liquidity effects due to known imbalances and trends in thenoise-trader supply w and α and the target imbalances ˜ a Σ lead to ex ante predictable17ntraday drifts in prices. Taking the conditional expectation of (2.17) gives us E [ˆ µ t | σ (˜ a Σ )]= 2 κ ( t ) − µ ( t ) M E [ w t ] + 2 κ ( t ) (cid:0) γ ( t ) − (cid:1) M w − κ ( t ) γ ( t ) M ˜ a Σ (2.28)= 2 κ ( t ) − µ ( t ) M (cid:0) w e − πt + απ (1 − e − πt ) (cid:1) + 2 κ ( t ) (cid:0) γ ( t ) − (cid:1) M w − κ ( t ) γ ( t ) M ˜ a Σ = 2 κ ( t ) − µ ( t ) M απ (1 − e − πt ) + 2 κ ( t ) (cid:0) e − πt + γ ( t ) − (cid:1) − µ ( t ) M w − κ ( t ) γ ( t ) M ˜ a Σ . Thus, prices should predictably drift up on days on which there is a ex ante predictableincrease in supply spread out over the trading day (expected prices rise to induce thestrategic investors to absorb α > a Σ < w on prices is more complicated since it has both adirect effect on E [ˆ µ t | σ (˜ a Σ )] and an indirect effect as the starting point for w t . The secondprediction is that if the speed-of-mean-reversion π is sufficiently low, then the volatility V [ S t − D t ] of the liquidity premium in (2.27) is non-monotone over the trading day. Thisfollows from (2.9) and the fact that V [ w t ] starts at 0 at time 0 and grows monotonically overthe trading day while, from (2.13), the absolute value of the price impact loading | σ w ( t ) | ismonotonically decreasing over the trading day when π is not too large.A natural question is about the impact of intraday trading targets on financial markets.There are two natural comparisons for our model. One comparison is a competitive market(see Example 3.1 below) with negligible trading targets. Given risk neutrality, this corre-sponds to an equilibrium where κ ( t ) is small, µ ( t ) := 0 for all t ∈ [0 , ϕ := ϕ := 0.In the limit in this case, the M risk-neutral strategic agents share the supply of shares fromthe noise traders equally in (2.8) and prices are just the current dividend value D t from(2.9). Comparing this case with our model in which κ ( t ) is large, shows that trading targetsinduce a random liquidity premium in prices and make markets less liquid. A second andmore nuanced comparison is a market in which the strategic investors have terminal tradingtargets ˜ a i at the end of the day but do not have intraday target trajectories. This corre-sponds to a penalty severity κ ( t ) that is negligible during the day but positive at the endof the day. Our next result shows that intraday TWAP benchmarking increases intradayprice volatility and market illiquidity. Corollary 2.4.
In the setting of Theorem 2.2, we have the following: For fixed ¯ t ∈ (0 , and ϕ ≤ , we let two penalty-severity functions κ ( t ) and κ ( t ) be ordered such that κ ( t ) < κ ( t ) for times t ∈ [0 , ¯ t ) and κ ( t ) = κ ( t ) for t ∈ [¯ t, . Then, given µ ( t ) < κ ( t ) , price volatility nd illiquidity are less in the market with κ ( t ) than in the market with κ ( t ) .Proof. The claim follows from Corollary 2.3(ii). ♦ This does not mean TWAP and VWAP incentives are inefficient. Rather, it means thatTWAP and VWAP as solutions to delegated-trading agency problems are not socially cost-less.
Our model has multiple equilibria, each uniquely pinned down by a choice of the price-impact function µ ( t ). This section considers some theoretically motivated examples of µ ( t ) functions. In addition, Section 4 constructs µ ( t ) based on an empirical calibration. Example 3.1 (Radner) . In a fully competitive Radner equilibrium, the perceived priceprocess ( S t ) t ∈ [0 , is unaffected by investor i ’s holdings θ i,t . In particular, the Radner price-taking condition says that the price dynamics perceived by investor i are independent of any amount θ i,t held by investor i . This case sets the investor order price-impact functionin (2.3) to µ ( t ) := 0 , for all t ∈ [0 , . (3.1)This independence is stronger than the equilibrium requirement (iii) in Definition 2.1, whichrequires the price dynamics perceived by investor i to be independent of investor i ’s stockholdings only for the equilibrium holdings ˆ θ i,t but not for arbitrary holdings.In the Radner case with µ ( t ) := 0, all M investors have, from (2.22), identical TWAPdeviations equal to the average deviation in (2.23). In other words, investors share thenoise-trader orders and the aggregate target shortfall equally. In contrast, when µ ( t ) (cid:54) = 0(as in the next two examples below), the TWAP deviations (2.22) differ across investorsdepending on their individual targets ˜ a i . ♦ Example 3.2 (Welfare maximization) . There are many ways to measure social welfare(see, e.g., Section 6.1 in Vayanos (1999)). We follow Du and Zhu (2017) and their Equation(42) and consider maximizing an expected aggregate certainty-equivalent criterion for the M strategic investors. The certainty equivalent CE i ∈ R for strategic investor i is definedas CE i := V ( X i, , w , ˜ a i , ˜ a Σ ) (3.2)where V is the value function defined in (1.13). The certainty equivalent (3.2) followsfrom the assumption that all strategic investors are risk neutral, (i.e., their utilities are19 i ( x ) := x ). We are interested in the price-impact function µ ∗ : [0 , → R that maximizesthe objective equal to total welfare of the M strategic investors µ ∗ ( t ) ∈ argmax µ ( t ) M (cid:88) i =1 E [CE i ] . (3.3)The objective (3.3) is ex ante in that the expectation E is taken over the random private-information variables ( θ (0)1 , − , ..., θ (0) M, − ) , ( θ , − , ..., θ M, − ), and (˜ a , ..., ˜ a M ). Theorem 5.1 in Sec-tion 5 gives sufficient conditions for existence of the welfare-maximizing equilibrium. ♦ Example 3.3 (Vayanos) . Vayanos (1999) considers investor demand curves in a discrete-time model with trading times 0 ≤ t < t < ... < t N ≤
1. The discretized version of ourmarket-clearing condition (2.5) is∆ w t n = (cid:88) k (cid:54) = i ∆ θ k,t n + ∆ θ i,t n . (3.4)Similar to Vayanos (1999), suppose the trades ∆ θ k,t n of investors k (cid:54) = i are linear functionsof S t n . Solving (3.4) for S t n gives that the price-impact function that investor i faces in heroptimization problem is linear in the difference ∆ w t n − ∆ θ i,t n . To convert this restrictioninto our continuous-time setting, we conjecture that equilibrium holdings for all investorsare symmetric and have the form θ k,t = F ( t )˜ a Σ + F ( t )( S t − D t ) + F ( t )˜ a k + F ( t ) (3.5)for common deterministic functions F ( t ) , ..., F ( t ). When all investors use (3.5), theclearing condition (1.2) ensures that the equilibrium price S t satisfies w t = (cid:0) M F ( t ) + F ( t ) (cid:1) ˜ a Σ + M F ( t )( S t − D t ) + M F ( t ) . (3.6)The solution S t of (3.6) whenever F ( t ) (cid:54) = 0 can be written as S t = g ( t ) + g ( t )˜ a Σ + σ w ( t ) w t + D t , (3.7)where the coefficient functions are g ( t ) := − F ( t ) F ( t ) , g ( t ) := − M F ( t ) + F ( t ) M F ( t ) , σ w ( t ) := 1 M F ( t ) . (3.8) For simplicity, we set w := 0 and θ i, − := 0 for all investors in this example. dS t = dD t + σ w ( t ) dw t + ˜ a Σ g (cid:48) ( t ) dt + w t σ (cid:48) w ( t ) dt + g (cid:48) ( t ) dt = dD t + σ w ( t )( αdt + ηdB t ) + ˜ a Σ g (cid:48) ( t ) dt + g (cid:48) ( t ) dt + (cid:0) σ (cid:48) w ( t ) − πσ w ( t ) (cid:1)(cid:16)(cid:0) M F ( t ) + F ( t ) (cid:1) ˜ a Σ + M F ( t )( S t − D t ) (cid:17) dt, (3.9)where the second equality uses (3.6).To verify that (3.9) fits into our price-impact model (2.2)-(2.3), we adapt an argumentfrom Vayanos (1999). When investor i is free to use any holding process but all otherinvestors use (3.5), the market-clearing condition (2.5) becomes w t = (cid:88) k (cid:54) = i (cid:16) F ( t )˜ a Σ + F ( t )( S t − D t ) + F ( t )˜ a k + M F ( t ) (cid:17) + θ i,t = ( M − F ( t )˜ a Σ + ( M − F ( t )( S t − D t ) + F ( t )(˜ a Σ − ˜ a i ) + ( M − F ( t ) + θ i,t . (3.10)Provided that F ( t ) (cid:54) = 0 and M ≥
2, we can solve for S t in (3.10) to get S t = D t + w t − ( M − F ( t )˜ a Σ − F ( t )(˜ a Σ − ˜ a i ) − ( M − F ( t ) − θ i,t ( M − F ( t ) . (3.11)Substituting (3.11) for S t − D t into the dynamics (3.9) gives dS t = dD t + σ w ( t )( αdt + ηdB t ) + ˜ a Σ g (cid:48) ( t ) dt + g (cid:48) ( t ) dt + (cid:0) σ (cid:48) w ( t ) − πσ w ( t ) (cid:1)(cid:16)(cid:0) M F ( t ) + F ( t ) (cid:1) ˜ a Σ + M w t − ( M − F ( t )˜ a Σ − F ( t )(˜ a Σ − ˜ a i ) − ( M − F ( t ) − θ i,t ( M − (cid:17) dt. (3.12)Therefore, in this example the coefficients in front of w t and − θ i,t in the drift of S t areidentical. In turn, this corresponds to requiring in (2.3) that µ ( t ) = − µ ( t ) , t ∈ [0 , . (3.13)When we add restriction (3.13), our model produces a unique equilibrium with µ ( t ) = 2 κ ( t )2 − M , t ∈ [0 , . (3.14)Consequently, the second-order condition (2.6) holds in this example whenever there are at To derive (3.14), we solve (A.17) and (3.13) simultaneously for µ ( t ) and µ ( t ). M ≥ µ ( t ) given by (3.14) is negative.We conclude by relating investor demand curves to the multiplicity of equilibria. To dothis, we extend (3.5) by adding a linear term in w t so that investor demand (3.5) becomes θ k,t = F ( t )˜ a Σ + F ( t )( S t − D t ) + F ( t )˜ a k + F ( t ) + F ( t ) w t , (3.15)for a deterministic function F ( t ). Adjusting the arguments above produces an extensionof (3.12) without the requirement (3.13). In other words, (3.15) leads to our model (2.3)with µ ( t ) now free. ♦ Our model can be calibrated empirically using two different approaches. One approach cali-brates an implied µ ( t ) function to make the optimal holding formula (2.8) match empiricaldata on how brokers divide parent trading targets ˜ a i − θ i, − into a series of child orders (seeO’Hara (2015)). However, such data are proprietary and typically known only by clientsand brokers.A second approach — which is our focus here — calibrates our model to a continuous-time empirical price-impact model where dS Mkt t := λ ( t )( µ Yt dt + dB t ) + dD t , (4.1)where λ ( t ) is a smooth deterministic function, and the arriving signed public aggregateorder-flow dynamics µ Yt dt + dB t have a drift µ Yt and are normalized to have a quadraticvariation of one. The goal is to take an estimated differentiable function λ ( t ) as a calibrationinput and derive the corresponding implied µ ( t ) function in (2.3). When π := 0 in (1.3)to match the un-autocorrelated order dynamics in (4.1), Theorem 2.2 gives the equilibriumODE for the loading σ w ( t ) as σ (cid:48) w ( t ) = 2 κ ( t ) − µ ( t ) M , σ w (1) = ϕ . (4.2)Consequently, given an estimated λ ( t ) function in (4.1), we need to solve λ ( t ) = σ w ( t ) η, λ (1) = ϕ η. (4.3)Differentiating (4.3) and using (2.10) and π = 0 give µ ( t ) in terms of λ (cid:48) ( t ) as µ ( t ) = 2 κ ( t ) − M λ (cid:48) ( t ) η , ϕ = λ (1) η , (4.4)22rovided that the solution in (4.4) satisfies the second-order condtion (2.6). Empirical prediction 3:
The calibration condition in (4.4) requires that the empiricalslope of the price-impact function λ (cid:48) ( t ) must be positive and sufficiently large in order forthe second-order condition (2.6) to hold. This restriction is interpreted as follow: Sinceour order-flow sign convention implies a negative price impact of the noise-trader supply ofshares w t to the market, this means λ ( t ) <
0, and so a positive derivative λ (cid:48) ( t ) means thatthe price impact is decreasing in absolute value over the day. This prediction is roughlyconsistent with an approximately declining pattern in the intraday trading-time-contingentprice impacts estimated in Barardehi and Bernhardt (2018, Table 4). Risk neutrality of the strategic investors lets us decompose the expected aggregate certaintyequivalent into two components tied to expected wealth and expected penalties M (cid:88) i =1 E [CE i ] = M (cid:88) i =1 E [ X i, ] − M (cid:88) i =1 E [ L i, ] . (5.1)To simplify the exposition, this section assumes parameter restrictions π := 0 , θ i, − := w M , ϕ := ϕ := 0 . (5.2)The market-clearing condition (2.5) and the ODEs (2.10) produce M (cid:88) i =1 E [ X i, ] = M (cid:88) i =1 (cid:18) S ˆ θ i, + ˆ θ (0) i, + (cid:90) E [ˆ θ i,t ˆ µ t ] dt (cid:19) = S w + (cid:90) E [ w t ˆ µ t ] dt = ... + (cid:0) g (0) + σ w (0) w (cid:1) w − (cid:90) E [ w t ] µ ( t ) M dt = ... + (cid:90) (cid:0) αw t + w − E [ w t ] (cid:1) µ ( t ) M dt, (5.3) The model has other parameters and functions to calibrate empirically. The TWAP functions κ ( t )and γ ( t ) can be assumed to be stable over time and can be estimated via GMM moment-matching. Theimbalances w and ˜ a Σ and the number of strategic investors M are not directly observable and presumablyvary from one day to the next. Thus, they can be estimated by daily implied calibration to match changingrealized price trends for each day in the sample. Some of the noise-trader parameters α , π , and η may beintertemporal constants and estimated via GMM, while others may take changing daily values and, thus,must be imputed by daily implied calibration. ... ” for terms that do not depend on µ ( t ). Consequently, we can findfunctions f = f ( µ, t ) and h = h ( µ, t ) such that M (cid:88) i =1 E [ X i, ] = (cid:90) f ( µ ( t ) , t ) dt, (5.4) M (cid:88) i =1 E [ L i, ] = (cid:90) E (cid:104) κ ( t ) (cid:16) γ ( t )(˜ a i − w M ) − (ˆ θ i,t − w M ) (cid:17) (cid:105) dt = (cid:90) h ( µ ( t ) , t ) dt. (5.5)To understand the difference between the Radner equilibrium with µ ( t ) = 0 and thewelfare-maximizing equilibrium with µ ( t ) = µ ∗ ( t ) maximizing (5.1), consider f ( µ ( t ) , t ) − f (0 , t ) = µ ( t ) w ( w + αt ) − E [ w t ] M , (5.6) h ( µ ( t ) , t ) − h (0 , t ) = µ ( t ) γ ( t ) κ ( t ) (cid:0) κ ( t ) − µ ( t ) (cid:1) (cid:16) M (cid:88) i =1 E [˜ a i ] − E [˜ a ] M (cid:17) . (5.7)There are two points to note about (5.7). First, (5.7) is independent of the noise-traderdynamics dw t , i.e., none of the constants ( w , α, η ) appear. Second, the difference (5.7) isnon-negative and minimized at µ ( t ) := 0, which is the Radner equilibrium. In contrast,the difference (5.6) is linear in µ ( t ) and its slope is determined by the constants ( w , α, η ).The following result describes properties of welfare-maximizing equilibria in differentregions of the parameter space (proof in Appendix A). Theorem 5.1.
Let γ : [0 , → [0 , ∞ ) be a continuous function, and let κ : [0 , → (0 , ∞ ) be continuous and square integrable. We assume that E [˜ a i ] < ∞ for i ∈ { , ..., M } , and welet (5.2) hold.(i) When the two additional parameter restrictions γ ( t ) (cid:16) E [˜ a ] − M M (cid:88) i =1 E [˜ a i ] (cid:17) < t ( η + α t + αw ) < , t ∈ (0 , , (5.8) hold, there exists a unique non-negative continuous price-impact function µ ∗ : [0 , → R that attains sup µ ( t ) M (cid:88) i =1 E [ CE i ] , (5.9) where the supremum is taken over all continuous and square integrable functions µ :[0 , → R satisfying the second-order condition (2.6) . Furthermore, the maximizer µ ∗ ( t ) is linear in κ ( t ) in that the time-dependent ratio µ ∗ ( t ) κ ( t ) is independent of κ ( t ) . ii) For αw ≥ , the critierion (5.9) does not have a maximizer and the welfare-maximizingequilibrium does not exist.(iii) For η := α := 0 , the critierion (5.9) is maximized by µ ∗ ( t ) = 0 .Remark . We note some features of this result:1. The two restrictions in (5.8) are sufficient conditions for a maximizer µ ∗ ( t ) to exist.The first inequality ensures that µ ∗ ( t ) satisfies the second-order condition (2.6). Thesecond inequality in (5.8) — which requires αw to be sufficiently negative — isa coercivity condition that ensures that very negative values of µ ∗ ( t ) can never beoptimal. While the simple TWAP target trajectory γ ( t ) := t is included in Theorem2.2, the first restriction in (5.8) prevents it from being included in Theorem 5.1.2. The competitive Radner equilibrium — which minimizes aggregate expected TWAPpenalties — does not always maximize aggregate strategic-investor total welfare be-cause the price-impact function µ ( t ) also affects aggregate expected strategic-investorwealth. In particular, initial strategic-investor wealth, given their initial holdings (cid:80) Mi =1 ˆ θ i, = w , depends on the initial market-clearing price S which, in equilibrium,depends on µ ( t ). In addition, the expected wealth the strategic investors can ex-tract from trading over the day with the noise traders, (cid:82) E [ w t ˆ µ t ] dt , also depends on µ ( t ). The parameter restrictions in (5.8) affect the existence and size of the pos-sible wealth effects. However, when the maximizer µ ∗ ( t ) exists and is non-zero, theexpected wealth gains from µ ∗ ( t ) > αw ≥
0, theRadner equilibrium still exists. In the next section on numerics we consider a casewith αw < w t = w for all t ∈ [0 , µ ∗ ( t ) in Theorem 5.1 implies that in the welfare-maximizing case in Example 3.2, the expected holdings (2.25) also do not depend on thepenalty severity κ ( t ). In addition, from (2.21) the drift in investor i ’s optimal holdingswhen θ i, − := w M is αM + (cid:18) κ ( t ) γ ( t )2 κ ( t ) − µ ( t ) (cid:19) (cid:48) (cid:16) ˜ a i − ˜ a Σ M (cid:17) . (5.10)25his is a deterministic function of investor i ’s private target ˜ a i and the public aggregatevariable ˜ a Σ defined in (1.5). In the welfare-maximizing case, (5.10) is independent of κ ( t ). This section compares model outcomes of the welfare-maximizing equilibrium (Example 3.2with µ ∗ ( t ) > µ ∗ ( t ) for (5.9); second, properties of the equilibrium price S t in (2.9); third, how the smart TWAP traders and market makers share the availablesupply w t given their individual heterogeneous target holdings ˜ a i ; and four, welfare. Thenumerical properties here all illustrate analytic derivations in Section 2.Our analysis uses the terminal stock-price restriction (1.6) with an initial dividend factornormalized to D := 20 and M := 10 strategic investors. For the dynamics of the noise-trader process w in (1.3), we use the parameter values w := 10 , α := − , π := 0 , η := 1 . (6.1)The strategic investors’ private information variables are θ (0) i, − := 0 , θ i, − := w M = 1 , ˜ a i ⊥ ˜ a j for i (cid:54) = j, E [˜ a i ] = 0 , E [˜ a i ] = 1 . (6.2)Under these assumptions, the aggregate variable ˜ a Σ in (1.5) has the moments E [˜ a Σ ] = 0 , E [˜ a ] = M = 10 . (6.3)The units here can be interpreted as follows: The initial expectation of future dividends is$20 per share ( D ), and the daily volatility of dividend value changes is normalized to $1.In aggregate, 10 strategic investors ( M ) share equally an initial supply of 10 million sharesfrom noise traders ( w ) who are expected to sell 1 million shares on this particular day ( α )with a daily trading volatility also of 1 million shares ( η ).For the penalty process L i,t in (1.1), the target-ratio function is γ ( t ) := 0 . . t, t ∈ [0 , . (6.4)This target ratio function γ ( t ) can be interpreted as a modified TWAP target trajectory The quantities ( θ (0) i, − , θ i, − , ˜ a i ) and the moments for ˜ a i are only used to (i) compute the welfare-maximizer µ ∗ ( t ), the optimal holdings, and other properties in the welfare-maximizing equilibrium and (ii) in the exante welfare analysis below.
26n which traders are initially impatient to get part of their trading done quickly, but thenbecome more patient during the rest of the day. We consider four different penalty-severityfunctions { κ ( t ) , ..., κ ( t ) } defined by κ ( t ) := 1 , t ∈ [0 , ,κ ( t ) := 1 + t, t ∈ [0 , ,κ ( t ) := 98 (1 − t ) − . , t ∈ [0 , ,κ ( t ) := . t ∈ [0 , . , . . t − .
95) for t ∈ [ . , . , (1 − t ) − . for t ∈ [ . , . (6.5)A natural baseline is κ ( t ) where the penalty severity is constant over the day. The nexttwo functions, κ ( t ) and κ ( t ), are both strictly greater than κ ( t ). Comparing results forthem vs. κ ( t ) illustrates how greater penalty severity affects market conditions. Note that κ ( t ) and κ ( t ) both integrate to over t ∈ [0 , κ ( t ) vs. κ ( t ) shows how market conditions change with different intraday penalty-severity patternswhile holding the total daily penalty severity fixed. In particular, κ ( t ) is bounded whereas κ ( t ) explodes as t ↑ κ ( t ) still satisfies the square integrability condition (2.7)).An exploding terminal penalty severity like κ ( t ) is another natural case of interest.With function κ ( t ), the end-of-day penalty severities are identical to κ ( t ), but theintraday penalty severities are negligible (close to zero) before t = 0 .
75. Comparing resultsfor κ ( t ) vs. κ ( t ) illustrates how intraday penalties affect market conditions beyond anyfixed terminal penalty severities. Consistent with Corollary 2.4, when early intraday tradingpenalties are negligible, then the early-in-the-day market liquidity is high (i.e., σ w ( t ) isclose to zero) and the liquidity premium S t − D t is smaller with lower volatility relativeto an otherwise similar market with meaningful intraday penalties in addition to terminalpenalties. This illustrates that intraday TWAP penalties can have a material impact onintraday price dynamics.Figure 1 shows the welfare-maximizing function µ ∗ ( t ) (Plot A) and the Vayanos µ ( t )in (3.14) (Plot B) for the different penalty-severities in (6.5). Comparing µ ∗ ( t ) for penalties κ ( t ) vs. κ ( t ) and for κ ( t ) vs. κ ( t ) in Plot A, we note that the stronger the penaltyseverity κ ( t ), the larger is the welfare-maximizing µ ∗ ( t ) function. For example, the positiveslopes of κ ( t ) and κ ( t ) in (6.5) imply that the penalty severity is greater later in theday relative to κ ( t ). As a result, we see that the welfare-maximizer µ ∗ ( t ) gets larger laterin the day, the steeper the slope is of the penalty-severity function κ ( t ). This effect is Because γ (0) >
0, a welfare-maximizing equilibrium exists with µ ∗ ( t ) > κ ( t ) which explodes toward the end of the day. For κ ( t ), the welfare-maximizing µ ∗ ( t ) is close to zero for most of the day. This follows directly from the second-order condition (2.6) and the non-negativity of µ ∗ ( t ) from Theorem 5.1. Thus, µ ∗ ( t ) for κ ( t ) is greater than for κ ( t ) until the end of the day when they converge.The above suggests a reason for why the welfare-maximizing equilibrium can differfrom the competitive Radner equilibrium. A larger penalty severity κ ( t ) increases expectedTWAP deviation costs from accommodating the inelastic noise-trader order-flow w t . Con-sequently, equilibrium prices must adjust more to induce the strategic investors to clear themarket. Intuitively, this is what a larger welfare-maximizer µ ∗ ( t ) does via its endogenouseffect on prices in (2.9) through g ( t ) and σ w ( t ) in (2.10).Figure 1: The welfare-maximizer µ ( t ) := µ ∗ ( t ) (Plot A) and the Vayanos µ ( t ) defined in(3.14) (Plot B). The parameters are given by (6.2)-(6.5), and the discretization divides theday into 1000 trading rounds. - - μ * ( t ) - - μ ( t ) A: [Welfare] κ (———) , B: [Vayanos] κ (———) ,κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) . κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) . Our second topic is pricing. Figure 2 shows the price-loading function σ w ( t ) in (2.9).The sign of σ w ( t ) is negative because a larger w t means that the strategic investors mustbuy more (i.e., our sign convention is that w t is the amount noise traders supply). Thegreater the penalty severity κ ( t ) is, the more sensitive prices are to shocks in the amount w t the strategic investors must absorb from the noise traders. For example, a greater supply w t depresses prices more (in order to induce the strategic traders to buy), and the amountprices must be depressed is increasing in the penalty for strategic-investor deviations fromtheir intraday target trading trajectory. In contrast, when the intraday penalty severityis low, as with κ ( t ) during the day, the price impacts are smaller. We also note thatthe patterns in the price impacts are robust across the Radner, welfare-maximizing, andVayanos equilibria for each of the four κ ( t ) penalty severities in (6.5). This robustness is a28eneral finding of other properties of pricing (see the Internet Appendix), but we will seenext that investor holdings can differ qualitatively across the three equilibria.Figure 2: Equilibrium price loading σ w ( t ) on noise-trader supply in the maximizing-welfareequilibrium with µ ( t ) := µ ∗ ( t ) (Plot A), in the competitive Radner equilibrium with µ ( t ) := 0 (Plot B), and in the Vayanos equilibrium with µ ( t ) defined in (3.14) (PlotC). The parameters are given by (6.2)-(6.5), and the discretization divides the day into1000 trading rounds - - - - σ w ( t ) - - - - σ w ( t ) A: [Welfare] κ (———) , B: [Radner] κ (———) ,κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) . κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) . - - - - σ w ( t ) C: [Vayanos] κ (———) ,κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) . Our third topic is the strategic-investor holdings in equilibrium. From (2.5), their ag-gregate holdings are constrained by market clearing to equal the inelastic supply w t fromthe noise traders. However, there is heterogeneity in individual investors’ holdings givenimbalances in their initial holdings θ i, − and differences in their trading targets ˜ a i . Thisheterogeneity is scaled by the coefficient κ ( t )2 κ ( t ) − µ ( t ) in equation (2.19). These coefficientfunctions are independent of κ ( t ) in Figure 3 because Theorem 5.1 ensures that µ ∗ ( t ) is29inear in κ ( t ) in the welfare-maximizing case and by (3.14) in the Vayanos case. In thecompetitive Radner case the scaling coefficient is one because µ ( t ) = 0.Figure 3: Scaling ratio κ ( t )2 κ ( t ) − µ ( t ) for ˆ θ i,t in (2.19) for the welfare-maximizer µ ( t ) := µ ∗ ( t ),the competitive equilibrium with µ ( t ) := 0, and the Vayanos µ ( t ) defined in (3.14). Theparameters are given by (6.2)-(6.5), and the discretization divides the day into 1000 tradingrounds. κ ( t )/( κ ( t )- μ ( t )) Welfare (———) , Radner( − − − ) , Vayanos ( − · − · − ) . Next, we turn to investor i ’s expected trades. Combining (2.25) with the initial position θ i, − := w M from (6.2) gives E [ˆ θ i,t | σ (˜ a i , ˜ a Σ )] − θ i, − = αtM + 2 κ ( t )2 κ ( t ) − µ ( t ) γ ( t ) (cid:16) ˜ a i − ˜ a Σ M (cid:17) . (6.6)Consequently, from (2.22) with parameters in (6.1)-(6.5), investor i expects to deviate fromher target trajectory by E [ˆ θ i,t | σ (˜ a i , ˜ a Σ )] − (cid:104) θ i, − + γ ( t ) (cid:0) ˜ a i − θ i, − ) (cid:105) = αtM + 2 κ ( t ) γ ( t )2 κ ( t ) − µ ( t ) (cid:16) ˜ a i − ˜ a Σ M (cid:17) − γ ( t ) (cid:0) ˜ a i − w M ) . (6.7)In the competitive Radner equilibrium from Example 3.1 where µ ( t ) := 0, the difference(6.7) does not depend on the target ˜ a i when ˜ a Σ is fixed and also does not depend on theseverity κ ( t ) of the penalty. Remarkably, Theorem 5.1 ensures that the difference (6.7) alsoremains independent of κ ( t ) for the welfare-maximizer µ ∗ ( t ).Figure 4 shows the expected deviation between a strategic investor’s holdings up throughtime t ∈ [0 ,
1] and their corresponding target. In this figure, we change the target ˜ a i of aparticular individual investor i while holding the targets of the other M − a j := 1, j (cid:54) = i . Thus, both ˜ a i and ˜ a Σ = ˜ a i + 9 change in these plots. The figureshows that if investor i wants to hold a large target quantity (e.g., ˜ a i = 5 or 15), then inthe welfare-maximizing equilibrium she trades ahead of her target early in the day but theneventually falls behind. This pattern is noticeably different from the Radner and Vayanosequilibria.Figure 4: Conditional expected TWAP deviation E [ˆ θ i,t | σ (˜ a i , ˜ a Σ )] − (cid:0) θ i, − + γ ( t ) (cid:0) ˜ a i − θ i, − ) (cid:1) with the welfare-maximizer µ ( t ) := µ ∗ ( t ) (Plot A), the competitive equilibrium with µ ( t ) := 0 (Plot B), and the Vayanos µ ( t ) defined in (3.14) (Plot C). The parametersare given by (6.2)-(6.5), ˜ a j := 1 for j (cid:54) = i , ˜ a Σ := 9 + ˜ a i , and the discretization divides theday into 1000 trading rounds. - - - - - - [ θ i , t | σ ( a ˜ i , a ˜ Σ )]-( θ i , - + γ ( t )( a ˜ i - θ i , - )) - - - - - - [ θ i , t | σ ( a ˜ i , a ˜ Σ )]-( θ i , - + γ ( t )( a ˜ i - θ i , - )) A: [Welfare] ˜ a i := 0 (———) , B: [Radner] ˜ a i := 0 (———) , ˜ a i := 5 ( − − − ) , ˜ a i := 15 ( − · − · − ) . ˜ a i := 5 ( − − − ) , ˜ a i := 15 ( − · − · − ) . - - - - - - [ θ i , t | σ ( a ˜ i , a ˜ Σ )]-( θ i , - + γ ( t )( a ˜ i - θ i , - )) C: [Vayanos] κ (———) ,κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) . Lastly, we turn to welfare in the three equilibria. Table 1 illustrates how the equilibriaperform in terms of the welfare objective (5.9). As discussed in Section 5, the investorwelfare gains in the welfare-maximizing equilibrium are due to profits from initial price31ffects and to trading with the noise traders.Table 1: Expected welfare objective (5.9) for the welfare-maximizer µ ( t ) := µ ∗ ( t ) (Column2), the competitive equilibrium with µ ( t ) := 0 (Column 3), and the Vayanos µ ( t ) definedin (3.14) (Column 4). The parameters are given by (6.2)-(6.5) and D := 20. κ ( t ) Welfare (Ex. 3.2) Radner (Ex. 3.1) Vayanos (Ex. 3.3) κ ( t ) = 1 196.052 196.008 195.863 κ ( t ) = 1 + t κ ( t ) = (1 + t ) − . κ ( t ) from (6.5) 196.340 196.316 196.205Table 2 uses the decompositions in (5.1) and the second line of (5.3) to break thetotal strategic-investor welfare into contributions due the initial stock valuation effect, theexpected profits from trading with the noise investors, and the TWAP penalties. For brevity,the results reported here are just shown for the welfare-maximization case. The initialwealth component is large, which is not surprising given that the strategic investors are long10 million shares of a stock paying expected future dividends of $20. More interestingly,for these κ ( t ) penalty severities, the TWAP penalties are roughly a third to a half of theexpected intraday trading profits.Table 2: Components of the expected welfare objective (5.9) for the welfare-maximizingequilibrium. The parameters are given by (6.2)-(6.5) and D := 20. κ ( t ) Welfare (5.9) S w (cid:82) E [ w t ˆ µ t ] dt (cid:80) Mi =1 E [ L i, ] κ ( t ) = 1 196.052 192.157 7.383 3.494 κ ( t ) = 1 + t κ ( t ) = (1 + t ) − . κ ( t ) from (6.5) 196.340 193.992 5.548 3.167 This section extends our model to include stochastic intraday target trajectories and thenrelates this extension to a stylized version of VWAP benchmarking. For simplicity of expo-sition, in this section we set θ i, − := w := 0 . (7.1)The following model is based on Frei and Westray (2015). Appendix B contains moredetails about this extension including the analogue of Theorem 2.2 and its proof. In this32xtension, the deterministic TWAP target ratio γ ( t ) is replaced with a stochastic targetratio γ = ( γ t ) t ∈ [0 , , which is a c`adl`ag gamma bridge process that takes values in [0,1]. Theprocess γ t starts at 0 at t = 0, and converges to 1 as t ↑
1. See ´Emery and Yor (2004) formore details about gamma bridge processes. We assume the strategic investors all use anidentical target-ratio process. Thus, in this stochastic target extension, the penalty (1.1) isreplaced with L i,t := (cid:90) t κ ( s ) (cid:16) ˜ a i γ s − − θ i,s (cid:17) ds, t ∈ [0 , . (7.2)Because γ s = γ s − for all but a countable subset of [0 , γ s − with γ s in (7.2)without changing the resulting Lebesgue integral.In this model, investor i ’s filtration (1.7) is redefined as F i,t := σ (˜ a Σ , ˜ a i , w u , D u , γ u ) u ∈ [0 ,t ] , t ∈ [0 , , (7.3)where the processes ( w, D ) are as before. Consequently, the strategic investor’s objective(1.13) becomes sup θ i ∈A i E (cid:20) X − (cid:90) κ ( t ) (cid:16) ˜ a i γ t − − θ i,t (cid:17) dt (cid:12)(cid:12)(cid:12) F i, (cid:21) . (7.4)Because of γ t ’s inaccessible jumps, the controls in A i must be predictable ( A i from Definition1.1 is redefined in Appendix B).Any gamma process (and therefore any gamma bridge process) is necessarily indepen-dent of the Brownian motions ( D, B ); see, e.g., Lemma 15.6 of Kallenberg (2002). Therefore,to allow for dependency between the stock supply and the target ratio γ t , we redefine thenoise-trader stock supply to be w t + ρ ˜ a Σ γ t − , which now consists of two parts: The Ornstein-Uhlenbeck process w t in (1.3) plus a scaled component driven by the gamma bridge γ t − .Thus, the clearing condition (1.2) becomes w t + ρ ˜ a Σ γ t − = M (cid:88) i =1 θ i,t , t ∈ [0 , , (7.5)for some constant ρ ∈ R . The constant ρ controls the dependence between the targetratio and the stock supply (see Eq. (B.3) in Appendix B for the precise statement).The economic interpretation of our stochastic target ratio process is that γ t is an ob-servable state variable used to benchmark order execution. For example, it might be the Multiplying ρ by ˜ a Σ in (7.5) produces simpler expressions in what follows and, when ˜ a Σ (cid:54) = 0, can bedone with no loss of generality. E [ vol t vol |F i,t ] given a suitable definition of publicinformation for the broad market (i.e., not just for the single particular stock being tradedhere but for all stocks). In this sense, the stochastic target model represents a type of gener-alized VWAP benchmarking. However, if we want VWAP for one particular stock, this canbe done but then there are mathematical challenges. In particular, because Brownian mo-tions have infinite first variation, volume is non-trivial to define. Thus, for a stock-specificVWAP, we would need η = 0. In this case, γ t is exactly the time t expectation of the dailyvolume curve. Because γ t is a gamma bridge, there is randomness in the total supply duringthe day but the total supply at the end of the day is non-random since γ = 1. Thus, theextension of our model to single-stock VWAP is an interesting topic for future research.Frei and Westray (2015) model the realized relative volume curve vol t vol used for VWAPbenchmarking by γ t . However, as discussed on page 617 in Frei and Westray (2015), thispresents a potential problem because the realized volume curve cannot be observed prior tothe end of the trading day. Later in this section, we show how the gamma bridge processcan be used to model the investors’ expected relative volume curve E [ vol t vol |F i,t ] which — bydefinition — is observed at time t .Appendix B uses the arguments behind Theorem 3.1 in Frei and Westray (2015) to showthat the value function corresponding to the optimization problem (7.4) remains quadraticwhen γ t is added as a state-process and (2.2)-(2.3) are replaced by dS t : = µ i,t dt + σ w ( t ) ηdB t + dD t + σ γ ( t )˜ a Σ (cid:0) dγ t − (1 − γ t − ) ψ ( t ) dt (cid:1) ,µ i,t : = µ ( t ) θ i,t + µ ( t )˜ a Σ γ t − + µ ( t ) w t + µ ( t )˜ a i γ t − . (7.6)Given a continuous function µ ( t ) satisfying (2.6), Appendix B gives formulas for smoothdeterministic functions σ w ( t ), σ γ ( t ) , µ ( t ) , µ ( t ), and µ ( t ) in terms of µ ( t ) which producea unique equilibrium in the sense that (i)-(iii) hold in Definition 2.1. One new feature ofthis extended model is that now there are two stochastic process, w t and γ t , affecting pricesand strategic-investor holdings.Theorem B.2 in Appendix B is the VWAP analogue of Theorem 2.2 in the TWAPmodel. In particular, the ODEs (B.11) determining the coefficient functions σ w ( t ) and σ γ ( t ) appearing in (7.6) show that σ w ( t ) is still given by (2.13) whereas σ γ ( t ) is given by σ γ ( t ) = 11 − t (cid:90) t (1 − u ) 2 κ ( u )(1 − ρ ) + µ ( u ) ρM du, t ∈ [0 , , (7.7)with limit σ γ ( t ) → t ↑
1. The following corollary of Theorem B.2 is the VWAP analogueof Corollary 2.3: Another complication is that defining volume in terms of the noise-trader supply omits crossed orders. orollary 7.1. In the setting of Theorem B.2, we have:(i) For ρ ≥ , the functions σ w ( t ) and σ γ ( t ) are increasing in µ ( t ) , decreasing in κ ( t ) ,and independent of γ ( t ) , α , and η .(ii) The dynamics of the predictable quadratic variation of the liquidity premium are d (cid:104) S − D (cid:105) t = σ w ( t ) η dt + σ γ ( t ) ˜ a (1 − γ t − ) ψ ( t ) dt, (7.8) where the deterministic function ψ ( t ) is defined in (B.2) . Empirical Predictions 4:
Price volatility and market liquidity have qualitatively similarrelations to the underlying model inputs as predicted in the TWAP model.We conclude the section by linking the stochastic target ratio model to VWAP bench-marking. To this end, we let vol = (vol t ) t ∈ [0 , be an exogenous stochastic process for thestock’s cumulative volume over time such that the relative cumulative volume processvol t vol , t ∈ [0 , , (7.9)is zero initially, has non-decreasing paths, and has terminal value one. We note that vol t vol isnot observable at time t <
1, so the ratio (7.9) cannot be used as a state-process.The VWAP objective replacing (1.13) issup θ i ∈A i E (cid:20) X − (cid:90) κ ( t ) (cid:16) vol t vol ˜ a i − θ i,t (cid:17) dt (cid:12)(cid:12)(cid:12) F i, (cid:21) . (7.10)While investor i ’s position θ i,t is adapted to the filtration F i,t , investor i cannot use herholdings θ i,t to manipulate the intraday contractual volume weights vol t vol (this is commonpractice, see, e.g., in Madhavan (2002, Exhibit 1)). In other words, we make the assumptionthat the strategic investors’ individual orders do not affect the volume ratio vol t vol . In thiscase, (7.10) can be replaced with the equivalent problem:sup θ i ∈A i E (cid:20) X − (cid:90) κ ( t ) (cid:16) E (cid:20) vol t vol (cid:12)(cid:12)(cid:12) F i,t (cid:21) ˜ a i − θ i,t (cid:17) dt (cid:12)(cid:12)(cid:12) F i, (cid:21) , (7.11)where F i,t is investor i ’s filtration. We model directly E (cid:104) vol t vol (cid:12)(cid:12)(cid:12) F i,t (cid:105) as the gamma bridge γ t . In that case, (7.11) becomes (7.4) when F i,t is defined by (7.3). Even though (7.10) and (7.11) yield different objective values, they are equivalent in the sense that theyshare the same maximizer. Alternatively, we can define vol t vol := (cid:0) γ t + ¯ γ t (cid:1) where γ t and ¯ γ t are independent gamma bridge processes Extension to exponential utilities
Appendix C extends our TWAP model with linear preferences for the strategic investorsas in (1.13) to exponential preferences. In other words, we replace U i ( x ) := x used inthe previous sections with U i ( x ) := − e − x/τ for a common risk tolerance parameter τ > κ ( t ) for divergences from the target tradingtrajectory — and general risk aversion to both wealth and trading risk captured by thecommon exponential risk tolerance parameter τ >
0. The analysis in Appendix C showsthat prices and stock holdings are again linear (but with modified ODEs) and that there isagain an infinite number of equilibria associated with different price-impact functions µ ( t ).See Theorem C.2 in Appendix C below for details. There are good reasons to think that the qualitative properties of our analysis are robusteven though the specific functional forms of prices and trading strategies depend on modelingassumptions (e.g., Brownian motion dynamics and linear or exponential preferences). First,flexibility in investor perceptions about how prices would respond to off-equilibrium ordersis likely a key factor leading to multiple equilibria. Second, the absence of manipulativepredatory trading is still likely with rational forward-looking liquidity provision. Third,intraday liquidity is likely to be impaired by intraday trading target penalties relative tojust terminal end-of-day target penalties. Fourth, strategic investors with the ability totake advantage of less flexible investors may be better off in less competitive equilibria.Two aspects of the mathematical structure of our model play particularly importantroles in the tractability of our model. The first is that there are no random unknown jumpsin the noise-trader supply w at time t = 0 (i.e., the initial value w is a constant). Withthis assumption, the aggregate target ˜ a Σ is inferable at time 0 and, as a result, there is noneed for filtering over time to learn the strategic-investor aggregate target. The second isthat the strategic investors are homogeneous in their TWAP or VWAP target ratios γ ( t ) or γ t and penalty severities κ ( t ) and differ only in their individual realized targets ˜ a , ..., ˜ a M .This leads to symmetry in the trading strategy coefficients used by the strategic investors.Lastly, we comment on the numerical implementation. The model is characterized bylow dimensional state-processes that makes numerics fast to perform. Furthermore, the that are zero at t = 0 and one at t = 1. Then (7.3) produces E [ vol t vol |F i,t ] = (cid:0) γ t + t ) which — modulo adeterministic function of time — fits into our setting. In addition, based on the analysis in the previoussections, it is straightforward to modify our VWAP analysis to include VWAP target trajectories of the form (cid:0) γ ( t ) + γ t (cid:1) where γ t is a gamma bridge process and γ ( t ) is a deterministic function of time.
10 Conclusion
This paper has solved for continuous-time equilibria with endogenous liquidity provisionand intraday trading targets. We show how intraday target trajectories in trading induceintraday patterns in investor positions and in prices. There are also potential extensions ofour model. First, it would be interesting to extend the model to allow for heterogeneity inthe strategic investors’ γ ( t ) and κ ( t ) penalty functions. Second, perhaps the most pertinentextension would be to allow for randomness in w appearing in (1.3) in which case the initialequilibrium stock price S cannot fully reveal the aggregate target ˜ a Σ . Such an extensionwould naturally involve elements from filtering theory. Third, our gamma bridge analysiscan be extended to an integrated model of single-stock VWAP. A Proofs
We start with a technical lemma, which is used in the proof of Theorem 2.2. The argumentsused in lemma’s proof are standard and can be found in, e.g., Chapter 7 in Lipster andShiryeav (2001) as well as in the appendix of Cheridito, Filipovi´c, and Kimmel (2007). Weinclude the lemma for completeness.
Lemma A.1.
Let the functions γ, κ, µ , and σ w be as in Theorem 2.2. The strictly positivelocal martingale (hence, also a supermartingale) N = ( N t ) t ∈ [0 , defined by N t := e − (cid:82) t λ u dZ u − (cid:82) t λ u du , t ∈ [0 , , (A.1) is a martingale with respect to the (augmented) filtration F t := σ (˜ a Σ , D u , B u ) u ∈ [0 ,t ] where ˆ µ t is defined by (2.17) and λ t := ˆ µ t (cid:112) σ w ( t ) η + 1 , (A.2) dZ t := σ w ( t ) ηdB t + dD t (cid:112) σ w ( t ) η + 1 , Z := 0 . (A.3) Proof.
For ( t, x, a ) ∈ [0 , × R we start by defining the linear function H ( t, x, a ) := 2 κ ( t ) − µ ( t ) M x + 2 κ ( t ) (cid:0) γ ( t ) − (cid:1) M w − κ ( t ) γ ( t ) M a, (A.4)37nd note that ˆ µ t = H ( t, w t , ˜ a Σ ) from (2.17). We define the auxiliary process dv t := ( α − πv t ) dt + ηdB t − H ( t, v t , ˜ a Σ ) σ w ( t ) η + 1 σ w ( t ) η dt, v := w . (A.5)Inserting H from (A.4) into (A.5) produces the various drift-coefficient functions in dv t tobe constant coefficient: α − κ ( t ) (cid:0) γ ( t ) − (cid:1) M σ w ( t ) η σ w ( t ) η + 1 w ,v t coefficient: − π − κ ( t ) − µ ( t ) M σ w ( t ) η σ w ( t ) η + 1 , ˜ a Σ coefficient: 2 κ ( t ) γ ( t ) M σ w ( t ) η σ w ( t ) η + 1 . (A.6)Because these deterministic coefficient functions (A.6) are integrable (indeed, they aresquare integrable by (2.7)), the linear SDE (A.5) has a unique non-exploding strong so-lution v t for t ∈ [0 ,
1] which is a Gaussian Ornstein-Uhlenbeck process.For n ∈ N , we define the stopping times τ wn := inf { t > (cid:90) t H ( t, w s , ˜ a Σ ) ds ≥ n } ∧ , (A.7) τ vn := inf { t > (cid:90) t H ( s, v s , ˜ a Σ ) ds ≥ n } ∧ , (A.8)where w s appearing in (A.7) is defined in (1.3) and v s appearing in in (A.8) is defined in(A.5). Because (A.5) has a unique strong and non-exploding solution for t ∈ [0 , n →∞ P ( τ vn = 1) = 1 . (A.9)By Novikov’s condition, the processes ( N t ∧ τ wn ) t ∈ [0 , are martingales for each n ∈ N , andso we can define on F τ wn the P -equivalent probability measure Q ( n ) by the Radon-Nikodymderivative d Q ( n ) d P := N τ wn . (A.10)For each n ∈ N , Girsanov’s theorem produces the Q ( n ) Brownian motion dB ( n ) t := dB t + H ( t, w t , ˜ a Σ ) σ w ( t ) η + 1 σ w ( t ) ηdt, t ∈ [0 , τ wn ] . (A.11)38herefore, the Q ( n ) dynamics of dw t defined in (1.3) become dw t = ( α − πw t ) dt + ηdB ( n ) t − H ( t, w t , ˜ a Σ ) σ w ( t ) η + 1 σ w ( t ) η dt, t ∈ [0 , τ wn ] . (A.12)By comparing (A.5) with (A.12) and using strong uniqueness of (A.5) we see that thedistribution of ( v t ∧ τ vn ) t ∈ [0 , under P is identical to the distribution of ( w t ∧ τ wn ) t ∈ [0 , under Q ( n ) . Consequently, by using the definitions (A.7) and (A.8), we have Q ( n ) ( τ wn ≤ x ) = P ( τ vn ≤ x ) , x > . (A.13)Then we have E [ N ] = lim n →∞ E [ N τ wn =1 ]= lim n →∞ E [ N τ wn τ wn =1 ]= lim n →∞ Q ( n ) ( τ wn = 1)= lim n →∞ P ( τ vn = 1) = 1 . (A.14)The first equality in (A.14) follows from the (A.9) and the Dominated Convergence Theoremwhich is applicable because N τ wn =1 ≤ N and E [ N ] ≤
1. The third equality uses (A.10),the fourth equality uses (A.13), and the fifth and final equality uses (A.9). Consequently, N t defined in (A.1) is a positive supermartingale with constant expectation and is thereforealso a martingale. ♦ Proof of Theorem 2.2:
We conjecture (and verify) the following equilibrium price-driftfunctions in (2.3) defined in terms of a continuous function µ ( t ) satisfying (2.6): µ ( t ) := 4 κ ( t ) γ ( t ) (cid:0) µ ( t ) − κ ( t ) (cid:1) M (cid:0) κ ( t ) − µ ( t ) (cid:1) , (A.15) µ ( t ) := − κ ( t ) γ ( t ) µ ( t )2 κ ( t ) − µ ( t ) , (A.16) µ ( t ) := 2 (cid:0) κ ( t ) − µ ( t ) (cid:1) M , (A.17) µ ( t ) := 4 (cid:0) γ ( t ) − (cid:1) κ ( t ) (cid:0) κ ( t ) − µ ( t ) (cid:1) M (cid:0) κ ( t ) − µ ( t ) (cid:1) , (A.18) µ ( t ) := 2 (cid:0) γ ( t ) − (cid:1) κ ( t ) µ ( t )2 κ ( t ) − µ ( t ) . (A.19)We split the proof into two steps. 39 tep 1 (individual optimality): Given the price-impact function µ ( t ) and the con-jectured associated functions (A.15)-(A.19) for the price-drift relation (2.2), we derive theindividual investor’s value function V for the maximization problem (1.13) and the associ-ated optimal control process ˆ θ i,t . To this end, for a i , a Σ , X i , w ∈ R , t ∈ [0 , L i ≥ V ( t,X i , w, L i , a i , a Σ ):= X i − L i − (cid:16) β ( t ) + β ( t ) a i + β ( t ) a i a Σ + β ( t ) a + β ( t ) w + β ( t ) wa i + β ( t ) a Σ w + β ( t ) w + β ( t ) a i + β ( t ) a Σ (cid:17) , (A.20)where the deterministic coefficient functions ( β j ) j =0 are given by the ODEs β (cid:48) = − αβ − β η + ( γ − κ ( κ − µ )(4 w − M w θ i, − ) − M ( γ − θ i, − κµ M ( µ − κ ) ,β (cid:48) = − γ κµ ( µ − κ ) ,β (cid:48) = 8 γ κ ( µ − κ ) M ( µ − κ ) ,β (cid:48) = 4 γ κ ( κ − µ ) M ( µ − κ ) ,β (cid:48) = κ − µ M + 2 β π, (A.21) β (cid:48) = 4 γκ ( κ − µ ) M (2 κ − µ ) + β π,β (cid:48) = 4 γκ ( µ − κ ) M (2 κ − µ ) + β π,β (cid:48) = − αβ + β π + 4( γ − w − M θ i, − ) κ ( κ − µ ) M (2 κ − µ ) ,β (cid:48) = − αβ + 2( γ − γκ (4 w κ ( κ − µ ) + M θ i, − µ ) M ( µ − κ ) ,β (cid:48) = − αβ + 8( γ − γκ ( M θ i, − − w )( κ − µ ) M ( µ − κ ) , together with the terminal conditions β j (1) = 0 for j ∈ { , ..., } . Even though some of the β ( t ) functions depend on investor i ’s initial holdings θ i, − , this dependence does not affectthe following arguments and so is suppressed in the notation that follows. We start byshowing that V defined in (A.20) is investor i ’s value function. The terminal conditions for40he ODEs describing ( β j ) j =0 produce the terminal condition V (1 , X i , w, L i , a i , a Σ ) = X i − L i , a i , a Σ , X i , w ∈ R , L i ≥ . (A.22)For an arbitrary strategy θ i ∈ A i , Itˆo’s lemma produces the dynamics dV = (cid:16) V t + 12 V ww η + 12 V XX θ i,t ( σ w η + 1) + V Xw θ i,t σ w η + V w ( α − πw t ) + V X θ i,t µ i,t + V L κ ( t ) (cid:0) γ ( t )(˜ a i − θ i, − ) − ( θ i,t − θ i, − ) (cid:1) (cid:17) dt + V X θ i,t ( σ w ηdB t + dD t ) + V w ηdB t ≤ V X θ i,t ( σ w ηdB t + dD t ) + V w ηdB t . (A.23)The inequality in (A.23) comes from the HJB-equation0 = sup θ i,t ∈ R (cid:16) V t + 12 V ww η + 12 V XX θ i,t ( σ w η + 1) + V Xw θ i,t σ w η + V w ( α − πw t ) + V X θ i,t µ i,t + V L κ ( t ) (cid:0) γ ( t )(˜ a i − θ i, − ) − ( θ i,t − θ i, − ) (cid:1) (cid:17) , (A.24)which the function V defined in (A.20) satisfies given the ODEs for β j , j ∈ { , ... } , in(A.21). In integral form, (A.23) reads V (1 , X i, , w , L i, , ˜ a i , ˜ a Σ ) − V (0 , X i, , w , L i, , ˜ a i , ˜ a Σ ) ≤ (cid:90) (cid:16) V X θ i,t ( σ w ηdB t + dD t ) + V w ηdB t (cid:17) . (A.25)To see that the Brownian integral (which is always a local martingale) on the right-hand-side in (A.25) is indeed a martingale, we first compute the two partial derivatives in (A.25)using the definition of V in (A.20): V X = 1 , V w = − (cid:0) β w + ˜ a i β + ˜ a Σ β + β (cid:1) . (A.26)Because the coefficient functions β j are bounded, the integrability condition (1.12) in thedefinition of the admissible set A i (see Definition 1.1) ensures the stochastic integral on theright-hand-side in (A.25) is a martingale. Consequently, the terminal condition (A.22) andthe inequality in (A.25) produce E [ X i, − L i, ] = E [ V (1 , X i, , w , L i, , ˜ a i , ˜ a Σ )] ≤ V (0 , X i, , w , L i, , ˜ a i , ˜ a Σ ) . (A.27)Because the right-hand side in (A.27) does not depend on θ i,t ∈ A i (from V ’s definition41A.20)), we have sup θ i ∈A i E [ X i, − L i, ] ≤ V (0 , X i, , w , L i, , ˜ a i , ˜ a Σ ) . (A.28)From (A.28) we see that V is an upper bound for the maximization problem (1.13). To getequality in (A.28), we show that ˆ θ i,t defined in (2.8) is optimal. To this end, we re-write(2.8) as ˆ θ i,t = G ( t )˜ a Σ + G ( t ) w t + G ( t )˜ a i + G ( t ) θ i, − + G ( t ) w , (A.29)where, from (2.8), we have defined the deterministic functions G ( t ) := − κ ( t ) γ ( t )2 κ ( t ) − µ ( t ) 1 M ,G ( t ) := 1 M ,G ( t ) := 2 κ ( t ) γ ( t )2 κ ( t ) − µ ( t ) ,G ( t ) := 2 κ ( t ) (cid:0) − γ ( t ) (cid:1) κ ( t ) − µ ( t ) ,G ( t ) := − κ ( t ) (cid:0) − γ ( t ) (cid:1) κ ( t ) − µ ( t ) 1 M . (A.30)The coefficient in front of θ i,t in (A.24) equals µ ( t ) − κ ( t ) . (A.31)Therefore, the second-order condition (2.6) comes from requiring negativity of (A.31). Be-cause µ ( t ) is assumed to satisfy (2.6), we see that ˆ θ i,t defined in (2.8) belongs to theadmissible set A i as defined in Definition (1.1). Furthermore, ˆ θ i,t produces equality in(A.24) and (A.25). Therefore, the upper bound (A.28) ensures that ˆ θ i,t is optimal. Step 2 (equilibrium):
This step of the proof establishes the equilibrium properties inDefinition 2.1. We start by using (A.29) and (1.5) to rewrite the clearing condition (1.2) as w t = M (cid:88) i =1 ˆ θ i,t = M G ( t )˜ a Σ + M G ( t ) w t + G ( t )˜ a Σ + G ( t ) w + M G ( t ) w . (A.32)This gives us the following three restrictions for the w t -coefficients, the ˜ a Σ -coefficients, and42he constants: 1 = M G ( t ) , M G ( t ) + G ( t ) , G ( t ) + M G ( t ) (A.33)which the functions in (A.30) satisfy. Next, to ensure that the last restriction (iii) inDefinition 2.1 holds, we substitute (A.29) into (2.3) to getˆ µ t := µ ( t )˜ a Σ + µ ( t )ˆ θ i,t + µ ( t )˜ a i + µ ( t ) w t + µ ( t ) w + µ θ i, − = µ ( t )˜ a Σ + µ ( t ) (cid:16) G ( t )˜ a Σ + G ( t ) w t + G ( t )˜ a i + G ( t ) θ i, − + G ( t ) w (cid:17) + µ ( t )˜ a i + µ ( t ) w t + µ ( t ) w + µ θ i, − . (A.34)The requirement in (iii) that the ˜ a i and θ i, − coefficients in ˆ µ t are zero in equilibrium canbe stated as 0 = µ ( t ) G ( t ) + µ ( t ) , µ ( t ) G ( t ) + µ ( t ) . (A.35)The formulas for µ ( t ), µ ( t ) , ..., µ ( t ) in (A.15)-(A.19) ensure that the two requirements in(A.35) hold. In particular, inserting (A.30) into ˆ µ t in (A.34) gives the equilibrium price-driftin (2.17). Because the private information variables (˜ a i , θ i, − , θ (0) i, − ) do not appear in (2.17),the last requirement (iii) in Definition 2.1 holds.Finally, we need to establish the terminal price condition (1.4). To this end, weneed the ODEs for ( g , g, σ w ) in (2.10). We define the (augmented) filtration by F t := σ (˜ a Σ , w u , D u ) u ∈ [0 ,t ] for t ∈ [0 , P -equivalentprobability measure Q can be defined on F by the Radon-Nikodym derivative d Q d P := e − (cid:82) λ u dZ u − (cid:82) λ u du , (A.36)where ( λ, Z ) are defined by (A.2) and (A.3) given ˆ µ t in (2.17). Girsanov’s theoremproduces the Q -Brownian motions dD Q t := dD t + ˆ µ t σ w ( t ) η + 1 dt, (A.37) dB Q t := dB t + ˆ µ t σ w ( t ) η + 1 σ w ( t ) ηdt. (A.38) Because our model has more sources of randomness than stocks, our model is necessarily incomplete.Consequently, there are infinitely many P -equivalent probability measures under which the equilibrium stockprice process is a martingale. The minimal measure Q defined in (A.36) is sufficient for our purpose and itsprominent history is detailed in F¨ollmer and Schweizer (2010). Q -dynamics of ( D, w ) then become dD t = dD Q t − ˆ µ t σ w ( t ) η + 1 dt, (A.39) dw t = ( α − πw t ) dt + ηdB Q t − ˆ µ t σ w ( t ) η + 1 σ w ( t ) η dt. (A.40)These dynamics (A.39)-(A.40) ensure that the pair ( D, w ) remains a Markov process under Q . We will now show the identity: E Q [ D + ϕ ˜ a Σ + ϕ w |F t ] = g ( t ) + g ( t )˜ a Σ + σ w ( t ) w t + D t , t ∈ [0 , . (A.41)To see this, note that the terminal conditions for the ODEs in (2.10) ensure that (A.41)holds at t = 1. Furthermore, the conditional expectation on the left-hand-side of (A.41) isa martingale under the minimal martingale measure Q . Therefore, to see that (A.41) alsoholds for t ∈ [0 , Q . To this end, we apply Ito’s lemma to the right-hand-side of (A.41) to producethe P -dynamics d (cid:0) g ( t ) + g ( t )˜ a Σ + σ w ( t ) w t + D t (cid:1) = (cid:16) g (cid:48) ( t ) + g (cid:48) ( t )˜ a Σ + σ (cid:48) w ( t ) w t (cid:17) dt + dD t + σ w ( t ) dw t . (A.42)The risk-neutral drift (i.e., the drift under the minimal martingale measure Q ) is g (cid:48) ( t ) + g (cid:48) ( t )˜ a Σ + σ (cid:48) w ( t ) w t − ˆ µ t σ w ( t ) η + 1+ (cid:16) ( α − πw t ) − ˆ µ t σ w ( t ) η + 1 σ w ( t ) η (cid:17) σ w ( t )= g (cid:48) ( t ) + g (cid:48) ( t )˜ a Σ + σ (cid:48) w ( t ) w t + ( α − πw t ) σ w ( t ) − ˆ µ t = g (cid:48) ( t ) + g (cid:48) ( t )˜ a Σ + σ (cid:48) w ( t ) w t + ( α − πw t ) σ w ( t ) − (cid:16) µ ( t )˜ a Σ + µ ( t )ˆ θ i,t + µ ( t )˜ a i + µ ( t ) w t + µ ( t ) w + µ θ i, − (cid:17) = 0 , (A.43)where the last equality follows from inserting ˆ θ i,t from (2.8) and using the ODEs in (2.10).This shows that the right-hand-side of (A.41) is given by g (0) + g (0)˜ a Σ + σ w (0) w + D Q t + η (cid:90) t σ w ( u ) dB Q u , t ∈ [0 , . (A.44)Because the function σ w ( t ) is bounded, the process (A.44) is a Q -martingale.44 Remark
A.1 . We make the following notes about the above proof:1. The proof is that of a “backward engineer’s” in that it guesses a solution and then veri-fies that the equilibrium conditions are met. Instead of conjecturing (A.15)-(A.19), wecould alternatively let µ j ( t ), j ∈ { , , , , } , be arbitrary continuous functions andadjust (A.30) appropriately. Then (A.33) and (A.35) would produce five restrictionswhich would in turn produce (A.15)-(A.19).2. Because the investors’ utilities are risk-neutral, Step 1 can be greatly simplified bynoticing that for θ i ∈ A i we have the representation E [ X i, − L i, |F i, ]= X i, + E (cid:20)(cid:90) (cid:16) θ i,s µ i,s − κ ( s ) (cid:16) γ ( s )(˜ a i − θ i, − ) − ( θ i,s − θ i, − ) (cid:17) (cid:17) ds (cid:12)(cid:12)(cid:12) F i, (cid:21) . By inserting µ i,s from (2.3), the integrand in the ds -integral becomes quadratic in θ i,s .Consequently, we can optimize pointwise over θ i,s to produce (2.8). The reason wegive a proof using the HJB-equation is because we can re-use it in the exponentialutility case considered in Appendix C. Proof of Theorem 5.1: (i): We will write “ ... ” for terms that do not depend on µ . We firstneed (recalling that the money market account is in zero net supply) M (cid:88) i =1 X i, = S M (cid:88) i =1 θ i, = (cid:16) g (0) + g (0)˜ a Σ + σ w (0) w + D (cid:17) w = ... + αw (cid:90) u µ − κM du + w (cid:90) µ − κM du, (A.45)45here the second equality follows from (cid:80) Mi =1 θ i, = w . Then we have M (cid:88) i =1 CE i = M (cid:88) i =1 X i, − (cid:16) M β (0) + β (0) M (cid:88) i =1 ˜ a i + (cid:0) β (0) + M β (0) (cid:1) ˜ a + M β (0) w + (cid:0) β (0) + M β (0) (cid:1) w ˜ a Σ + M β (0) w + ( β (0) + M β (0))˜ a Σ (cid:17) = ... + (cid:90) (cid:110) − u (cid:0) η + α u + αw (cid:1) µ M − γ κµ ( µ − κ ) M (cid:88) i =1 ˜ a i − γ κ ( κ − µ ) M ( µ − κ ) ˜ a (cid:111) du. (A.46)We define the constants c := E [˜ a ] − M M (cid:88) i =1 E [˜ a i ] , c := t ( η + α t + αw ) , (A.47)in which case the two conditions in (5.8) become4 γ ( t ) c < c < , t ∈ (0 , . (A.48)Based on the above, we seek to maximize − c µ M − γ κµ (2 κ − µ ) M (cid:88) i =1 E [˜ a i ] − γ κ ( κ − µ ) M (2 κ − µ ) E [˜ a ] . (A.49)By changing variables to y := κγ k − µ so that µ = y − γ ) κy , the maximization problembecomesmax y ∈ (0 , γ ) G ( y ) , where G ( y ) := ( c y − c )( y − γ ) − c γM y κ − γ κ M (cid:88) i =1 E [˜ a i ] . (A.50)The inequalities in (A.48) produce G (cid:48)(cid:48) ( y ) = 2 κM y (cid:0) c y + 2 c γ ) < , y ∈ (0 , γ ) . (A.51)Therefore, the first-order condition is sufficient. We observe that G (cid:48) ( y ) = 2 κM y (cid:0) c y − γ ( c y + c ) (cid:1) . (A.52)46hen, (A.48) produces G (cid:48) ( γ ) = − κc M γ > , (A.53) G (cid:48) (2 γ ) = κ M γ (4 γ c − c < . (A.54)By the intermediate value theorem and the strict concavity of G , we conclude that theunique solution of G (cid:48) ( y ) = 0 satisfies γ < ˆ y < γ . This ˆ y corresponds to 0 < µ ∗ < κ .Finally, (A.52) says that ˆ y = κγ k − ˆ µ is the solution of c y − γ ( c y + c ) = 0 and here κ does not appear.(ii) and (iii): Because E [ w t ] = w + (2 w α + η ) t + α t , the differences (5.6) and (5.7)produce the following objective which is equivalent to (5.1): f ( µ ( t ) , t ) − f (0 , t ) − (cid:0) h ( µ ( t ) , t ) − h (0 , t ) (cid:1) = − µ ( t ) (cid:16) t α M + t ( w α + η ) M (cid:17) − µ ( t ) γ ( t ) κ ( t ) (cid:0) κ ( t ) − µ ( t ) (cid:1) (cid:16) M (cid:88) i =1 E [˜ a i ] − E [˜ a ] M (cid:17) . (A.55)The claims in (ii) and (iii) follow from (A.55). ♦ B VWAP equilibrium
This section relies heavily on the analysis in Frei and Westray (2015). The c`adl`ag process γ = ( γ t ) t ∈ [0 , is a gamma bridge process with γ = 0 and γ = 1 with probability one.The underlying gamma process is normalized to have both unit mean and unit variance.Corollary 1 in ´Emery and Yor (2004) ensures that γ t has predictable intensity (1 − γ t − ) ψ ( t )and that the quadratic variation process [ γ ] t has predictable intensity (1 − γ t − ) ψ ( t ). Inother words, the processes γ t − (cid:90) t (1 − γ s − ) ψ ( s ) ds, and [ γ ] t − (cid:90) t (1 − γ s − ) ψ ( s ) ds, t ∈ [0 , , (B.1)are martingales where the deterministic functions ψ ( t ) and ψ ( t ) are defined as ψ ( t ) := (cid:90) [0 , (1 − z ) − t dz, ψ ( t ) := (cid:90) [0 , z (1 − z ) − t dz, t ∈ [0 , . (B.2)The dynamics of the predictable cross quadratic variation process (also called thequadratic cross characteristics) between the stock supply w t + ρ ˜ a Σ γ t on the left-hand-side47f (7.5) and the target ratio γ t is given by d (cid:104) γ, w + ρ ˜ a Σ γ (cid:105) t = ρ ˜ a Σ d (cid:104) γ (cid:105) t = ρ ˜ a Σ (1 − γ t − ) ψ ( t ) dt. (B.3)This implies that ρ controls their dependency structure. In particular, ρ := 0 producesindependence between the stock supply and the target ratio.As in Frei and Westray (2015), we claim (and prove below) that the value functioncorresponding to the optimization problem (7.4) for price dynamics defined by (7.6) is V ( t, X i , w, L i , γ ) := X i − L i − J ( t, w, γ ) , X i , w ∈ R , L i ≥ , γ ∈ [0 , . (B.4)In (B.4), the function J ( t, w, γ ) is defined by J ( t, w, γ ) := (cid:16) β ( t ) + β ( t ) γ + β ( t ) γw + β ( t ) w + β ( t ) w + β ( t ) γ (cid:17) , (B.5)where the deterministic coefficient functions are given by the following linear ODEs β (cid:48) = − αβ − β η − β ψ − β ψ , β (1) = 0 ,β (cid:48) = 4˜ a κ ( κ − µ ) + 8˜ a i ˜ a Σ M κ ( µ − κ ) + M ( β ( µ − κ ) (2 ψ − ψ ) − ˜ a i κµ ) M ( µ − κ ) + ˜ a Σ ( κ − µ ) ρ (4˜ a i M κ + 2˜ a Σ κ ( ρ − − ˜ a Σ µ ρ ) M (2 κ − µ ) , β (1) = 0 , (B.6) β (cid:48) = 4˜ a Σ κ ( µ − κ ) + M (4˜ a i κ ( κ − µ ) + M β (2 κ − µ )( π + ψ ) M (2 κ − µ ) + 2˜ a Σ ( κ − µ ) ρM ,β (1) = 0 ,β (cid:48) = κ − µ M + 2 β π, β (1) = 0 ,β (cid:48) = β π − αβ − β ψ , β (1) = 0 ,β (cid:48) = β ψ + 2 β ( ψ − ψ ) − αβ , β (1) = 0 . The investor index i is suppressed in the following.We need to adjust the notion of admissibility given in Definition 1.1 to the current caseof noise generated by the Brownian motions ( B, D ) and the gamma bridge process γ t . Werecall that F i,t is defined in (7.3). Definition B.1. An F i,t predictable process θ i = ( θ i,t ) t ∈ [0 , is deemed admissible, and wewrite θ i,t ∈ A i , if E (cid:20)(cid:90) θ i,t (cid:0) − γ t − ) ψ ( t ) (cid:1) dt (cid:12)(cid:12)(cid:12) F i, (cid:21) < ∞ . (B.7)48 When µ ( t ) satisfies the second-order condition (2.6), we can define the deterministicpricing coefficients in (7.6) to be: µ ( t ) := 2 (cid:0) κ ( t ) − µ ( t ) (cid:1)(cid:0) κ ( t )( ρ − − µ ( t ) ρ (cid:1) M (cid:0) κ ( t ) − µ ( t ) (cid:1) ,µ ( t ) := 2 (cid:0) κ ( t ) − µ ( t ) (cid:1) M ,µ ( t ) := − κ ( t ) µ ( t )2 κ ( t ) − µ ( t ) . (B.8)The following result is the analogue of Theorem 2.2 for the gamma bridge process. Theorem B.2.
Let the parameter restrictions (7.1) hold and let κ : [0 , → (0 , ∞ ) , and µ : [0 , → R be continuous functions which satisfy the second-order condition (2.6) . Thenthe functions µ , µ , and µ defined in (B.8) form an equilibrium in which:(i) Investor optimal holdings in equilibrium are given by ˆ θ i,t := w t M + γ t − (cid:104) κ ( t )2 κ ( t ) − µ ( t ) (cid:16) ˜ a i − ˜ a Σ M (cid:17) + ρ ˜ a Σ M (cid:105) . (B.9) (ii) The equilibrium stock price is given by S t = g ( t ) + σ w ( t ) w t + D t + σ γ ( t )˜ a Σ γ t , (B.10) where the deterministic functions g , σ w and σ γ are the unique solutions of the follow-ing linear ODEs: g (cid:48) ( t ) = − ˜ a Σ σ γ ( t ) ψ ( t ) − ασ w ( t ) , g (1) = ϕ ˜ a Σ ,σ (cid:48) w ( t ) = 2 κ ( t ) − µ ( t ) M + πσ w ( t ) , σ w (1) = ϕ ,σ (cid:48) γ ( t ) = σ γ ( t ) ψ ( t ) + 2 κ ( t )( ρ − − µ ( t ) ρM , σ γ (1) = 0 . (B.11) Remark
B.1 . Before we give the proof, let us comment on the linear ODEs in (B.11).Because ψ ( t ) = − t explodes as t ↑
1, the zero terminal condition for σ γ produces theunique solution (7.7). Because both κ ( t ) and µ ( t ) are continuous for t ∈ [0 , σ γ ( t ) → t ↑
1. From (7.7), it follows that ψ ( t ) σ γ ( t ) is integrable over t ∈ [0 ,
1] which ensures that g ( t ) in (B.11) can be found by integration. Proof.
The proof of Theorem B.2 is similar to the proof of Theorem 2.2 and here we only49utline the needed changes. For an arbitrary strategy θ i,t ∈ A i , Itˆo’s lemma produces thedrift of V = V ( t, X i,t , w t , L i,t , γ t ) defined in (B.4) to be V t + (cid:90) [0 , (cid:0) J ( t, w t , γ t − + (1 − γ t − ) z ) − J ( t, w t , γ t − ) (cid:1) (1 − z ) − t z − dz + V w ( α − πw t ) + θ i,t µ i,t − κ ( t ) (cid:0) ˜ a i γ t − − θ i,t (cid:1) + 12 V ww η . (B.12)By using (B.5) and (B.2) we can write the dz -integral in the drift (B.12) as (cid:90) [0 , (cid:0) J ( t, w t , γ t − + (1 − γ t − ) z ) − J ( t, w t , γ t − ) (cid:1) (1 − z ) − t z − dz = − β ψ w t + (cid:0) ( β − β ) ψ + 2 β ψ (cid:1) γ t − + β (cid:0) ψ − ψ ) γ t − + β ψ w t γ t − − ψ β − ψ β . (B.13)Therefore, subject to µ ( t ) satisfying the second-order condition (2.6), we see that ˆ θ i,t defined in (B.9) maximizes the drift (B.12). To verify that ˆ θ i,t defined in (B.9) is admissiblein the sense of Definition B.1, we use γ t ∈ [0 ,
1] and (2.6) to get the bound | ˆ θ i,t | ≤ | w t | M + 2 (cid:12)(cid:12)(cid:12) ˜ a i − ˜ a Σ M (cid:12)(cid:12)(cid:12) + | ρ ˜ a Σ | M . (B.14)Based on this, the independence between w t and γ t as well as E [ γ t ] = t give us for t ∈ [0 , E (cid:104) ˆ θ i,t (cid:0) − γ t − ) ψ ( t ) (cid:1)(cid:12)(cid:12) F i, (cid:105) ≤ E (cid:20)(cid:16) | w t | M + 2 (cid:12)(cid:12)(cid:12) ˜ a i − ˜ a Σ M (cid:12)(cid:12)(cid:12) + | ρ ˜ a Σ | M (cid:17) (cid:12)(cid:12)(cid:12) F i, (cid:21) . (B.15)Here we also used that ψ ( t ) in (B.2) can be written as ψ ( t ) = − t for t ∈ [0 , t ∈ [0 ,
1] and the integrability property(B.7) follows by integration in time.Finally, we adjust the second part of the proof of Theorem 2.2 by changing the right-hand-side of (A.41) to E Q [ D + ϕ ˜ a Σ + ϕ w | σ (˜ a Σ , w u , γ u , D u ) u ∈ [0 ,t ] ]= g ( t ) + σ w ( t ) w t + D t + σ γ ( t )˜ a Σ γ t . (B.16)In (B.16), the probability measure Q is still defined by the Radon-Nikodym derivative50A.36). Itˆo’s product rule produces the dynamics dσ γ ( t ) γ t = σ (cid:48) γ ( t ) γ t − dt + σ γ ( t ) dγ t = (cid:0) σ (cid:48) γ ( t ) γ t − + σ γ ( t )(1 − γ t − ) ψ ( t ) (cid:1) dt + σ γ ( t ) (cid:0) dγ t − (1 − γ t − ) ψ ( t ) dt (cid:1) . (B.17)Based on this, we re-use (A.43) to see that the drift of the right-hand-side of (B.16) is g (cid:48) ( t ) + σ (cid:48) w ( t ) w t + ( α − πw t ) σ w ( t ) − ˆ µ t + ˜ a Σ σ (cid:48) γ ( t ) γ t − + ˜ a Σ σ γ ( t )(1 − γ t − ) ψ ( t ) , (B.18)where the analogue of ˆ µ t defined in (2.17) is given byˆ µ t : = 2 κ ( t ) − µ ( t ) M w t + 2 κ ( t )( ρ − − µ ( t ) ρM γ t − ˜ a Σ . (B.19)Consequently, by inserting (B.19) for ˆ µ t into (B.18), the ODEs in (B.11) follow from match-ing w t , ˜ a Σ γ t − , and the remaining terms. ♦ C Equilibrium with exponential utilities
This appendix extends our equilibrium analysis to strategic investors with exponential util-ities U i ( x ) := − e − x/τ with a common risk-tolerance parameter τ >
0. In other words, wereplace the risk-neutral objective (1.13) withinf θ i ∈A i E (cid:104) e − τ ( X i, − L i, ) (cid:12)(cid:12)(cid:12) F i, (cid:105) . (C.1)Here the processes ( L i,t , X i,t ) are still defined by (1.1) and (1.10); however, the admissible set A i needs to be altered (see Definition C.1 below). Unlike risk-neutral utilities, exponentialutilities produce coupled non-linear ODEs (see (C.4) and (C.5) below), which potentiallyexplode in finite time. While it is possible to work out the exponential utility model withoutthe parameter restrictions α := 0 , π := 0 , η := 1 , θ i, − := w M , (C.2)these restrictions greatly simplify the following presentation.We will consider continuous functions µ : [0 , → R which satisfy the following two51onditions. First, in the exponential case, the second-order condition (2.6) becomes µ ( t ) < σ w ( t ) τ + κ ( t ) , t ∈ [0 , . (C.3)Second, the following coupled Riccati ODEs β (cid:48) = 1 + σ w + 2 κτ − τ ( µ + 2 M β τ )2 M τ , β (1) = 0 , (C.4) σ (cid:48) w = 1 + σ w + 2 κτ − µ τ − M β σ w τM τ , σ w (1) = ϕ , (C.5)must have non-exploding solutions for t ∈ [0 , V ( t, X i ,w, L i , a i , a Σ ) := e − τ ( X i − L i )+ β ( t )+ β ( t ) a i + β ( t ) a i a Σ + β ( t ) a + β ( t ) w + β ( t ) wa i + β ( t ) a Σ w + β ( t ) a i . (C.6)We will show that V is the value function for the optimization problem (C.1) where, given(C.4) and (C.5), the deterministic coefficient functions β , ...β , β , β , and β are given bythe following linear ODEs β (cid:48) = − β − w ( γ − κM τ , β (1) = 0 ,β (cid:48) = 12 τ (1 + 2 τ κ − τ µ + σ w ) (cid:16) τ β γκσ w (1 + 2 τ κ − τ µ + σ w ) − τ β (cid:0) (1 + 2 τ κ − τ µ ) + (1 + 2 τ κ ) σ w (cid:1) − γ κ (cid:0) τ κ (1 + σ w ) + (1 − τ µ + σ w ) (cid:1)(cid:17) , β (1) = 0 ,β (cid:48) = − β β − (2 γκ + β σ w ) (1 + 2 τ κ − τ µ + σ w ) M (1 + 2 τ κ − τ µ + σ w ) , β (1) = 0 ,β (cid:48) = − β γκ + β σ w ) (1 + 2 τ κ − τ µ + σ w )2 M (1 + 2 τ κ − τ µ + σ w ) , β (1) = 0 , (C.7) β (cid:48) = − β β + (2 γκ + β σ w )(1 + 2 τ κ − τ µ + σ w ) τ M (1 + 2 τ κ − τ µ + σ w ) , β (1) = 0 ,β (cid:48) = − β β − (2 γκ + β σ w )(1 + 2 τ κ − τ µ + σ w ) τ M (1 + 2 τ κ − τ µ + σ w ) , β (1) = 0 ,β (cid:48) = 2 w ( γ − γκτ M , β (1) = 0 . Finally, we can adjust the notion of admissibility given in Definition 1.1 to the case ofexponential utilities. 52 efinition C.1.
A jointly measurable and F i,t adapted process ( θ i,t ) t ∈ [0 , is deemed to beadmissible, and we write θ i,t ∈ A i , if the local martingale (cid:90) t (cid:16) V X θ i,u (cid:0) σ w ( u ) dB u + dD u (cid:1) + V w dB u (cid:17) , t ∈ [0 , , (C.8)is well-defined and is a martingale. In (C.8), the terms V X and V w denote the partialderivatives of the function V defined in (C.6). ♦ The deterministic pricing coefficients (A.15)-(A.19) are replaced with µ ( t ) := − (2 γ ( t ) κ ( t ) + σ w ( t ) β ( t ))(1 + 2 τ ( κ ( t ) − µ ( t )) + σ w ( t ) M (cid:0) σ w ( t ) + 2 κ ( t ) τ − µ ( t ) τ (cid:1) − σ w β ,µ ( t ) := − µ ( t )(2 κ ( t ) γ ( t ) + β ( t ) σ w ( t )) τ σ w ( t ) + 2 κ ( t ) τ − µ ( t ) τ ,µ ( t ) := 1 + σ w ( t ) + 2( κ ( t ) − µ ( t ) − M β ( t ) σ w ( t )) τM τ , (C.9) µ ( t ) := 2 κ ( t )( γ ( t ) − M ,µ ( t ) := 0 . The analogue of Theorem 2.2 for the case of exponential utilities is:
Theorem C.2.
Let the parameter restrictions (C.2) hold and let γ : [0 , → [0 , ∞ ) be acontinuous function. Let κ : [0 , → (0 , ∞ ) and µ : [0 , → R be continuous and squareintegrable functions (i.e., (2.7) holds), satisfy the second-order condition (C.3) , and ensurethat the coupled Riccati ODEs (C.4) and (C.5) have well-defined non-explosive solutions on [0 , . Then the functions µ , µ , µ , µ , and µ defined in (C.9) together with σ w defined in (C.5) form an equilibrium in which:(i) Investor optimal holdings in equilibrium are given by ˆ θ i,t = w t M + (cid:0) κ ( t ) γ ( t ) + β ( t ) σ w ( t ) (cid:1) τ (cid:0) κ ( t ) − µ ( t ) (cid:1) τ + 1 + σ w ( t ) (cid:16) ˜ a i − ˜ a Σ M (cid:17) . (C.10) (ii) The equilibrium stock price is given by (2.9) where the deterministic functions g and g are the unique solutions of the following linear ODEs: g (cid:48) ( t ) = − γ ( t ) κ ( t ) + (cid:0) β ( t ) + M β ( t ) (cid:1) σ w ( t ) M , g (1) = ϕ ,g (cid:48) ( t ) = 2 w ( γ ( t ) − κ ( t ) M , g (1) = 0 . (C.11)53 roof. The proof of Theorem C.2 is similar to the proof of Theorem 2.2 and here we onlyoutline the two needed changes. First, to verify that (C.10) is admissible in the sense ofDefinition C.1, we re-write the local martingale dynamics (A.23) appearing in (C.8) as dV ( t, X i,t ,w t , L i,t , ˜ a i , ˜ a Σ ) = V ( t, X i,t , w t , L i,t , ˜ a i , ˜ a Σ ) (cid:0) J i,t dB t − τ ˆ θ i,t dD t (cid:1) . (C.12)In (C.12), the process ˆ θ i,t is defined in (C.10) and J i,t := (cid:0) β ( t ) w t + ˜ a i β ( t ) + ˜ a Σ β ( t ) (cid:1) − τ ˆ θ i,t σ w ( t ) . (C.13)Because the deterministic functions appearing in front of w t , ˜ a i , and ˜ a Σ in (C.10) and(C.13) are uniformly bounded, and because w t defined in (1.3) is Gaussian, the Dol´eans-Dade representation (C.12) combined with Corollary 3.5.16 in Karatzas and Shreve (1991)produces the wanted martingality of V .Second, we need to verify that the local martingale N = ( N t ) t ∈ [0 , in (A.1) is a martin-gale when ˆ µ t in (2.17) is replaced byˆ µ t := 1 + σ w ( t ) + (cid:0) κ ( t ) − µ ( t ) (cid:1) τ − M β ( t ) σ w ( t ) τM τ w t + 2 (cid:0) γ ( t ) − (cid:1) κ ( t ) M w − γ ( t ) κ ( t ) + (cid:0) β ( t ) + M β ( t ) (cid:1) σ w ( t ) M ˜ a Σ . (C.14)To this end, we note that w t remains a non-exploding Gaussian Ornstein-Uhlenbeck processunder the P -equivalent probability measures ( Q ( n ) ) n ∈ N defined in (A.10). Consequently, theproof of Lemma A.1 carries over to this exponential utility case where ˆ µ t is defined in (C.14). ♦ References [1] R. Almgren (2003):
Optimal execution with nonlinear impact functions and trading-enhanced risk , Applied Mathematical Finance , 10–18.[2] R. Almgren (2012): Optimal trading with stochastic liquidity and volatility , SIAM Jour-nal of Financial Mathematics , 163–181.[3] R. Almgren and N. Chriss (1999): Value under liquidation , Risk , 61–63.[4] R. Almgren and N. Chriss (2000): Optimal execution of portfolio transactions , Journalof Risk , 5–39.[5] K. Back (1992): Insider trading in continuous time , Review of Financial Studies ,387–409. 546] M. Baldauf, C. Frei, and J. Mollner (2018): Contracting for financial execution , workingpaper, University of Alberta.[7] Y. Barardehi and D. Bernhardt (2018):
Re-appraising intraday trading patterns: Whatyou didn’t know you didn’t know , working paper, Chapman University.[8] S. Berkowitz, D. Logue, and E. Noser (1988):
The total cost of transactions on theNYSE , Journal of Finance , 97–112.[9] B. Bouchard, M. Fukasawa, M. Herdegen, and J. Muhle-Karbe (2018): Equilibriumreturns with transaction costs , Finance and Stochastics , 569–601.[10] M. K. Brunnermeier, and L. H. Pedersen (2005): Predatory trading , Journal of Finance , 1825–1863.[11] B. I. Carlin, M. S. Lobo, and S. Viswanathan (2007): Episodic liquidity crises: Coop-erative and predatory trading , Journal of Finance , 2235–2274.[12] J. H. Choi and K. Larsen (2015): Taylor approximation of incomplete Radner equilib-rium models , Finance and Stochastics , 653–679.[13] J. H. Choi, K. Larsen, and D. Seppi (2018): Information and trading targets in adynamic market equilibrium , Journal of Financial Economics, to appear.[14] P. O. Christensen, K. Larsen, and C. Munk (2012):
Equilibrium in securities marketswith heterogeneous investors and unspanned income risk , Journal of Economic Theory , 1035–1063.[15] P. O. Christensen and K. Larsen (2014):
Incomplete continuous-time securities marketswith stochastic income volatility , Review of Asset Pricing Studies , 247–285.[16] I. Domowitz and H. Yegerman (2005): The cost of algorithmic trading: A first look atcomparative performance , In Brian R. Bruce (Ed.): Algorithmic Trading: Precision,Control, Execution (Institutional Investor London), 30–40.[17] S. Du and H. Zhu (2017):
What is the optimal trading frequency in financial markets?
Review of Economic Studies , 1606–1651.[18] D. Duffie (2001): Dynamical asset pricing theory , 3rd Ed., Princeton University Press.[19] M. ´Emery and M. Yor (2004):
A parallel between Brownian bridges and gamma bridges ,Publ. Res. Inst. Math. Sci. , 669–688.[20] Financial Insights (2006): Marching up the learning curve: The second buy side algo-rithmic trading survey, Financial Insights Special Report, December 2006.[21] H. F¨ollmer and M. Schweizer (2010): The minimal martingale measure , In Encyclope-dia of Quantitative Finance (R. Cont, ed.) 1200–1204. Hoboken, NJ: Wiley.5522] F. D. Foster and S. Viswanathan (1996):
Strategic trading when agents forecast theforecasts of others , Journal of Finance , 1437—1478.[23] C. Frei and N. Westray (2015): Optimal execution of a VWAP order: A stochasticcontrol approach , Mathematical Finance , 612–639.[24] N. Gˆarleanu and L. H. Pedersen (2016): Dynamic portfolio choice with frictions , Jour-nal of Economic Theory , 487–516.[25] M. B. Garman (1976):
Market microstructure , Journal of Financial Economics , 257–275.[26] J. Gatheral and A. Schied (2011): Optimal trade execution under geometric Brownianmotion in the Almgren and Chriss framework , International Journal of Theoreticaland Applied Finance , 353–368.[27] J. Gatheral and A. Schied (2013): Dynamical models for market impact and algorithmsfor optimal order execution , In: Handbook on Systemic Risk (Eds.: J.-P. Fouque andJ. Langsam), Cambridge University Press, 579–602.[28] S. J. Grossman and J. E. Stiglitz, (1980):
On the impossibility of informationallyefficient markets , American Economic Review , 393–408.[29] B. Hagstr¨omer and L. Nord´en (2013): The diversity of high-frequency traders , Journalof Financial Markets , 741–770.[30] B. Johnson (2010): Algorithmic trading and DMA , 4Myeloma Press, London.[31] O. Kallenberg (2002):
Foundations of modern probability , 2nd Ed., Springer.[32] I. Karatzas and S. E. Shreve (1991):
Brownian motion and stochastic calculus , 2ndEd., Springer.[33] I. Karatzas and S. E. Shreve (1998):
Methods of mathematical finance , Springer-Verlag,New York[34] A. Kyle (1985):
Continuous auctions and insider trading , Econometrica , 1315–1336.[35] K. Larsen and T. Sae-Sue (2016): Radner equilibrium in incomplete L´evy models , Math-ematics and Financial Economics , 321–337.[36] A. Madhavan (2002): VWAP strategies , Transaction Performance, Spring 2002, 32–38.[37] H. M. Markowitz (1952):
Portfolio selection , Journal of Finance , 77–91.[38] A. Menkveld (2013): High frequency trading and the new market makers , Journal ofFinancial Markets , 712–740 .[39] M. O’Hara (2015): High frequency market microstructure , Journal of Financial Eco-nomics , 257–270. 5640] A. F. Perold (1988):
The implementation shortfall: Paper versus reality , Journal ofPortfolio Management , 4–9.[41] S. Predoiu, G. Shaikhet, and S. Shreve (2011): Optimal execution in a general one-sidedlimit-order book , SIAM Journal of Financial Mathematics , 183–212.[42] H. R. Stoll (1978): The pricing of security dealer services: An empirical study ofNASDAQ stocks , Journal of Finance , 1153–1172.[43] U.S. Securities and Exchange Commission (2010): Concept release on equity marketstructure
Strategic trading and welfare in a dynamic market , Review ofEconomic Studies , 219–254.[45] B. Weller (2013): Intermediation chains and specialization by speed: Evidence fromcommodity futures markets.
Unpublished working paper. Duke University.[46] K. Weston (2018):
Existence of a Radner equilibrium in a model with transaction costs ,Mathematics and Financial Economics , 517–539.[47] H. Xing and G. ˇZitkovi´c (2018): A class of globally solvable Markovian quadratic BSDEsystems and applications , Annals of Probability , 491–550.[48] G. ˇZitkovi´c (2012): An example of a stochastic equilibrium with incomplete markets ,Finance and Stochastics , 177–206. 57 Internet Appendix
This appendix extends the numerical analysis of Section 6 to consider the impact of TWAPbenchmarking on the liquidity-premium S t − D t . From (2.9), the expected liquidity premiumis E [ S t − D t | σ (˜ a Σ )] = g ( t ) + g ( t )˜ a Σ + σ w ( t ) E [ w t ] , (D.1)which consists of three components. Consider g ( t ) first. From (2.11), g ( t ) depends linearlyon α and w . The first term in (2.11) implies that the liquidity premium increases when α < σ w ( t ) <
0. Thesecond term in (2.11) implies that, as long as the target ratio γ ( u ) is less than one duringthe day, the slope coefficient on w is positive. Therefore, equilibrium prices and, thus, theexpected liquidity premia, are increasing in the initial aggregate position w held by thestrategic investors. This is illustrated numerically in Plot A in Figure 5. Note that, sincethe slope coefficient on w in (2.11) does not depend on µ ( t ), the slope is the same in allthree equilibria.Next, consider the g ( t )˜ a Σ component of the expected liquidity premium in (D.1). TheODE for g ( t ) in (2.10) and a boundary condition g (1) ≥ g ( t ) is positive anddecreasing given ϕ ≥
0. Thus, the expected liquidity premium is increasing in the latentaggregate target imbalance ˜ a Σ for the strategic agents. This is intuitive since higher pricesdepress expected stock returns and, thus, suppress strategic-investor demand as requiredfor market clearing. This is illustrated numerically in Plot B in Figure 5. Once again, theplots are identical for all three equilibria, since g ( t ) from (2.10) is independent of µ ( t ).Lastly, the contribution from the third component of the expected liquidity premium(D.1) follows from the price-impact loading σ w ( t ) (as illustrated in Figure 2) and the ex-pected path of noise-trader order imbalances. For example, the expected liquidity premiumat time t is depressed by decreasing (given α <
0) expected supply of shares E [ w t ] fromnoise traders.As is intuitive, the expected premium is larger when the penalty severity κ ( t ) is greatersince the strategic investors require more compensation (i.e., larger price discounts for buy-ing and price premiums when selling) for deviating from their target trajectory. However,as the end of the day approaches, the terminal price constraint (1.6) forces the expectedliquidity premium to converge to 0 given ϕ = ϕ = 0. In particular, this is true even forthe exploding severity function κ ( t ).Figure 6 shows the volatility of the intraday liquidity premium induced by the randomnoise-trader imbalances. Initially, as expected, larger penalty severities κ ( t ) mean pricesneed to move more to compensate investors for deviating from their target trajectories,58igure 5: Expected liquidity premium slope coefficient on w in (2.11) (Plot A) and thefunction g ( t ) from (2.10) (Plot B). The parameters are given by (6.2)-(6.5), and the dis-cretization divides the day into 1000 trading rounds. w slope in g ( t ) ( t ) A: κ (———) , κ ( − − − ) , B: κ (———) , κ ( − − − ) ,κ ( − · − · − ) κ ( − · ·− ) . κ ( − · − · − ) κ ( − · ·− ) . which magnifies the effect of randomness in the noise-trader imbalance w t . The liquidity-premium variance initially increases due to the growing variance of w t , but eventually theterminal price condition (1.6) forces the liquidity-premium volatility to converge to zerogiven ϕ = ϕ = 0. 59igure 6: Standard deviation SD[ S t − D t | σ (˜ a Σ )] of the liquidity premium with the welfare-maximizer µ ( t ) := µ ∗ ( t ) (Plot A), the competitive equilibrium with µ ( t ) := 0 (PlotB), and the Vayanos µ ( t ) in (3.14) (Plot C). The parameters are given by (6.2)-(6.5),˜ a Σ = (cid:80) Mi =1 θ i, − = w = 10, and the discretization divides the day into 1000 trading rounds. [ S t - D t | σ ( a ˜ Σ )] [ S t - D t | σ ( a ˜ Σ )] A: [Welfare] κ (———) , B: [Radner] κ (———) ,κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) . κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) . [ S t - D t | σ ( a ˜ Σ )] C: [Vayanos] κ (———) ,κ ( − − − ) , κ ( − · − · − ) κ ( − · ·− ) ..