A Stationary Kyle Setup: Microfounding propagator models
Michele Vodret, Iacopo Mastromatteo, Bence Tóth, Michael Benzaquen
AA Stationary Kyle Setup: Microfounding propagator models
Michele Vodret , Iacopo Mastromatteo , Bence Tóth , and Michael Benzaquen Chair of Econophysics & Complex Systems, Ecole polytechnique, 91128 Palaiseau Cedex, France Ladhyx, UMR CNRS 7646, Ecole polytechnique, 91128 Palaiseau Cedex, France Capital Fund Management, 23-25, Rue de l’Université 75007 Paris, France
December 15, 2020
Abstract
We provide an economically sound micro-foundation to linear price impact models, by derivingthem as the equilibrium of a suitable agent-based system. In particular, we retrieve the so-called propagator model as the high-frequency limit of a generalized Kyle model, in which the assump-tion of a terminal time at which fundamental information is revealed is dropped. This allows todescribe a stationary market populated by asymmetrically informed rational agents. We investigatethe stationary equilibrium of the model, and show that the setup is compatible with universal pricediffusion at small times, and non-universal mean-reversion at time scales at which fluctuationsin fundamentals decay. Our model suggests that at high frequency one should observe a quasi-permanent impact component, driven by slow fluctuations of fundamentals, and a faster transientone, whose timescale should be set by the persistence of the order flow.
Keywords:
Market microstructure, price impact, statistical inference.
Contents [email protected] a r X i v : . [ q -f i n . T R ] D ec Conclusion 188 Acknowledgments 18A Numerical solver 21B Particular solutions of equilibrium condition in the Markovian case 21
B.1 The case of non-correlated Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.2 The case of Noise and Signal with equal autocovariance timescales . . . . . . . . . . . . . 23
C Solution of the Markovian case 24
C.1 Construction of the Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24C.2 Solving the ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Financial markets are designed to achieve two seemingly unrelated goals: they allow market partici-pants to find other agents with whom to transact (thereby solving a liquidity problem), and at the sametime they allow to discover the price at which such transactions should take place (thereby solving an information -related task).The Efficient Market Hypothesis (EMH) states that prices integrate all information that is publiclyavailable [ ] . If this is the case, there can be no forecastable structure in asset returns for agents inpossession of public information only. Historically, the EMH was first rationalized theoretically withthe introduction of the Rational Expectation Hypothesis (REH). According to the REH all agents arerational and perfectly informed about the other players’ strategies. This hypothesis is appealing sinceit allows to build analytically tractable setups [ ] in which financial markets are able to deliver thepromise they were conceived for, once some exogenous source of dynamics is injected into the system,thus preventing no-trade theorems. It has also important drawbacks: for example, the REH impliesthat the value of a risky asset is completely determined by its fundamental price, equal to the presentdiscounted value of the expected stream of future dividends. As already argued by Shiller [ ] , the excessvolatility puzzle, i.e., the fact that the price deviates substantially from the fundamental value, cannotbe explained by the REH. Nevertheless, the REH is still considered the main expectation formationparadigm in many economic circles [ ] .An important class of REH models is the so-called Information-Based Models. These models typi-cally involve the presence of agents that trade due to exogenous reasons ( noise traders ) that use finan-cial markets in order to find counterparties for satisfying needs that come from outside of the market,and arbitrageurs that possess privileged information on the traded goods ( informed traders ), and thuschoose to transact whenever they expect to use their informational advantage in their favor. Fromthis perspective, informed traders provide a service (making prices informative) that noise traders canchoose to pay in order to be granted access to liquidity. To lubricate this mechanism, dealers ( marketmakers ) are typically required: instead of letting noise traders and informed traders interact directly,market makers can temporarily incorporate the imbalance in the trading pressure, accepting to bearinventory risk for a limited time under the promise of some reward (bid-ask spread, rebate fees). Theiractivity allows to defer in time the moment at which the initial buyer and the final seller meet, thusenabling both informed and noise traders to find more easily possible counter-parties.When one tries to validate empirically how (or whether) this idealized mechanism takes place inreal markets, one is confronted with a very different picture: liquidity at the “efficient” prices tends tobe scarce [
5, 6 ] , so that both informed and noise traders are required to fragment their orders in longstreams of correlated trades in order to conceal their intentions. On the other hand, (statistical) priceefficiency is empirically supported to a large extent [
6, 7 ] , indicating that the information containedin the trade flows is quite effectively disentangled from its uninformed component through what isreferred to as price impact . The price to pay in order to have statistically efficient markets in presence2f vanishing liquidity is to introduce a non-trivial impact function, one that strongly reacts to smalltrades, potentially triggering market instabilities and flash crashes [
8, 9 ] . This is to be contrastedwith the more resilient view of the market that arise from classical Information-Based Models, thatprescribe an impact function linear in the size of the imbalance and constant along time [ ] . Linearimpact models are nevertheless a good starting point to investigate price impact.A particularly successful class of models to describe statistical regularities in financial markets in-volves the notion of propagator , a linear kernel used in autoregressive models that couples price changesto past order flow imbalances. In this setting, the (discounted) price of a good at time t , which we de-note p t , can be expressed as a function of the past signed order flow imbalance q t as: p t : = t (cid:88) t (cid:48) = −∞ G t − t (cid:48) q t (cid:48) , (1)where the causal kernel G t is the propagator. Propagator models were originally proposed in orderto solve the so-called diffusivity puzzle , namely the fact that price efficiency, and consequently pricediffusion, can be achieved even if the order flow imbalances q t display long-ranged correlation [ ] .Moreover, variations of these models have proven to be effective in order to paint an accurate pictureof the market at high frequency [
12, 13 ] , in the sense that a large fraction of the price fluctuations canbe explained by the past order flow [ ] .On the other hand, the perspective taken in order to construct such models is quite distinct fromthe one preferred in the literature of theoretical economics. The propagator setup is not properlymicrofounded. In fact, it builds on statistical stylized facts, rather than on an economic rationale. Thegoal of this paper is to bridge this gap in an economically standard setting by showing how propagator-like models can be rationalized as the equilibrium resulting from a set of rational agents seeking toachieve optimality. Along this line, our work is closely related to the classic Kyle setting [ ] , in whichthe price discovery mechanism emerges as a linear equilibrium between three representative agentswith asymmetric information.We establish a setting for an Information Based Model that gives rise to a stationary market, wherethe equilbrium pricing rule is given by Eq. (1). A similar setting has been considered in Ref. [ ] inthe special case of a stationary Markovian system. Here, instead, we keep the model general, so toaccount for memory effects (order flows are strongly correlated in real markets), thus extending someof the results of the aformentioned investigation. Our work goes beyond the purely theoretical aspect,since the framework we build allows to explicitly construct kernels G t that ensure price efficiency underdifferent circumstances.The organization of the paper is as follows. Section 2 introduces the notations we use throughoutthe paper. In Section 3 we present the model. Section 4 is devoted to the study of the equilibriumof the model. Section 5 discusses the relation of our model with its building blocks, namely the orig-inal propagator and the Kyle model. In Section 6 we further investigate the model we propose inthe paradigmatic Markovian case, whose tractable solution allows to gain intuition on the system. InSection 7 we conclude. Throughout the paper, we will alternate between scalar notations, in which the time dependence ofthe variables is explicit (e.g. X t ), vector notations, and matrix notations. We will use bold symbols forvectors and Sans Serif symbols for matrices.For convenience we introduce two types of vectors: XXX t : = { X t (cid:48) } tt (cid:48) = −∞ and XXX / t : = { X t (cid:48) } ∞ t (cid:48) = t . Further,for a given vector XXX t we define the associated Toeplitz matrix as X t , t : = { X t (cid:48) − t (cid:48)(cid:48) } tt (cid:48) , t (cid:48)(cid:48) = −∞ . In some Note that we are omitting from Eq. (1) a residual noise term, that can be easily restored in order to account for pricechanges that are not explained by the past order flow. X / t , / t = { X t (cid:48) − t (cid:48)(cid:48) } ∞ t (cid:48) , t (cid:48)(cid:48) = t or X / t , t = { X t (cid:48) − t (cid:48)(cid:48) } with t (cid:48) ≥ t and t (cid:48)(cid:48) ≤ t . The transpose operation will be denoted by the superscript (cid:62) .The identity matrix is denoted I , the vector with all components equal to one is written 111, the matrixwith all entries equal to 1 is denoted with U , the elements of the canonical basis are denoted with e t ,with a time subscript indicating the non-zero element and the lag operator, i.e., the operator that actson an element of a time series to produce the previous element, is denoted L . In this way we write XXX t − = L XXX t . Dimensionless quantities are signified with tildes. Consider a market in which agents exchange a risky asset (stock) against a safe asset (cash). The(discounted) transaction price of the risky asset at time t is denoted by p t . Each unit of the risky assetentitles its owner to a stochastic payoff µ t in cash (dividend) at each unit of time t . The dividendprocess µ t is modeled as an exogenous, stationary, zero-mean Gaussian process with autocovariancefunction (ACF): Ξ µτ : = (cid:69) [ µ t µ t + τ ] . (2)The portfolio of each agent comprises a combination of risky and safe assets. The position of agent i in the risky asset at time t is given by Q it , whereas his trades are denoted by q it : = Q it − Q it − . Withthese conventions, the equations for the evolution of cash C it , stock-position Q it , and wealth W it foreach agent can be written down respectively as: ∆ C it : = µ t Q it − p t q it (3) ∆ Q it : = q it (4) ∆ W it : = ∆ C it + Q it p t − Q it − p t − . (5)We consider an agent-based market model with asymmetric information akin to the well knownKyle model [ ] , in which the agents take actions at discrete time steps t . A strategic agent possessingprivileged information about the realizations of the stochastic dividend process ( informed trader , or IT)trades with a non-strategic and non-informed trader ( noise trader , or NT) that accesses the market forexogenous reasons. Both the IT and NT are modeled as liquidity takers. A liquidity provider ( marketmaker , or MM) provides liquidity for both the NT and the IT and sets the transaction price p t .At the beginning of each time interval [ t , t + ] both the IT and the NT build a demand for the riskystock q it (with i ∈ { IT,NT } ). The IT builds his demand without exploiting equal-time information oneither p t nor on the decision of their peer. In order to maximize his wealth, the IT exploits privilegedinformation on realized dividends. After the excess demand q t : = q IT t + q NT t is formed, the MM clearsthe excess demand of the liquidity takers, executing a trade q MM t : = − q t and setting the transactionprice p t . The price p t arises endogenously as the result of the action of the agents, described in whatfollows.Before discussing the strategies of the different agents, let us highlight that both the IT and the MMknow the statistical properties of the exogenous processes µ t and q NT t , as well as each other’s strategy,and that realized prices and excess demands are public information. Noise trader
The NT acts in a purely stochastic fashion. His demand process q NT t is a zero-mean,stationary Gaussian process with ACF given by: Ω NT τ : = (cid:69) [ q NT t q NT t + τ ] . (6)4 nformed trader The IT is a strategic, risk-neutral (expected) utility maximizer. His access to privi-leged information about the dividend process is modeled by assuming that he observes past realizationsof the process µ t and uses such information to maximize his future expected wealth. Moreover, sincerealized excess demand is public information, the IT can trivially infer the NT’s past trades. The infor-mation accessible to the IT at time t is thus given by: (cid:73) IT t = (cid:8) q t − , q NT t − , µµµ t − (cid:9) . (7)Since the IT is risk-neutral and assuming that the price is a linear function of realized excess de-mands (we shall discuss why this is the case in a moment), his demand q IT t at time t is a linear functionof his current information set (cid:73) IT t : q IT t = R q t − + R NT q NT t − + R µ µµµ t − , (8)where we have introduced the demand kernels ( R , R NT , R µ ). Let us give here a first description of thesedemand kernels. Since we discuss a market with multiple trading periods, the IT strategically takesinto account past trades and past dividends in order to determine his demand. The demand kernel R accounts for the dependence on past order flow which arises from price impact of past traded volumes.The kernel R NT accounts for the dependence that comes from the price impact induced by expectedfuture trades of the NT, while the kernel R µ accounts for the dependence arising from expected futuredividends. The demand kernels are the result of a Model Predictive Control (MPC) [ ] strategy.Indeed, as soon as a new piece of information is available to the IT (i.e. at each time-step t ), he willconstruct an updated long-term strategy, and he will trade accordingly. More details about the IT’s MPCstrategy are provided in Sec. 3.3, with explicit expressions of the demand kernels. Market maker
The MM is risk-neutral and competitive. He sets a pricing rule that allows him tostatistically break even on every trade, without controlling the inventory that he might accumulatewhile matching the demand. The realization of the dividend process µ t is unknown to the MM, andso is the proportion of the demand due respectively to the IT and the NT. Thus, the information setavailable to the MM at time t is solely given by realized aggregate excess demand: (cid:73) MM t : = { q t } . (9)An important point is that the resulting excess demand q t conveys information to the MM about theasset’s fundamental value, via the information set used by the IT (Eq. (7)) to construct his tradingschedule (Eq. (8)). Note also that the information set of the MM is not contained in the informationset of the IT, due to the fact that the excess demand q t is only available to the IT at time t + t and past dividends and it is given by: q t = ( I − RL ) − (cid:2)(cid:0) I + R NT L (cid:1) q NT t + R µ µµµ t − (cid:3) . (10)Due to the Gaussian nature of both µ t and q NT t and the risk-neutral nature of market participants,the choice of considering a linear (instead of a general) equilibrium implied by Eq.(1) is not restrictive.Thus, the market can be modeled by the MM as a Linear Gaussian State-Space Model (LG-SSM) [ ] .Actually, while the state of the market, i.e. realized dividends µµµ t and NT’s trades q t , are not observableby the MM, he can infer these quantities, and in particular realized dividends, filtering them out fromhis information set. This procedure in the LG-SSM literature is referred to as Kalman filtering technique.More details about these important aspects of the model will be given in the following section.5 .2 Competitive pricing rule As anticipated above, we assume the MM to be competitive and risk neutral. Thus, by a Bertrandauction type of argument [ ] , we postulate a break even condition for the MM for each T -periodholding strategy built as follows: buy q t units of stock by matching the demand at time t at a price p t and sell them back at time t + T at a price (cid:69) [ p t + T |(cid:73) MM t ] , earning the dividends in the meanwhile. Notethat even though the MM cannot choose to execute with certainty at t + T , we can see T as the timelag at which the MM decides to mark-to-market his position, even if he might not be actually able toliquidate it. Imposing competitiveness of the MM, this trajectory should have zero payoff on average,leading us to postulate a pricing-rule of the form: p t = t + T − (cid:88) t (cid:48) = t (cid:69) [ µ t (cid:48) |(cid:73) MM t ] + (cid:69) [ p t + T |(cid:73) MM t ] . (11)Thus, the price at time t is given by the long-term sum of future dividends plus a boundary term whichin general is non-zero. Stationary dividends with zero mean
If the boundary term in Eq. (11) evaluated at T = ∞ is equal to zero, i.e., the transversality conditionholds, one obtains the standard EMH fundamental rational expectation pricing-rule: p t = (cid:69) (cid:2) p F t |(cid:73) MM t (cid:3) , where p F t = (cid:62) / t µµµ / t . (12)In case of mean-reverting dividends process with zero mean, the transversality condition is justified. Wewill investigate the model with this assumption, for simplicity reasons. Under this prescription the jobof the MM is to provide the optimal forecast of discounted future cash flows from infinity to the presenttime t , given his current information set. Notice that restoring a fundamental price with non-zero meanwould simply amounts to a rigid (although, infinite) shift of the price process, since the mean of thefundamental price is assumed to be public information and so it is immediately incorporated into theprice.It will be interesting to compare the result of the MM’s estimate, given by Eq. (12), with the oneconstructed by the IT, which is not distorted by the the noise induced by the NT: p IT t = (cid:69) (cid:2) p F t (cid:12)(cid:12) (cid:73) IT t (cid:3) . (13)Let us note here that the dividends have to be predictable for the market to be non trivial. In fact,if the dividend process is not correlated, i.e., Ξ µτ = Ξ µ δ τ , then p IT t =
0, i.e., the IT does not have anyinformational advantage over the MM. Thus, in this case, the MM would simply set the price equal tozero.With the pricing rule given by Eq. (12) the MM statistically breaks even for each buy or sell trade,if he waits enough time for the income due to the dividends to restore his cash account to zero. Thislocal constraint is thus given by: (cid:69) [ ∆ C MM t ] =
0. (14)As a consequence (cid:69) [ ∆ C IT t ] + (cid:69) [ ∆ C NT t ] =
0, i.e. the gain of the IT is balanced by the losses of the NT.This is what typically happens in models where NT are uninformed and non-rational [ ] .In the following we give the explicit expression of the pricing rule (12) in terms of the IT’s tradingschedule, i.e. in terms of the IT’s demand kernels introduced in Eq. (8).6 ividends regression from observed excess demand The pricing rule given by Eq. (12) prescribes that the MM should estimate the sum of future dividendsby observing realized excess demand. This problem can be solved in two steps. First the MM estimatesrealized dividends applying a filter on realized excess demand. The optimal estimator of realized div-idends is well known in the LG-SSM literature as Kalman filter and it is linear in the measurements,i.e., the realized excess demand in our model. Then, the MM computes the expected sum of futuredividends summing over the forecasts of future dividends. In the following we detail these two steps.The MM’s estimate of realized dividends ˆ µµµ t : = (cid:69) [ µµµ t |(cid:73) MM t ] is given by:ˆ µµµ t = K q t , (15)where we have implicitly defined the (steady-state) Kalman gain K . This matrix can be constructed ina standard way [
17, 18 ] given the dynamics of the MM’s measurements, i.e., Eq. (10). The Kalmangain is proportional to the signal noise, i.e., Ξ µ , and inversely proportional to the measurement noise,which is the ACF of the excess demand Ω τ : = (cid:69) [ q t q t + τ ] and it is explicitly given by: K = Ξ µ ( J µ ) (cid:62) Ω − , (16)where J µ = ( I − RL ) − R µ LΩ = J µ Ξ µ ( J µ ) (cid:62) + D NT . (17) J µ is the matrix that multiplies the dividends in the r.h.s. of Eq. (10) and D NT is the NT’s dressed ACF,given by: D NT = ( I − RL ) − (cid:0) I + R NT L (cid:1) Ω NT (cid:0) I + R NT L (cid:1) (cid:62) (cid:2) ( I − RL ) − (cid:3) (cid:62) . (18)The noise ACF is dressed since the noise (i.e., the NT’s trade process) not only affects the excess demanddynamics by construction ( q t = q IT t + q NT t ), but also because the IT’s optimal trading strategy dependsupon past and future realizations of the noise (see Eq. (8)).Using the Woodbury identity on Eq. (16), one obtains the alternative expression of the gain matrix K : K = (cid:148) ( Ξ µ ) − + ( J µ ) (cid:62) (cid:0) D NT (cid:1) − J µ (cid:151) − ( J µ ) (cid:62) (cid:0) D NT (cid:1) − . (19)This alternative expression gives a complementary interpretation of the gain matrix K : in fact thematrix inside the square bracket is the dividends posterior information matrix. This matrix is given bythe dividends prior information matrix ( Ξ µ ) − summed to the information added by the measurement,i.e., ( J µ ) (cid:62) (cid:0) D NT (cid:1) − J µ .From estimated realized dividends ˆ µµµ t , the MM has to estimate the fundamental price p F t , defined inEq. (12). To do so, he builds the forecast of future dividends as (cid:69) [ µµµ / t | ˆ µµµ t ] = F µ ˆ µµµ t , where we introducedthe dividends forecast matrix F µ . Since the dividends process is Gaussian with zero-mean, F µ dependsonly on the ACF of the dividends Ξ µ . Finally, by summing over the estimated future dividends weobtain the following equation for the price at time t : p t = (cid:62) / t F µ K q t . (20)Notice, that Eq. (20) explicitly gives the rule for propagator, Eq. (1). In the following section weconstruct the IT’s optimal trading strategy based on the maximization of his expected future wealth, as afunction of the MM’s pricing rule. This means that, as anticipated, the IT’s demand kernels ( R , R NT , R µ )are functions of the propagator G introduced in Eq. (1), and so is the Kalman gain matrix K introducedin Eq. (19). Because of this, Eq. (20) will turn out to be a self-consistent equation for the propagator G .7 .3 Optimal insider trading The utility function U IT t , whose expectation is maximized by the IT at each time step t , is defined bythe value of his wealth account at a terminal time t + T (where T is not related to that introduced inSec. 3.2), given by W IT t + T , in which his position Q IT t in the risky asset is flattened. Thus, U IT t = W IT t + T subject to the constraint Q IT t (cid:48) = t (cid:48) ≥ t + T .At each time step t , the IT optimizes his expected utility function over the whole future trajectory q IT / t given the information set at the current time (cid:73) IT t given by Eq. (7), and trades the first step of theoptimal strategy. The IT’s trade at time t is thus calculated as follows: q IT t = e (cid:62) t argmax q IT / t (cid:69) (cid:2) U IT t (cid:12)(cid:12) (cid:73) IT t (cid:3) , (21)where e (cid:62) t explicits the fact that only the first step of the future trajectory is executed. Notice that thepresence of a finite liquidation time does not break the assumption of the time-translational invarianceof the model, because the terminal condition is also receding as time moves on. Indeed, the IT will ingeneral hold a non-zero position Q IT t up to t → ∞ despite the presence of the liquidation constraint.The constraint should then be seen as a device used by the IT in order to properly mark-to-marketthe value of his current stock positions at time t by taking into account the forecast of their future liquidation value p t + T , rather than as a measure taken to prevent him from trading at large times.In the following we analyze the case in which T = ∞ with mean-reverting dividends. Stationary demand kernels of the insider with infinite horizon If T = ∞ in Eq. (21), the IT can neglect the round-trip constraint, since liquidation costs are pushedto the far away future and, due to the assumptions of zero-mean and mean-reverting dividends, theexpected price at infinity is zero. Because of this, the actual trading profile of the IT that we willconsider in the following is given by Eq. (21) with U IT t = C IT ∞ . In doing so, the maximization programis given by q IT t = e (cid:62) t argmax q IT / t (cid:69) (cid:2) C IT ∞ (cid:12)(cid:12) (cid:73) IT t (cid:3) , where C IT ∞ = C IT t − − (cid:128) q IT / t (cid:138) (cid:62) (cid:128) p / t − p F / t (cid:138) . (22)In order to keep the discussion simple we consider the dividend process with integrable auto-covariance,such that the IT’s estimate of the fundamental price p F t is finite. One can in fact relax this hypothesis,with a suitable renormalization of the price and dividends process.Notice that the introduction of a non-zero mean for the fundamental price does not affects nor theIT’s strategy nor the price impact function. In fact, since the mean of the fundamental price is assumedto be public information, the MM could immediately incorporate it in the price, as discussed previously.Then, since the IT’s gain in Eq. (22) is proportional to the difference between the price and the IT’sestimate of the fundamental price, it follows that the IT’s trading strategy does not depend on the meanof the fundamental price. To conclude, since the propagator depends only on IT’s demand kernels andfluctuations of dividends and NT’s trades via Eq. (20), it follows that the mean of the fundamental priceis immaterial in shaping the the price impact function.The expression for the demand kernels ( R , R NT , R µ ) at equilibrium can be determined as solutionof the quadratic optimization program defined by Eq. (22). The expected gain at infinity C IT ∞ dependson estimated future dividends (via p F / t ) and on estimated future NT’s trades (via p / t ). Thus, in order towrite it down explicitly, we need the dividends forecast matrix F µ introduced in the previous section,and the forecast matrix of NT’s trades, F NT , defined similarly by (cid:69) [ q NT / t | q NT t ] = F NT q NT t .Since (cid:69) [ C IT ∞ |(cid:73) IT t ] depends on past realizations and forecasts, we insert time subscripts over matrix8ymbols in order to avoid ambiguities. We obtain: (cid:69) [ C IT ∞ |(cid:73) IT t ] = − (cid:128) q IT / t (cid:138) (cid:62) G sym / t , / t q IT / t − (cid:128) q IT / t (cid:138) (cid:62) (cid:148) G / t , t − q t − + G / t , / t F NT / t , t − q NT t − − U / t , / t F µ/ t , t − µµµ t − (cid:151) , (23)where we dropped C IT t − , since it does not depend on IT’s future trades q IT / t , and we introduced thesymmetric propagator G sym = ( G + G (cid:62) ) in order to write in a compact form the quadratic term in q IT / t .The quadratic term in q IT / t of Eq. (23) is the cost term that the IT will face due to his own future marketimpact, while the the linear term in q IT / t is his signal term. The first term of the signal comes from priceimpact due to known order flow realizations, the second one comes from the expected price impact offuture NT’s trades, while the third one comes from his private information about p F / t .Inserting Eq. (23) in Eq. (22), allows to obtain the expression for the IT’s demand kernels in termsof the propagator G and the forecast matrices F NT and F µ : R t = − e (cid:62) t (cid:148) G sym / t , / t (cid:151) − G / t , t − , (24a) R NT t = − e (cid:62) t (cid:148) G sym / t , / t (cid:151) − G / t , / t F NT / t , t − , (24b) R µ t = e (cid:62) t (cid:148) G sym / t , / t (cid:151) − U / t , / t F µ/ t , t − . (24c)Finally, we have all the ingredient to write down explicitly the functional equation for the equilib-rium pricing rule, which will be given in the following section. The linear equilibrium of the model can be found by self-consistently taking into account the compet-itive pricing rule of the MM and the strategy of the IT, given respectively in Eqs. (20) and (24). Theself-consistent equation for the propagator, in scalar notation, reads: G t − s = ∞ (cid:88) t (cid:48) = t t (cid:88) t (cid:48)(cid:48) = −∞ F µ t (cid:48) , t (cid:48)(cid:48) K t (cid:48)(cid:48) , s [ G ] (25)where we made explicit that the filter K , given in terms of IT’s demand kernel by Eq. (19), is a functionof the propagator itself, as can be seen from Eqs. (24).The linear equilibrium equation (25) is a non-linear functional equation for the propagator G t . Assuch it is not amenable for analytical treatment in the general case of arbitrary Gaussian, zero-meanand stationary dividends and NT’s trades process. Nevertheless, we have been able to solve Eq. (25)iteratively, as illustrated in Appendix A. In two special cases we have been able to validate the result ofthe iterative numerical solver by means of the analytical solution of Eq. (25) (see Appendix B).Via an extensive analysis of the model based on the iterative numerical solver of Eq. (25) we foundthat the market at equilibrium exhibits some robust properties, that hold in case of an integrable andstationary ACF of the NT’s trades and dividends, regardless of the exact structure of the ACFs. Theseproperties are listed below. Return covariance
The equilibrium is characterized by a return ACF Ξ τ : = (cid:69) [ ∆ p t ∆ p t + τ ] with thesame temporal structure as that related to the IT’s price estimate p IT t , given by Eq. (13), which will bereferred to as Ξ IT τ . In formula: Ξ τ = Ξ ˜ Ξ IT τ , with ˜ Ξ IT0 =
1. (26)9he price distortion induced by the noise injected into the system by the NT is thus completely encodedin a scalar, the return variance Ξ .The left panels of Figs. 1, 2 and 3 display numerical results that do confirm Eq. (26). In particular,in top panels, bullet points correspond to Ξ τ / Ξ obtained by means of the numerical solver of Eq. (25)and show a good collapse on the dashed line, which corresponds to Ξ IT τ / Ξ IT0 calculated semi-analytically.In the bottom part of the panels instead we show the relative cumulative absolute error between thetwo curves, defined as: er r Ξ τ = (cid:80) τ i = | Ξ i / Ξ − Ξ IT i / Ξ IT0 | (cid:80) τ i = | Ξ i / Ξ | . (27)In Figs 1 and 2, where non-markovian ACFs are examined, these errors are larger than in Fig. 3,where ACFs decay exponentially. This is due to the fact that in the former case the forecast of futuredividends suffers from finite size effects. The estimation of these effects is carried on in detail inAppendix A.The inset of the left-top panels shows the variogram of the price, defined by V τ : = (cid:69) (cid:2) ( p t − p t + τ ) (cid:3) ,which, as expected, is linear at high frequencies and mean-reverting at low frequencies. Excess demand covariance
The equilibrium is characterized by an excess demand ACF Ω τ : = (cid:69) [ q t q t + τ ] with the same temporal structure as the one related to NT’s trades, plus an extra contribution at lag 0.In formula: Ω τ = a ( ˜ Ω NT τ + ˜ b δ τ ) , with ˜ Ω NT0 =
1, (28)where the symbol δ τ denotes the discrete delta function, while a and b are scalars. The excess demandvariance is given by Ω = a ( + ˜ b ) .Since IT’s information at time t does not include the current trade of the NT q NT t (see Eq. (7)), thebest that the IT can do in order to hide his trades is to create a trading strategy such that the excessdemand ACF resembles that of the NT apart from the lag 0 term. Because of the distortion at lag 0, wecall this property quasi-camouflage strategy . Indeed, in order to prolong his informational advantageover the MM, the IT hides his trades in the excess demand process by creating a strategy that resemblesthat of the NT alone.Right panels of Figs. 1, 2 and 3 display numerical results that confirm the quasi-camouflage prop-erty. In top panels bullet points correspond to Ω τ / Ω obtained by means of the numerical solverof Eq. (25) which show a good collapse for positive lags on the dashed line, which corresponds to Ω NT τ / Ω NT1 . It is clear, from the insets of the plots on the left, that the collapse is not reached at lag 0. Asit will be shown in the next section, this extra contribution at lag 0 depends in a non trivial way on theACFs of the dividends and NT’s trades. On the bottom, the relative cumulative absolute error betweenthe two curves is presented. In this case, it starts from lag 1, so: er r Ω τ = (cid:80) τ i = | Ω i / Ω − Ω NT i / Ω NT1 | (cid:80) τ i = | Ω i / Ω | . (29)Again, these errors are larger in the case where non-markovian ACFs are examined.From the properties given by Eqs. (26) and (28), together with the MM’s break even condition, oneis in principle able to find the propagator. In fact, introducing the price ACF Σ τ : = (cid:69) [ p t p t + τ ] , from thedefinition of the propagator (1) follows that: Σ τ = t + τ (cid:88) t (cid:48) = −∞ t (cid:88) t (cid:48)(cid:48) = −∞ G t + τ − t (cid:48) G t − t (cid:48)(cid:48) Ω | t (cid:48) − t (cid:48)(cid:48) | , with τ >
0, (30)where the price ACF Σ τ can be computed from Eq. (26) and the excess demand ACF is given by Eq. (28).This program can be accomplished in the case of a Markovian system and it is described in full detail Camouflage is also called inconspicuous strategy in the economics literature [
15, 19, 20 ]
25 50 75 100 125 150 175 2000.00.20.40.60.81.0 / e rr V / V / + 1 e rr Figure 1:
Numerical check of equilibrium properties with ACFs given by ( + | τ | /τ k ) − γ k where k = { µ ,NT } .We arbitrarily choose ( τ NT , τ µ , γ NT , γ µ ) = ( ) . The numerical solver has been implemented with T cut = · and T it = Ξ τ / Ξ (bullet points) and Ξ IT τ / Ξ IT0 (dashed line). The collapse between these two ACFs is quantified in the bottom panel, where the relativecumulative absolute error between the two curves is displayed. The inset in the top panel shows the collapse onthe variogram. (Right) In the main top panel we show the good collapse for positive lags between Ω τ / Ω (bulletpoints) and Ω NT τ / Ω NT1 (dashed line), whereas in the inset we show that the collapse doesn’t involve the lag 0term. In the bottom panel the collapse between these two ACFs is quantified, calculating the relative cumulativeabsolute error starting from lag 1. / e rr V / V / e rr Figure 2:
Numerical check of equilibrium properties with ACFs given by exp − τ/τ k sin ( x /τ k + π/ ) where k = { µ ,NT } . We arbitrarily choose ( τ , τ µ , τ , τ µ ) = ( ) . The numerical solver has beenimplemented with T cut = and T it = Ξ τ / Ξ (bullet points) and Ξ IT τ / Ξ IT0 (dashed line). The collapse between these two ACFs is quantified in thebottom panel, where the relative cumulative absolute error between the two curves is displayed. The inset inthe top panel shows the collapse on the variogram. (Right) In the main top panel we show the good collapse forpositive lags between Ω τ / Ω (bullet points) and Ω NT τ / Ω NT1 (dashed line), whereas in the inset we show that thecollapse doesn’t involve the lag 0 term. In the bottom panel the collapse betweeen these two ACFs is quantified,calculating the relative cumulative absolute error starting from lag 1.
20 40 60 80 1000.00.20.40.60.81.0 / e rr
1e 140 1000.00.51.0 V / V / e rr
1e 12 0 50.81.01.2
Figure 3:
Numerical check of equilibrium properties for ACFs given by e − τ/τ k where k = { µ ,NT } . We arbitrarilyfixed ( τ NT , τ µ ) = ( ) . The numerical solver has been implemented with T cut = · and T it = Ξ τ / Ξ (bullet points) and Ξ IT τ / Ξ IT0 (dashed line). Thecollapse between these two ACFs is quantified in the bottom panel, where the relative cumulative absolute errorbetween the two curves is displayed. The inset in the top panel shows the collapse on the variogram. (Right)In the main top panel we show the good collapse for positive lags between Ω τ / Ω (bullet points) and Ω NT τ / Ω NT1 (dashed line), whereas in the inset we show that the collapse doesn’t involve the lag 0 term. In the bottom panelthe collapse between these two ACFs is quantified, calculating the relative cumulative absolute error startingfrom lag 1. in Sec. 6. There, we shall provide semi-analytical results for all of the parameters introduced in theequations listed above, which do share qualitative features with the general non-Markovian case. Aninteresting finding of this analysis is given by the fact that as the predictability of the NT’s trades processincrease, the IT’s camouflage becomes exact, allowing him to reduce the cost due to price impact of histrading schedule.But before discussing the Markovian case, let us highlight similarities and differences with respectto existing models.
While strongly inspired by the one-period Kyle model, our model is quite different on several grounds.First, instead of exogenously postulating the presence of a fundamental price, in our setting it is theintegrated-dividend process that plays the role of the fundamental price, mechanically relating it tothe payoff of the asset. Second, we do not have explicit fundamental price revelation, thus allowingto consider a stationary setting in the model. Such a stationary regime is relevant in practice becausein order to analyze the behavior of the market at short time scales (minutes, hours) one would liketo abstract away the non-stationary effects potentially induced by the dynamics of the fundamentalinformation (e.g, dividends, earning announcements, scheduled news) at slower time scales. Third,we introduced (integrable) serial correlations both in the dividends – equivalently, in the fundamentalprice – and in the order flow.Let us also point out how we can recover the Kyle model in our setting. Assuming that ( i ) the NT’strades are uncorrelated, ( ii ) the sum of future dividends p F t follows a random walk process, ( iii ) theIT knows the value of p F t at the beginning of each period and ( i v ) p F t becomes public information oncethe MM has set the price, we recover exactly an iterated version of the single period Kyle model.12 .2 Propagator model Equation (30) is the cornerstone equation when dealing with propagator models. It is typically usedin the literature in order to extract a propagator G t from empirical data given the order flow correla-tion and the price volatility. Hence, our framework allows us to recover the propagator model in aneconomically standard setting, with three important caveats:• The excess demand ACF Ω τ function observed in real markets is typically non-integrable, due tothe strongly persistent nature of the order flow [
11, 21 ] .• The price process observed in real markets is close to be diffusive at high frequency.• The propagator observed in real markets is found to be a slowly decaying function of time.Let us address these empirical facts, showing how one can account for them within our stylized model.First, the non-integrability of the excess demand ACF can be retrieved by extending our frameworkto the case in which the NT’s trades ACF are themselves non-integrable. This is due to the fact that thecamuflage condition relating excess demand and noise trading is also expected to extend to the settingof non-integrable NT’s trades.Second, price diffusivity also can be recovered in our model as the limiting regime in which divi-dends are much slower than any other time scales in the model. In order to prove this, note that thevariogram of the price can be written in terms of the price ACF Σ τ as follows: V τ = V ∞ ( − ˜ Σ τ ) , where ˜ Σ =
1, (31)where the first equality holds in stationary conditions, as the one described by the model introducedhere. Thus, we do recover price diffusivity at high frequency if ˜ Σ τ − ∝ τ/τ ∗ in the high frequencylimit of the model, i.e., τ (cid:28) τ ∗ , where τ ∗ is some typical timescale. Instead, in the opposite lowfrequency limit τ (cid:29) τ ∗ , because of the assumption of mean-reverting dividends, which translates intohaving mean-reverting fundamental price, the price ACF decays to zero, i.e., Σ τ ∼
0, and we recovera flat variogram. For example, in the Markovian case described below, where the dividends ACF is anexponential decay function with timescale τ µ , one has τ ∗ = τ µ . To wrap up, if the hypothesis of linearprice ACF Σ τ in the high frequency limit holds, the price in our model interpolates between two verydifferent situations: when the model is probed in its high frequency limit it describes a market withdiffusive price, while in the low frequency limit the price is mean-reverting. This is very satisfactorysince it is obtained with a single propagator, which is the solution of Eq. (25). At high frequency, wherethe dividend process appears highly persistent, price diffusivity stems from IT’s surprises in dividendsvariations: this is the universal mechanism which originates the diffusive behavior in our model. Infact, in this limit, the IT’s estimate of the fundamental price is a martingale, and thus it is describedby a diffusive process. From Eq. (26) follows that the price process itself it is described by a diffusiveprocess.Since the first two properties can be retrieved, the third one follows from standard scaling argu-ments. Thus, in the high frequency limit price impact has to be a slowly decaying function of time inorder to ensure price diffusion while having a strongly correlated order flow process via Eq. (30).It is interesting to notice that in order to observe any impact at all in the model, one is forced tointroduce a non-trivial dividend process: the introduction of fundamental information that gives theIT an informational advantage over the MM is enough in order to induce non-trivial dynamics into theprice, and to typically induce a diffusive behavior of prices at high frequency. Hence, the price paid inorder to micro-found the propagator model is the introduction of an auxiliary dividend process, whosedetailed shape is inessential at high enough frequency, but whose fluctuations sets the scale of the priceresponse. See the brief discussion under Eq. (13) Markovian case
Significant simplifications of the equilibrium condition (25) are possible in the case in which both thedividend and the NT flow are Markovian processes, where their ACFs are given by: Ξ µτ = Ξ µ α τµ , (32a) Ω NT τ = Ω NT0 α τ NT . (32b)One of these simplifications comes from the fact that the price estimate p IT t given by Eq. (13) is pro-portional to the current dividend realization, i.e. p IT t = µ t − α µ / ( − α µ ) . Thus, the price efficiencyproperty given in Eq. (26) becomes: Ξ τ = Ξ ˜ Ξ F τ , with ˜ Ξ F0 =
1, (33)where Ξ F is the return ACF of the fundamental price p F t . From Eq. (33) follows that the ACF of theprice process Σ τ is a decaying exponential with timescale given by τ µ : = − / log ( α µ ) . As a result,the price process in the Markovian case is a discrete Ornstein-Uhlenbeck process with timescale τ µ : = − / log ( α µ ) .We validated the result of the iterative numerical solution exposed in the previous section by solvingexplicitly the equilibrium condition in two peculiar Markovian cases: the case of non-correlated NTtrades, obtained by replacing the equation for the NT’s trades ACF by Ω NT τ = Ω NT0 δ τ , and the case inwhich the ACF timescale of NT’s trades is the same as the dividends’ one, i.e., the case given by Eq. (32)with α µ = α NT . These findings are reported in Appendix B.Furthermore, we found the explicit solution of the equilibrium condition by imposing the genericequilibrium properties listed in the previous section, together with the MM’s break even condition givenby Eq. (14). Details about the outcome of this procedure are given in the following sections.Let us point out that even though the choice of Markovian dividends and NT’s trades processes ismade in order to obtain analytical results and build an intuition about the system in a simple case, themain qualitative conclusions found in this section do extrapolate to generic stationary, mean revertingprocesses with integrable ACFs. Non-correlated NT trades
This case is particularly simple since the quasi-camouflage property givenby Eq. (28) becomes exact. Eq. (30) is solved by an exponential decay propagator with the sametimescale as the dividends ACF, i.e., τ µ . The amplitude of the propagator is derived in App. B.1. Correlated NT trades
The solution of Eq. (25) is obtained in two steps. First, we build an ansatzbased on the quasi-camouflage strategy property, i.e. Eq. (28) and the property about return ACF givenby Eq. (26). Details about this are given in Appendix C.1. Then we fix the ansatz by imposing the MM’sbreak even condition (see Appendix C.2). The results of this procedure, described below, do matchwith the results of the iterative numerical solver of Eq. (25).The propagator we find reads: G τ = G (cid:20) α µ − α NT α µ − ρ α τµ + (cid:18) − α µ − α NT α µ − ρ (cid:19) ρ τ (cid:21) , (34)where a new timescale τ ρ : = − / log ( ρ ) appears. This new timescale is given, in the general Markoviancase, by a non-linear combination of the two fundamental timescales τ µ and τ NT : = − / log ( α NT ) (theimplicit expression for ρ and G is obtained as illustrated in Appendix C.2). From the left panel ofFig. 4, it is clear that in the regime in which τ µ , τ NT (cid:29) τ ρ approaches a value close to the time-step, i.e. τ ρ ∼
1, thus being much smaller than the two fundamental timescales. This finding and14
10 15 20 25510152025 N T N T b Figure 4: (Left) Endogenously generated timescale τ ρ as a function of τ µ and τ NT . τ ρ is never larger than ∼ Σ τ , i.e., ˜ b introduced in Eq. (28), as a function of τ µ and τ NT . As one can see in the inset, ˜ b attains its maximum value for small timescale, while it decrease to zeroas τ µ and τ NT increase, thus recovering exact camouflage for the IT strategy. the one related to the case with non-correlated NT’s trades indicate that when dividends are highlypersistent ( α µ →
1) the propagator exhibits a quasi-permanent component and a non-zero transientcomponent. The former stems from the apparent persistency of the fundamental process when probedat high frequency, while the latter arises from non-trivial predictability of the NT’s trades process.As we shall see below, the large timescales behavior of τ ρ is related to the behavior of the excessdemand ACF distortion at lag 0, i.e. ˜ b , introduced in Eq. (28). In fact, in the derivation of Eq. (34)(see Appendix C.1) one finds: ˜ b = ρ ( − α ) α NT ( + ρ ) − ρ ( + α ) . (35)In the right panel of Fig. 4 we display ˜ b as function of τ µ and τ NT . This amplitude is close to 1 inthe limit of small dividends and NT’s trades timescales and decreases to zero as these increase. Thus,the excess demand ACF temporal structure resembles more and more the NT’s one as soon as the NT’strades or dividends are strongly correlated.The interpretation of this finding is the following: the IT wants to hide his own trades in the excessdemand process, by shaping the ACF to resemble the NT’s trades one. However the IT knows onlyup to time t − t .Therefore, the IT is not able to hide his current trade. Instead, if the NT’s trades are strongly correlated,the IT’s information about NT’s past trades allows him to accurately predict the current NT’s trade, andthus the IT is able to hide his current trade. Briefly, we find that: Ω τ → Ω ˜ Ω NT τ as α NT →
1, (36)thus recovering an exact camouflage trading strategy of the IT, exhibited by many Kyle-like models [
15, 20, 22, 23 ] .The limit α NT → α µ → Ω τ = Ω ˜ Ω NT τ , and Eq. (33) in continuous-time one can solve the continuous-time analogof Eq. (30), finding: G τ = G (cid:18) δ τ + τ µ − τ NT τ µ τ NT e − τ/τ µ (cid:19) . (37)From this equation we can see that the term in the propagator that depends on the endogenouslygenerated timescale (see Eq. (34)) approaches a Dirac delta function in the continuum limit of themodel, as a result of the IT’s exact camouflage strategy.15
10 15 20 25510152025 N T / NT0 N T / IT0
Figure 5: (Left) Ratio between variance of the excess demand and variance of the NT’s order flow as a functionof τ µ and τ NT . When the timescale τ µ and τ NT are small (inset), the excess demand is higher than the one ofthe NT’s trades alone. Conversely, when the timescales τ µ and τ NT are large, the excess demand variance islower than the one of the NT’s alone. (Right) Ratio between the variance of the price and the variance of theIT’s fundamental price estimate as a function of τ µ and τ NT . The variance ratio in this case is very small when τ µ is close to zero, while it increase as τ µ increases. The result for the ratio Ω / Ω NT0 as a function of τ µ and τ NT is presented in the left panel of Fig. 5. Thevariance ratio is bounded between 2, for small timescales, and 0.5, for large timescales. The increaseof the ratio of variances, Ω / Ω NT0 , for small τ NT can be understood as follows. In this regime, the NT’scurrent trade is almost unpredictable, thus the IT’s current trade is independent of the current trade ofthe NT. As a consequence, the excess demand variance increases with respect to the NT’s variance. Assoon as the NT component of the order flow is predictable, the IT uses this information.In particular, the IT’s current trade is on average anti-correlated with the current NT trade. Thisenable the IT to move less the price, founding liquidity in the NT’s trade and reducing the typicalaggregate volume demanded to the MM. When the predictability of the NT’s trades and dividendsprocess increase, the current IT’s trade is more anti-correlated with the current NT’s trade, thus enablinghim to loose less money due to price impact. The current IT’s trade is instead positively correlated withthe current dividend. Fig. 6 shows these findings. In our model price variance is directly linked to price efficiency, as argued below Eq. (26). As alreadynoted by Shiller, in a Rational Expectation Model where the price is the expected fundamental price,using the principle from elementary statistics that the variance of the sum of two uncorrelated vari-ables is the sum of their variances, one then has Σ / Σ IT0 ≤ Σ / Σ F0 ≤
1, where Σ F0 is the variance of thefundamental price.We display the results for the ratio Σ / Σ IT0 in the right panel of Fig. 5, as a function of the dividendsand NT’s timescales, that confirm the fundamental constraint exposed before. Moreover, we find thatthe ratio of variances strongly depends on τ µ . In particular, if the dividends are weakly correlated theprice variance poorly reflects the IT’s price estimate variance Σ IT0 . Instead, in the limit of large dividendtimescales with respect to the one of the NT’s trades, the price variance better reflects the IT’s priceestimate p IT t . In the regime of small τ NT and large τ µ the price variance accounts for all the variance ofthe IT’s price estimate, Σ IT0 , as indeed found analytically from the calculations reported in Appendix B.1.16
10 15 20 25510152025 N T [ q IT t q NT t ]/ NT0 N T [ q IT t t ]/( IT0 NT0 ) Figure 6: (Left) Ratio of the covariance between equal-time IT’s and NT’s trades, and the variance of NT’strades as a function of τ µ and τ NT . The IT’s trades are anti-correlated with the equal time NT’s trades. (Right)Properly rescaled covariance of current IT’s trade and dividend as a function of τ µ and τ NT . The IT’s tradesare on average positively correlated with the equal time dividend. When the predictability of the NT’s tradesand dividends process increase, the current IT’s trade is more positively (negatively) correlated with the currentdividend (NT’s trade), thus enabling him to gain more (loose less). As explained around Eq. (14), the payoff of the different agents is, on average, the following: the MMbreaks even, the NT loses and the IT gains what the NT loses.If the dividend process is completely unpredictable (but still stationary with zero-mean), then theprice is set to zero by the MM; thus the IT won’t trade anymore and the NT’s losses are reduced to zero.When the τ µ becomes large with respect to τ NT (bottom right corner of main left panel of Fig. 7), theprice is more and more efficient as we have seen in the previous section. In this case, the IT’s gains arelowered, as well as the NT’s losses. These findings are reported in the left panel of Fig. 7, where weplot the ratio − (cid:69) [ δ q NT t C NT t ] / ( Ξ µ Ω NT0 ) / , with δ q NT t C NT t = − q NT t (cid:0) p t − (cid:80) t (cid:48) ≥ t µ t (cid:48) (cid:1) . N T [ q NT t C NT t ]/( IT0 NT0 ) N T r MM t /( IT0 NT0 ) Figure 7: (Left) Properly rescaled loss per trade of the NT (or gain per trade of the IT) as a function of τ µ and τ NT . When τ µ is close to zero (inset) the loss per trade of the NT are close to zero, while these increase asthe predictability of the dividends and NT’s trades increase. (Right) Properly rescaled MM’s risk per trade as afunction of τ µ and τ NT . From the inset we can see that the risk is higher when the NT’s trades and dividends areclose to be unpredictable, whereas the risk is lower as the predictability increase. Another interesting quantity is the risk per trade experienced by the MM, i.e. r MM t = (cid:69) [( δ q t C MM t ) ] ,where δ q t C MM t = q t (cid:0) p t − (cid:80) t (cid:48) ≥ t µ t (cid:48) (cid:1) . We find: r MM t = (cid:69) [ q t ] (cid:69) (cid:148)(cid:0) p t − p IT t (cid:1) (cid:151) , (38)17here we used the break even condition (Eq. (14)) together with Wick’s theorem to calculate higherorder correlations of a Gaussian process. The analytical solution is given in the right panel of Fig. 7.As we can see, the risk experienced by the MM is high when both the timescales of the two funda-mental processes are small, while it decreases when both the dividends and the NT’s trades becomespredictable. The aim of this paper was to provide an economically standard micro-foundation for linear price impactmodels, customarily used in the econophysics literature. To do so, we presented a multi-period Infor-mation Based Model and we analyzed its equilibrium. The model is built by generalizing the seminalKyle model, which constitutes a theoretical cornerstone of market microstructure. First, we removedthe assumption of fundamental price revelation, assuming that a stock pays dividends to the ownerbut only the insider collects and exploits information about past dividends. Then, we modeled thedividends process and the noise trader trading schedule as stationary stochastic processes. In order toregularise the model we assumed that the dividend ACF was integrable, to ensure a bounded funda-mental price of the traded stock. The model appeared to exhibit a stationary equilibrium, which wehave investigated in detail. A self-consistent equation for the pricing-rule set by the market-maker hasbeen derived and solved numerically. Two robust properties have been found: the price ACF retains thesame temporal structure as the insider’s fundamental price estimate and the insider strategy respectsa quasi-camouflage condition, i.e., the ACF of the excess demand retains the temporal structure of thenoise trader’s one apart from the lag 0 term.As a consequence of these findings, we have been able to establish a precise correspondance be-tween the propagator model and the Kyle model: the propagator model arises here as the high-frequency limit of a suitably stationarized Kyle model. The price impact function that is found in thisregime displays a quasi-permanent component related to the timescale of variation of the fundamentalinformation, and a transient one whose timescale is set by the persistence of the order flow.The assumption of stationary dividends with integrable ACF translates into having a mean-revertingprice process. Since price diffusivity can be retrieved in the high frequency limit, the model is able toprovide a stylized picture of what happens in real markets at high and low frequency. The model alludesalso to a relation between the diffusion constant of the price process and the timescale over which thefundamental price mean reverts. We leave the empirical check of this finding as an interesting follow-upof the present investigation.
We thank J.-P. Bouchaud and C.-A. Lehalle for fruitful discussions. This research was conducted withinthe Econophysics & Complex Systems Research Chair, under the aegis of the Fondation du Risque, theFondation de l’Ecole polytechnique, the Ecole polytechnique and Capital Fund Management.18 eferences [ ] B. G. Malkiel. The efficient market hypothesis and its critics.
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IMA Journal of Applied Mathematics , 84(4):679–695, 2019.20
Numerical solver
The iterative numerical scheme is as follows:1. Choose a maximum time T cut − T cut elements.2. Choose a “seed” propagator.3. Plug this seed in the r.h.s. of Eq. (25).4. Insert the result obtained with this procedure in the r.h.s. for a number of iterations equal to T it ,checking for convergence.The only issue of this procedure is the following: as one can see from the first of Eqs. (24), in orderto compute R t one has to evaluate the block matrix given by G / t , t . This matrix has entries that cannotbe calculated, due to the truncation constraint of our numerical procedure. Nevertheless, because ofthe mean reverting assumption of the dividends, we know that the propagator should decay to zero atlarge times, so the large lags terms in G / t , t can be simply set to zero. Convergence
In this section we give further details about the convergence of the results of the iterative numericalsolution of Eq. (12).In Fig. 8 we show results about the relative cumulative absolute error for the price ACF Σ τ and theexcess demand ACF Ω τ . The first one is calculated as in Eq. (27), while the second one is given by(29).We choose T it = T cut , i = ∆ t × i , for different i . The plots on the left are obtained with a power law ACF that decays faster thanthe one used to obtain the plots on the right. We can see, as expected, that the slower is the decay ofthe power law, the slower is the convergence.We have investigated the behavior of the error for higher T it , but we didn’t find quantitative differ-ences. B Particular solutions of equilibrium condition in the Markovian case
B.1 The case of non-correlated Noise
In the case of non correlated NT’s trades, the IT’s forecast of future NT’s trades is zero, and so thedemand kernel R NT , explicitly given in Eqs. (24), is zero. Since we are dealing with a Markoviandividend process, the IT’s forecast at time t of future dividends relies only on the last known dividend,i.e., µ t − and so R µ = R µ I , where R µ is a scalar.The self-consistent equilibrium condition given by Eq. (25) for the dimensionless propagator isgiven by: ˜ G t − t (cid:48) = − α µ e (cid:62) t ˜ Γ ( I − RL ) , (39)where ˜ Γ = (cid:2) ( ˜ Ξ µ ) − + ( ˜ R µ L ) (cid:62) ˜ R µ L (cid:3) − ( ˜ R µ L ) (cid:62) . (40)The solution of Eq. (39) is constructed in three steps. i) First we analyze the vector e (cid:62) t ˜ Γ and weshow that it is related to the inverse of a tri-diagonal matrix with modified corner elements, for which21
100 200 300 400 500 6000.00000.00250.00500.00750.01000.01250.01500.0175 e rr e rr e rr e rr Figure 8:
Numerical check of equilibrium properties with ACFs given by ( + | τ | /τ k ) − γ k where k = { µ ,NT } . Wearbitrarily choose τ NT = τ µ =
10. (Left panels) γ NT = γ µ = ∆ t = γ NT = γ µ = ∆ t = the explicit expression is known [ ] . Then, ii) we prove that a single exponential propagator solvesEq. (39) and we identify the amplitude and the timescale of the propagator in terms of α µ and R µ .iii) Finally, we can calculate the expression of R µ in terms of α µ from its general expression given inEqs. (24). In this way we fix completely the shape of the propagator only in terms of α µ .i) Since in the Markovian case R µ is proportional to the idenitity matrix, from Eq. (40) we obtain: e (cid:62) t ˜ Γ = ( a t , b t − )( ˜ R µ L ) (cid:62) = ˜ R µ b t − , (41)where the vector b t − can be found by means of the block matrix inverse formula applied to the matrixinside the square brackets of Eq. (40), given by: M = ( ˜ Ξ µ ) − + ( ˜ R µ L ) (cid:62) ˜ R µ L = (cid:20) a t B t − B (cid:62) t − C (cid:21) . (42)In particular, using the block inverse formula, the vector b t − is given by b t − = − a − t B (cid:62) t − ( M / a t ) − = α µ e (cid:62) t − ( M / a t ) − , (43)where the last equality has been obtained with the following property (checked by direct inspection ofEq. (42)): B (cid:62) t − ∝ e (cid:62) t − . ( M / a t ) is the Schur’s complement of M with respect to a t , which is given by ( M / a t ) = C − B (cid:62) t − a − t B t − = ( ˜ Ξ µ ) − + ( ˜ R µ ) I . (44) ( M / a t ) is a tri-diagonal matrix with modified corner elements. Thus, the inverse of the Schur’s com-plement of M with respect to a t can be calculated explicitly (see Ref. [ ] ). The explicit expression ofEq. (43) is given by a single decaying exponential: b t − = b { γ τ } ∞ τ = , (45)where b = α µ ( ˜ R µ ) − g ( ˜ R µ ) , γ = g α µ ( ˜ R µ ) (46)22nd g is given by: g = β − (cid:112) β − ( ˜ R µ ) − α µ , β = ( ˜ R µ ) − + + ( ˜ R µ ) − α µ − α µ ( ˜ R µ ) − α µ , (47)so that b t − is completely specified by α µ and ˜ R µ .ii) We are going to proof that an ansatz for the propagator given by a decaying exponential towardszero actually solves Eq. (39). The ansatz for the propagator reads: G t − t (cid:48) = G ρ t − t (cid:48) . (48)As a preliminary result, from this ansatz, one can compute the elements of the vector R t , which appearin Eq. (39), by means of the first equation in Eqs. (24) . This is given by: R t − t (cid:48) = − R ρ t − t (cid:48) , (49)where R = − g s , g s = − (cid:112) − ρ ρ . (50)Equipped with this result, together with Eq. (45) one can easily show that Eq. (39) is solved withthe ansatz given by Eq. (48). The ansatz is constraint to satisfy the following equations:˜ G = b − α µ , ρ = γ ( − R ) . (51)iii) Since we proved that G is of the exponential form we are now able to compute the explicit formof R µ , starting from its definition in Eq. (24). The explicit expression for ˜ R µ , which completely specifies R µ , is given by: R µ = ( − α ) G (cid:130) α µ g s − α µ ρ − ( − ρ ) g s − g s ρα µ (cid:140) (52)Now, we can use insert in the above equation the expression for g , g s , G , ρ given respectively byEqs. (47), (50) and (51) and solve for ˜ R µ . In doing so we find˜ R µ =
1. (53)Finally, reintroducing the variance terms, i.e., using Ω NT τ = Ω NT0 δ τ and Ξ µτ = Ξ µ α τµ , then the solutionto Eq. (39) is given by G τ = (cid:18) Ξ µ Ω NT0 (cid:19) / α µ − α µ (cid:32) − − (cid:113) − α µ α µ (cid:33) α τµ . (54) B.2 The case of Noise and Signal with equal autocovariance timescales
In this section we deal with the Markovian case specified by Eqs. (32) with α µ = α NT . A difference withthe previous case is given by the fact that now R NT = R NT I , where R NT is a nonzero scalar. The solutionof the self-consistent equilibrium condition (25) is akin to the one exposed in the previous section, dueto a simplification induced by the assumption given by α µ = α NT . In order to show this we define E NT as: E NT t = (cid:0) I + R NT L (cid:1) Ω NT (cid:0) I + R NT L (cid:1) (cid:62) . (55)The simplification is the following: (cid:166)(cid:2) ( Ξ µ ) − + ( R µ L ) (cid:62) ( E NT ) − R µ L (cid:3) − ( R µ L ) (cid:62) ( E NT ) − (cid:169) t , t (cid:48) = α µ R µ L (cid:166)(cid:2) E NT ( Ξ µ ) − + ( R µ ) I (cid:3) − (cid:169) t , t (cid:48) , (56)23here the matrix inside the square bracket on the r.h.s. is a tri-diagonal matrix with modified cornerelements, for which, as seen before, analytical results are available. Thus, akin to the previous case, apropagator given by a single exponential decay term given by Eq. (48) is a solution of the self-consistentequation for the propagator (25). The result of the calculation that we do not report here is given by R NT : ( R NT ) − ( R NT ) α µ + R NT (cid:128) α µ + α µ (cid:138) − α µ =
0, (57)where one has to retain the only positive real solution. Then, R µ = (cid:118)(cid:117)(cid:116) Ω NT0 Ξ µ (cid:113) + ( R NT ) + R NT α µ , (58) ρ = R NT + ( R NT ) + R NT α µ (59)and finally G = (cid:118)(cid:117)(cid:116) Ξ µ Ω NT0 α µ (cid:198) ( R NT ) − α µ R NT + ( − α µ ) R NT − α µ + R NT − ( ( R NT ) − α µ R NT + ) (cid:130)(cid:118)(cid:116) − ( R NT ) ( ( R NT ) − αµ R NT + ) + (cid:140) + R NT . (60) C Solution of the Markovian case
C.1 Construction of the Ansatz
In this section we prove the results presented in Sec. 6.1, in particular Eqs. (34) and (35). i) First,we rewrite the property exposed in Eq. (33) in expectation form. ii) Then we inject in this form thequasi-camouflage property and we find a simple finite-difference equation for the propagator whosesolution gives the formulas presented in Eqs. (34) and (35).i) If the price ACF is exponentially decaying with the dividends timescale, as found by means of thenumerical solver, then the following relation holds: (cid:69) [ p t + |(cid:73) MM t ] = α µ p t . (61)Equation (61) gives us a relation between the excess demand ACF and the propagator. In fact usingthe equation that defines the propagator model, i.e., p t = (cid:80) t (cid:48) ≤ t G t − t (cid:48) q t (cid:48) , it can be rewritten as: G (cid:69) [ q t + |(cid:73) MM t ] = α µ t (cid:88) t (cid:48) = −∞ G t − t (cid:48) q t (cid:48) − t (cid:88) t (cid:48) = −∞ G t + − t (cid:48) q t (cid:48) . (62)This equation is particularly interesting and it holds regardless the structure of the NT’s trades auto-covariance.Let us give a first example of how the above equation can be used in order to derive the result aboutnon correlated NT’s trades. The camouflage is exact in this case, so the excess demands are uncorre-lated, i.e., the l.h.s. of the above equation is zero, then we can see that G decays itself exponentiallywith the dividends time-scale. This is precisely what happens if the NT are not correlated, where thepropagator is given by Eq. (54).ii) In the following we deal with the case of arbitrary Markovian NT’s trades process. Using theexpression of the general forecast matrix of a Gaussian process with zero mean, we can rewrite Eq. (62),as (cid:2) ( ˜ Ω ) − (cid:3) − ( ˜ Ω ) − t + − t (cid:48) = ˜ G t + − t (cid:48) − α µ ˜ G t − t (cid:48) , ˜ G τ = G τ / G . (63)24ince we found that in generic situations an approximate camouflage relation holds, we know thatthe structure of the excess demand ACF matrix is given by Eq. (28). The inverse of the excess demandACF can be computed, and it is given by: ( ˜ Ω ) − = ω − (cid:112) ω − b α NT , ω = ˜ b + + ˜ b α − α ˜ b α NT , (64)and ( ˜ Ω ) − t + − t (cid:48) = − α NT ˜ b (cid:0) − ( + ˜ b − α )( ˜ Ω ) − (cid:1)(cid:2) ( ˜ Ω ) − α NT ˜ b (cid:3) t − t (cid:48) . (65)Then, we can rewrite Eq. (63) as ˜ G t + − t (cid:48) = α µ ˜ G t − t (cid:48) + P ρ t − t (cid:48) , (66)where we defined P = − α NT ˜ b (cid:0) − ( + ˜ b − α )( ˜ Ω ) − (cid:1)(cid:2) ( ˜ Ω ) − (cid:3) − , ρ = ( ˜ Ω ) − α NT ˜ b . (67)The solution of Eq. (66) is Eq. (34) introduced in the main text. Moreover the second equation inEqs. (67) gives Eq. (35). C.2 Solving the ansatz
In this appendix we present the calculations which allowed us to obtain the results presented in thefigures of Secs. 6.2, 6.3 and 6.4.From the expression of the propagator given by Eq. (34), one is able to derive the inverse of thesymmetrized propagator, which is given by ( ˜ G sym ) − t , t (cid:48) = Γ γ t − t (cid:48) + Γ γ t − t (cid:48) + δ ( t − t (cid:48) ) , (68)where Γ and Γ are the solution of the following set of equations: Γ α µ α µ − γ + Γ α µ α µ − γ + = Γ ρρ − γ + Γ ρρ − γ + =
0, (69)whereas γ and γ are the two real positive solution of the equation below: α µ − α NT α µ − ρ (cid:18) − α µ γ − α µ α µ − γ (cid:19) + (cid:18) − α µ − α NT α µ − ρ (cid:19)(cid:129) − ργ − ρρ − γ (cid:139) + =
0. (70)With the explicit expression of G s ym given above one is able to calculate the IT’s demand Kernelsgiven by Eqs. (24). These are given by R t − t (cid:48) = − α t − t (cid:48) α µ − α NT α µ − ρ (cid:18) Γ − γ α µ + Γ − γ α µ + (cid:19) − ρ t − t (cid:48) (cid:18) − α µ − α NT α µ − ρ (cid:19)(cid:18) Γ − γ α µ + Γ − γ α µ + (cid:19) R NT t − t (cid:48) = δ t (cid:48) − t R NT R µ t − t (cid:48) = δ t (cid:48) − t R µ (71)25here R NT = − α NT (cid:20) α µ − α NT α µ − ρ (cid:18) Γ ( − α µ γ )( − α NT γ ) + Γ ( − α µ γ )( − α NT γ ) (cid:19) + (cid:18) − α µ − α NT α µ − ρ (cid:19)(cid:129) Γ ( − α NT γ )( − ργ ) + Γ ( − α NT γ )( − ργ ) (cid:139) + (cid:21) , R µ = α µ G ( − α µ ) (cid:18) Γ − γ α µ + Γ − γ α µ + (cid:19) . (72)Moreover, by a careful inspection of previous formulas and numerical solver results of Eq. (25) in themarkovian case, one realize that the following property holds: R µ = (cid:118)(cid:117)(cid:116) Ω NT0 Ξ µ (cid:198) ( R NT ) + α NT R NT +
1. (73)From the equation above one is able to deduce the expression of G , by inverting the previous equationfor R µ .Finally, imposing the break even condition per trade of the MM given by Eq. (14), one is able toderive the following identity: Ω = Ξ µ ( R µ ) α µ ργ γ (cid:18) ˜ b + α µ − α NT α µ − ρ − α µ α NT + (cid:18) − α µ − α NT α µ − ρ (cid:19) − α NT ρ (cid:19) . (74)In order to close the ansatz on itself we have to compute the total order flow ACF. To do this, we needto calculate the first row of the inverse ( I − RL ) − which appear in Eq. (10). This is given by { ( I − RL ) − } t − t (cid:48) = { ( G s ym ) − } t , t (cid:48) { ( G s ym ) − } t , t = α µ ργ γ { ˜ G s ym } t − t (cid:48) . (75)The explicit expression of the excess demand at time t is given by q t = α µ ργ γ (cid:168)(cid:150) q NT t + t (cid:88) t (cid:48) = −∞ (cid:128) Γ γ t − t (cid:48) + Γ γ t − t (cid:48) (cid:138) q NT t (cid:48) (cid:153) + R NT (cid:150) q NT t − + t − (cid:88) t (cid:48) = −∞ (cid:128) Γ γ t − t (cid:48) − + Γ γ t − t (cid:48) − (cid:138) q NT t (cid:48) (cid:153) + R µ (cid:150) µ t − + t − (cid:88) t (cid:48) = −∞ (cid:128) Γ γ t − t (cid:48) − + Γ γ t − t (cid:48) − (cid:138) µ t (cid:48) (cid:153)(cid:171) . (76)With this equation one is able to compute explicitly the excess demand ACF. In particular, by com-paring the lag-0 term of it with the functional form given in Eq. (28) and using Eqs. (35) and (74) oneis able to compute an implicit very complicated equation for ρ , fixing completely the ansatz given byEq. (34).The figures presented in Sec. 6 have been obtained by fitting the result of the numerical solver withEq. (34), obtaining numerical values for ρ which have been cross-validated using the aformentionedanalytical implicit equation for ρρ