A Mean Field Game of Portfolio Trading and Its Consequences On Perceived Correlations
aa r X i v : . [ q -f i n . T R ] J a n A MEAN FIELD GAME OF PORTFOLIO TRADING AND ITS CONSEQUENCESON PERCEIVED CORRELATIONS
CHARLES-ALBERT LEHALLE AND CHARAFEDDINE MOUZOUNI
Version: February 27, 2019 A BSTRACT . This paper goes beyond the optimal trading Mean Field Game model intro-duced by Pierre Cardaliaguet and Charles-Albert Lehalle in [13]. It starts by extending itto portfolios of correlated instruments. This leads to several original contributions: firstthat hedging strategies naturally stem from optimal liquidation schemes on portfolios.Second we show the influence of trading flows on naive estimates of intraday volatilityand correlations. Focussing on this important relation, we exhibit a closed form formulaexpressing standard estimates of correlations as a function of the underlying correlationsand the initial imbalance of large orders, via the optimal flows of our mean field game be-tween traders. To support our theoretical findings, we use a real dataset of 176 US stocksfrom January to December 2014 sampled every 5 minutes to analyze the influence of thedaily flows on the observed correlations. Finally, we propose a toy model based approachto calibrate our MFG model on data.
1. I
NTRODUCTION
Optimal liquidation emerged as an academic field with two seminal papers: one [5]focussed on the balance between trading fast (to minimize the uncertainty of the obtainedprice) and trading slow (to minimize the “ market impact ”, i.e. the detrimental influence ofthe trading pressure on price moves) for one representative instrument; while the other[10] focussed on a portfolio of tradable instruments, shedding light on the interplay withcorrelations of price returns and market impact. The last twenty years have seen a lotof proposals to sophisticate the single instrument case (see these reference books [12,17, 23] for typical models and references) but very few on extending it to portfolios ofmultiple assets (with the notable exception of [25]). Moreover, the usual framework foroptimal execution is the one of one large privileged agent facing a “mean-field” or a“background noise” made of the sum of behaviours of other market participants, and
Mathematics Subject Classification.
Key words and phrases. mean field games, market microstructure, crowding, multi-asset portfolio, optimaltrading, optimal stochastic control.
Acknowledgement.
C. Mouzouni was supported by LABEX MI- LYON (ANR-10-LABX-0070) of Universit´ede Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French Na-tional Research Agency (ANR), and partially supported by project (ANR-16-CE40-0015-01) on Mean FieldGames. Authors would like to thank Pierre Cardaliaguet for a careful reading of large parts of this paper, andJean-Philippe Bouchaud to have insisted on the fact that results of [13] should be observable using intradaydata. academic literature seldom tackles the strategic interaction of many market participantsseeking to execute large orders.More recently, game theory has been introduced in this field. First around cases withfew agents, like in [33], and then by [13, 20, 28] relying on Mean Field Games (MFG)to get rid of the combinatorial complexity of games with few players, considering a lotof agents, such that their aggregated behaviour reduces to a “ anonymous mean field ofliquidity ”, shared by all of them.In this paper, we clearly start within the framework and results obtained by [13] andextend them to the case of a portfolio of tradable instruments. Our agents are the sameas in this paper: optimal traders seeking to buy and sell positions given at the start ofthe day. That for, they rely on the stochastic control problem well defined for one instru-ment in [17], which result turns to be deterministic because of its linear-quadratic nature:minimize the cost of the trading under risk-averse conditions and a terminal cost. Thisframework can be compared to the one used by [10] in their section on portfolio, with adiagonal matrix for the market impact, and in a game played by a continuum of agents.Note that in all these papers, including ours, the time scale is large enough to not takeinto account orderbook dynamics, and small enough to be used by traders and deal-ing desks; our typical terminal time goes from one hour to several days, and time stepshave to be read in minutes. In their paper, Cardaliaguet and Lehalle have shown howa continuum of such agents with heterogenous preferences can emulate a mix of typicalbrokers (having a large risk aversion and terminal cost), and opportunistic traders (witha low risk aversion). It will be the same for us. But while their paper only addressesthe strategic behavior of investors on a one single financial instrument this one handlesthe case of a portfolio of correlated assets. In the real applications, a financial instrumentis rarely traded on its own; most investors construct diversified or hedged portfolios orindex trackers by simultaneously buying and selling a large number of assets.This has motivated the present work in which we introduce an extension of the initialCardaliaguet-Lehalle framework to the case of a multi-asset portfolio. On the one hand,this extension allows to cover a new type of trading strategies, such as Program Trading(executing large baskets of stocks), Arbitrage Strategies (which aims to benefit from dis-crepancies in the dynamics of two or more assets), Hedging Strategies (where a roundtrip on a second asset – typically a very liquid one – can be used to partially hedge theprice risk in the execution process of a given asset), and Index Tracking (i.e. following thecomposition given by a formula, like in factor investing, or simply following the marketcapitalization of a list of instruments). On the other hand, it enables us to understand thedependence structure between the market orders flows at “equilibrium”, and assess theirinfluence on standard estimates of the covariance (or correlation) matrix of asset returns.These questions were independently raised by some authors and studied in seldom em-pirical and theoretical works (see e.g. [9, 11, 18, 26, 32] and references therein).Following the seminal paper [13], we assume that the market impact is either instan-taneous or permanent, and that the public prices – of all assets – are influenced by thepermanent market impact of all market participants. Conversely, since the agents are af-fected by the public prices, they aim to anticipate the “market mean field” (i.e. the markettrend due to the market impact of the mean field of all agents) by using all the information
MEAN FIELD GAME OF PORTFOLIO TRADING 3 they have in order to minimize their exposition to the other agents’ impact. As explainedin [13] this leads to a Nash equilibrium configuration of MFG type, in which all agentsanticipate the average trading speed of the population and adjust their execution accord-ingly. We refer the reader to Section 2 for a more detailed explanation of the Mean FieldGame model. In the context of a MFG with multi-asset portfolio, the strategic interactionbetween the agents during a trading day leads to a non-trivial relationship between theassets’ order flows, which in turn generates a non-trivial impact on the intraday covari-ance (or correlation) matrix of asset returns. In Section 3, we provide an exact formulafor the excess covariance matrix of returns that is endogenously generated by the tradingactivity, and we show that the magnitude of this effect is more significant when the mar-ket impact is large. This means for an highly crowded market, illiquid products or largeinitial orders (cf. Section 3). These results can be related to the ones of [18], except that inthis paper we do not focus our attention on distressed sells only; we are able to capturethe influence of the usual variations of trading flows to deformations of the naive esti-mate of the covariance matrix of a portfolio of assets that are simultaneously traded. Wealso carry out several numerical simulations and apply our results in an empirical analy-sis which is conducted on a database of market data from January to December 2014 fora pool of 176 US stocks. At first, we exhibit the theoretical relation between the intradaycovariance matrix of net traded flows and the standard intraday covariance matrix is in-creasing, then we use this relation to estimate some parameters of our model, includingthe market impact coefficients (cf. Section 3). Next, we normalize the covariance matrixof returns to compute the intraday median diagonal pattern (across diagonal terms), andthe intraday median off-diagonal pattern (across off-diagonal terms) (cf. Section 3.3), as away of characterizing the typical intraday evolution for diagonal and off-diagonal terms.It allows us to obtain empirically the well-known intraday pattern of volatility that is inline with our model, and we show that it flattens out as the typical size of transactionsdiminishes. In such a case the empirical volatility is close to its “fundamental” value (cf.Figure 4). Finally, we propose a toy model based approach to calibrate our MFG modelon data.This paper is structured as follows: in Section 2 we formulate the problem of optimalexecution of a multi-asset portfolio inside a Mean Field Game. We derive a MFG systemof PDEs and prove uniqueness of solutions to that system for a general Hamiltonianfunction. Then we construct a regular solution in the quadratic framework, which willbe considered throughout the rest of the paper. Next, we provide a convenient numericalscheme to compute the solution of the MFG system, and present several examples of anagent’s optimal trading path, and the average trading path of the population. Section 3 isdevoted to the analysis of the crowd’s trading impact on the intraday covariance matrixof returns. At the MFG equilibrium configuration, we derive a formula for the impactof assets’ order flows on the dependence structure of asset returns. Next, we carry outnumerical simulations to illustrate this fact, and apply our results inan empirical analysison a pool of
US stocks.
CHARLES-ALBERT LEHALLE AND CHARAFEDDINE MOUZOUNI
2. O
PTIMAL P ORTFOLIO T RADING W ITHIN T HE C ROWD
The Mean Field Game Model.
Consider a continuum of investors (agents), whichare indexed by a parameter a . Each agent has to trade a portfolio corresponding to in-structions given by a portfolio manager. Think about a continuum of brokers or dealingdesks executing large orders given by their clients. The portfolios are made of desiredpositions in a universe of d different stocks (or any financial assets). The initial positionof any agent a is denoted by q a := ( q a , ..., q d , a ) . For any i , when the initial inven-tory q i , a is positive, it means the agent has to sell this number of shares (or contracts)whereas when it is negative, the agent has to sell this amount. Given a common horizon T > , we suppose that all the investors have to sell or buy within the trading period [ T ] . This means the agent has to sell this number of shares (or contracts) whereas whenit is negative, the agent has to buy this amount.The intraday position of each investor a is modeled by a R d -valued process ( q at ) t ∈ [ T ] which has the following dynamics: d q at = v at d t , q a ( ) = q a . The investor controls its trading speed ( v at ) t := ( v at , ..., v d , at ) t through time, in orderto achieve its trading goal. Following the standard optimal liquidation literature, weassume that, for each stock, the dynamics of the mid-price can be written as:(2.1) d S it = σ i d W it + α i µ it d t , i =
1, ..., d ; where σ i > is the arithmetic volatility of the i th stock, and α , ..., α d are nonnegativescalars modeling the magnitude of the permanent market impact. Here ( W t , ..., W dt ) t > are d correlated Wiener processes, and the process ( µ t ) t ∈ [ T ] := ( µ t , ..., µ dt ) t correspondsto the average trading speed of all investors across the portfolio of assets. Throughout,we shall denote by Σ the covariance matrix of the d -dimensional process ( W t ) t ∈ [ T ] :=( σ W t , ..., σ d W dt ) t ∈ [ T ] and suppose that Σ is not singular. The performance of any investor a is related to the amount of cash generated through-out the trading process. Given the price vector ( S t ) t ∈ [ T ] := ( S t , ..., S dt ) t ∈ [ T ] , the amountof cash ( X at ) t ∈ [ T ] on the account of the trader a is given by: X at = − Z t v as · S s d s − d X i = Z t V i L i v i , as V i ! d s , where the positive scalars V , ..., V d denotes the magnitude of daily market liquidity (inpractice the average volume traded each day can be used as a proxy for this parameter)of each asset. Here L , ..., L d are the execution cost functions (similar to the ones of [25]),modelling the instantaneous component of market impact, which takes part in the aver-age cost of trading. The family of functions L i : R → R are assumed to fulfil the followingset of assumptions: • L i ( ) = ; • L i is strictly convex and nonnegative; • L i is asymptotically super-linear, i.e. lim | p | → + ∞ L i ( p ) | p | = + ∞ . MEAN FIELD GAME OF PORTFOLIO TRADING 5
The initial Cardaliaguet-Lehalle model [13], corresponds to d = , and a quadraticliquidity function of the form L ( p ) = κ | p | .In this paper, we consider a reward function that is similar to [13], and correspondingto Implementation Shortfall (IS) orders . In this specific case the reward function of anyinvestor a is given by:(2.2) U a ( t , x , s , q ; µ ) := sup v E x , s , q X aT + q aT · ( S T − A a q aT ) − γ a Z Tt q as · Σ q as d s ! , where A a := diag ( A a , ..., A ad ) , A ai > , and γ a is a non-negative scalar which quantifiesthe investor’s risk aversion. That is when γ a = the investor is indifferent about holdinginventories through time, while when γ a is large the investor attempt to liquidate as fastas possible. The quadratic term q aT · ( S T − A a q aT ) penalizes non-zero terminal inventories.One should note that the expression of the reward function (2.2) is derived by consideringthat agents are risk-averse with CARA utility function. We omit the details and refer thereader to [23, Chapter 5].The Hamilton-Jacobi equation associated to (2.2) is = ∂ t U a − γ a q · Σ q +
12 Tr (cid:0) ΣD s U a (cid:1) + A µ · ∇ s U a + sup v (cid:14) v · ∇ q U a − v · s + d X i = V i L i (cid:18) v i V i (cid:19)! ∇ x U a (cid:15) , with the terminal condition U a ( T , x , s , q ; µ ) = x + q · ( s − A a q ) . In all this paper we set A := diag ( α , ..., α d ) . Due to the simplifications that we willobtain afterwards, we suppose that µ = ( µ t ) t ∈ [ T ] is a deterministic process, so that theHJB equation above is deterministic. When µ is a random process, that is adapted tothe natural filtration of ( W t ) t ∈ [ T ] , we obtain a stochastic backward HJB equation whichrequires a specific treatment (cf. [14]).Following the approach of [13], we consider the following ersatz: U a ( t , x , s , q ; µ ) = x + q · s + u a ( t , q ; µ ) , which entails the following HJB equation for u a :(2.3) γ a q · Σ q = ∂ t u a + A µ · q + sup v (cid:14) v · ∇ q u a − d X i = V i L i (cid:18) v i V i (cid:19) (cid:15) , endowed with the terminal condition: u aT = − A a q · q . For any i =
1, ..., d , let H i be the Legendre-Fenchel transform of the function L i that isgiven by H i ( p ) := sup ρ pρ − L i ( ρ ) . CHARLES-ALBERT LEHALLE AND CHARAFEDDINE MOUZOUNI
Since the maps ( L i ) i d are strictly convex, ( H i ) i d are functions of class C , and theoptimal feedback strategies associated to (2.3) are given by v i , a ( t , q ) := V i ˙ H i ( ∂ q i u a ( t , q )) , where ˙ H i denotes the first derivative of H i . Therefore, the Mean Field Game systemassociated to the above problem reads:(2.4) γ a q · Σ q = ∂ t u a + A µ · q + d X i = V i H i ( ∂ q i u a ( t , q )) ∂ t m + d X i = V i ∂ q i (cid:16) m ˙ H i ( ∂ q i u a ( t , q )) (cid:17) = µ it = Z ( q , a ) V i ˙ H i ( ∂ q i u a ( t , q )) m ( t , d q , d a ) m (
0, d q , d a ) = m ( d q , d a ) , u aT = − A a q · q . The Mean Field Game system (2.4) describes a Nash equilibrium configuration, with in-finitely many well-informed market investors: any individual player anticipates the rightaverage trading flow on the trading period [ T ] , and computes his optimal strategy ac-cordingly. Observe that we make a strong assumption by supposing that the consideredgroup of investors has a precise knowledge of market mean field. In reality this knowl-edge is only partial or approximate.Well-posedness for system (2.4) is investigated in [13] within the general frameworkof Mean Field Games of Controls . In this work, we provide simpler arguments to deal withthe specific cases of our study. We shall suppose that ( H i ) i d are of class C and satisfythe following condition:(2.5) ∀ i =
1, ..., d , ∀ p ∈ R , C − ¨ H i ( p ) C , for some C > , and m is a probability density with a finite second order moment . More-over, we suppose that the investors’ index varies in a closed subset D ⊂ R .We say that ( u a , m ) a ∈ D is a solution to the MFG system (2.4) if the following hold: • u a ∈ C ([ T ] × R ) , for a.e a ∈ D , and m in C ([ T ] ; L ( R × D )) ; • the equation for u a holds in the classical sens, while the equation for m holds inthe sense of distribution; • for any t ∈ [ T ] ,(2.6) Z R × D | q | d m ( t , d q , d a ) < ∞ , and | ∇ q u a ( t , q ) | C ( + | q | ) , for some C > .Let us start with the following remark on the uniqueness of solutions to (2.4). Proposition 2.1.
Under the above assumptions, system (2.4) has at most one solution.
MEAN FIELD GAME OF PORTFOLIO TRADING 7
Proof.
Let ( u a , m ) a ∈ D and ( u a , m ) a ∈ D be two solutions to (2.4), and set ¯ u a := u a − u a , ¯ m := m − m . At first, let us assume that m , m are smooth so that the computationsbelow holds. By using system (2.4), we have:(2.7) dd t Z ( q , a ) ¯ u a ¯ m = − Z ( q , a ) ¯ m (cid:14) d X i = V i ( H i ( ∂ q i u ) − H i ( ∂ q i u )) + A ( µ − µ ) · q (cid:15) − Z ( q , a ) ¯ u (cid:14) d X i = V i (cid:16) ∂ q i (cid:16) m ˙ H i ( ∂ q i u ) (cid:17) − ∂ q i (cid:16) m ˙ H i ( ∂ q i u ) (cid:17)(cid:17) (cid:15) , where µ , µ correspond respectively to ( u a , m ) a ∈ D and ( u a , m ) a ∈ D .On the one hand, note that Z ( q , a ) ¯ m A ( µ − µ ) · q =
12 dd t A ¯ E · ¯ E , where ¯ E ( t ) := Z ( q , a ) q d ¯ m ( t ) . This follows from dd t ¯ E = µ − µ , which is in turn obtained from system (2.4) after an integration by parts.On the other hand, by virtue of (2.5) we have d X i = V i Z (cid:16) ¯ m ( H i ( ∂ q i u ) − H i ( ∂ q i u )) − ∂ q i ¯ u (cid:16) m ˙ H i ( ∂ q i u ) − m ˙ H i ( ∂ q i u ) (cid:17)(cid:17) = − d X i = V i Z (cid:16) m (cid:16) H i ( ∂ q i u ) − H i ( ∂ q i u ) − ˙ H i ( ∂ q i u ) ∂ q i ( u − u ) (cid:17)(cid:17) − d X i = V i Z (cid:16) m (cid:16) H i ( ∂ q i u ) − H i ( ∂ q i u ) − ˙ H i ( ∂ q i u ) ∂ q i ( u − u ) (cid:17)(cid:17) − min i d V i Z ( q , a ) ( m + m ) C | ∇ q u − ∇ q u | . Therefore, (2.7) provides(2.8) min i d V i Z T Z ( q , a ) | ∇ q u ( s ) − ∇ q u ( s ) | d ( m + m ) d s + C A ¯ E ( T ) · ¯ E ( T ) = By using a standard regularization process, identity (2.8) holds true for any solutions ( u a , m ) a ∈ D and ( u a , m ) a ∈ D of (2.4). Thus, one can use this identity to deduce that ∇ q u ≡ ∇ q u on { m > } ∪ { m > } , so that m , m solve the same transport equation: ∂ t ν + d X i = V i ∂ q i (cid:16) ν ˙ H i ( ∂ q i u a ( t , q )) (cid:17) = ν t = = m . This entails m ≡ m and so u ≡ u , by virtue of our regularity assumptions. (cid:3) CHARLES-ALBERT LEHALLE AND CHARAFEDDINE MOUZOUNI
Quadratic Liquidity Functions.
In practice the liquidity function is often chosen asstrictly convex power function of the form: L ( p ) = η | p | + φ + ω | p | , with η , φ , ω > . Theadditional term ω | p | captures proportional costs such as the bid-ask spread, taxes, feespaid to brokers, trading venues and custodians[23]. The quadratic case ( φ = ) – that isalso considered in [13] – is particularly interesting because it induces some considerablesimplifications and allows to compute the solutions at a relatively low cost. Throughoutthe rest of this paper, we suppose that the liquidity functions take the following simpleform:(2.9) L i ( p ) = η i | p | where η i > i =
1, ..., d . Following the approach of [13], we start by setting ¯ m ( d a ) := R q m ( d q , d a ) . We shallsuppose that(2.10) ¯ m ( a ) = for a.e a ∈ D , and that investors do not change their preference parameter a over time. Thus, we alwayshave R q m ( t , d q , d a ) = ¯ m ( d a ) , so that we can disintegrate m into m ( t , d q , d a ) = m a ( t , d q ) ¯ m ( d a ) , where m a ( t , d q ) is a probability measure in q for ¯ m -almost any a . Let us now definethe following process which plays an important role in our analysis: E a ( t ) := Z q q m a ( t , d q ) ∀ t ∈ [ T ] , for a.e a ∈ D , and we shall denote by E a ,1 , ..., E a , d the components of E a . By virtue of the PDE satisfiedby m , observe that E a satisfies the following: ˙ E a ( t ) = Z q q ∂ t m a ( t , d q ) (2.11) = Z q (cid:18) V i η i ∂ q i u a ( t , q ) (cid:19) i d m a ( t , d q ) , so that(2.12) µ t = Z a ˙ E a ( t ) d ¯ m ( a ) . Due to the existence of linear and quadratic terms in the equation satisfied by u a , weexpect the solution to have the following form:(2.13) u a ( t , q ) = h a ( t ) + q ′ · H a ( t ) + q ′ · H a ( t ) · q where h a ( t ) is R -valued function, H a ( t ) := ( H ia ( t )) i d is R d -valued function, andthe map H a ( t ) := ( H i , ja ( t )) i , j d take values in the set of R d × d -symmetric matrices.Inserting (2.13) in the HJB equation of (2.4) and collecting like terms in q leads to the MEAN FIELD GAME OF PORTFOLIO TRADING 9 following coupled system of BODEs:(2.14) ˙ h a = − V H a · H a ˙ H a = − A µ − H a V H a ˙ H a = − H a VH a + γ a Σh a ( T ) = H a ( T ) = H a ( T ) = − A a , where V := diag (cid:16) V η , ..., V d η d (cid:17) . In order to solve completely (2.14) we need to know µ ,or the process ˙ E a thanks to (2.12). Thus, one needs an additional equation to completelysolve the problem.By virtue of (2.11), we have(2.15) ˙ E a = V H a + VH a E a . By combining this equation with system (2.14) one obtains the following FBODE:(2.16) ¨ E a = − VA Z a ˙ E a d ¯ m ( a ) + γ a V Σ E a E a ( ) = E a := Z q q m ( q , a ) / ¯ m ( a ) ˙ E a ( T ) + V A a E a ( T ) = This system is a generalized form of the one that is studied in [13], and summarizes thewhole market mean field. Observe that the permanent market impact acts as a frictionterm while the market risk terms act as a pushing force toward a faster execution. Theinvestors heterogeneity is taken into account in the first derivative term, which meansthat the contribution of all the market participants to the average trading flow is alreadyanticipated by all agents.System (2.16) is our starting point to solve the MFG system (2.4) in the quadratic case.Due to the forward-backward structure of system (2.16), we need a smallness conditionon A in order to construct a solution. This assumption is also considered in [13], and isnot problematic from a modeling standpoint since | A | is generally small in applications(cf. Section 3.3). Let us present the construction of solutions to system (2.16). Proposition 2.2.
Suppose that A a , γ a ∈ L ∞ ( D ) , then there exists α > such that, for | A | α , the following hold:(i) there exists a unique process E a in L m ( D ; C ([ T ])) which solves system (2.16) ;(ii) there exists a constant C > , such that (2.17) sup w T | µ w | C (cid:18) + Z a | E a | d ¯ m (cid:19) e C T , where ( µ t ) t ∈ [ T ] is given by (2.12) .Proof. At first, note that the solution H a to the matrix Riccati equation in (2.14) exists on [ T ] , is unique, depends only on data, and satisfies (see e.g. [29])(2.18) − A a − T γ a Σ H a where the order in the above inequality should be understood in the sense of positivesymmetric matrices. Moreover, note that Σ V and V Σ are both diagonalizable with non-negative eigenvalues. Thus by using the ODE satisfied by H a , we know that H a V and VH a are both diagonalizable with a constant change of basis matrix. In particular, it holdsthat(2.19) (cid:20) H a ( t ) V , Z wt H a ( u ) V d u (cid:21) = (cid:20) VH a ( t ) , Z wt VH a ( u ) d u (cid:21) = for any t , w T , where the symbol [ B , A ] denotes the Lie Bracket: [ B , A ] = BA − AB .Given H a , we aim to construct ˙ E a in L m ( D ; C ([ T ])) by solving a fixed point relation,and then deduce E a . For that purpose, we start by deriving a fixed point relation for ˙ E a .By virtue of (2.19), observe that any solution E a to (2.16) fulfills (2.15) with (see e.g. [31]) H a ( t ) = Z Tt exp (cid:12)Z wt H a ( s ) V d s (cid:13) A Z a ˙ E a ( w ) d ¯ m ( a ) d w , so that E a ( t ) = exp (cid:12)Z t VH a ( w ) d w (cid:13) E a + V Z t exp (cid:12)Z tτ VH a ( w ) d w (cid:13) Z Tτ exp (cid:12)Z wτ H a ( s ) V d s (cid:13) A Z a ˙ E a ( w ) d ¯ m ( a ) d w d τ . By combining this relation with (2.15), we deduce that ˙ E a satisfies the following fixedpoint relation:(2.20) x a ( t ) = Φ A ( x a )( t ) := VH a ( t ) exp (cid:12)Z t VH a ( w ) d w (cid:13) E a + VH a ( t ) V Z t exp (cid:12)Z tτ VH a ( w ) d w (cid:13) Z Tτ exp (cid:12)Z wτ H a ( s ) V d s (cid:13) A Z a x a ( w ) d ¯ m ( a ) d w d τ + V Z Tt exp (cid:12)Z wt H a ( w ) V d w (cid:13) A Z a x a ( w ) d ¯ m ( a ) d w . Conversely, one checks that if x a is a solution to the fixed point relation (2.20), for a.e. a ∈ D , then E a ( t ) = E a + R t x a ( s ) d s is a solution to system (2.16).To solve the fixed point relation (2.20), one just uses Banach fixed point Theorem on Φ A : X → X , where X := L m ( D ; C ([ T ])) . It is clear that Φ A is a contraction for | A | smallenough: indeed, given x , y ∈ X , it holds that: | Φ A ( x a )( t ) − Φ A ( y a )( t ) | C | A | k x − y k X where C > depends only on T , k γ k ∞ , k A k ∞ , | V | and | Σ | . Thus, given the solution x a to(2.20), the function E a ( t ) = E a + R t x a ( s ) d s solves (2.16), and belongs to L m ( D ; C ([ T ])) given that m have a finite first order moment. Estimate (2.17) ensues from Gr ¨onwall’sLemma. (cid:3) We are now in position to solve the MFG system (2.4) in the case of quadratic liquidityfunctions (2.9).
MEAN FIELD GAME OF PORTFOLIO TRADING 11
Theorem 2.3.
Under (2.9) , (2.10) , and assumptions of Proposition 2.2, the Mean Field Gamesystem (2.4) has a unique solution.Proof. Since (2.16) is solvable thanks to Proposition 2.2, we can now solve completelysystem (2.14) and deduce u a ( t , q ; µ ) thanks to (2.13). In fact, owing to (2.19) we knowthat (cf. [31]): H a ( t ) = Z Tt exp (cid:12)Z wt H a ( s ) V d s (cid:13) A µ w d wh a ( t ) = Z Tt V H a ( w ) · H a ( w ) d w , so that the function u a ( t , q ; µ ) , that is given by (2.13), is C ([ T ] × R ) . Furthermore, byvirtue of (2.17)-(2.18), note that(2.21) | ∇ q u a ( t , q ) | C ( + | q | ) for some constant C > which depends only on T and data.Now, as u a is regular and satisfies (2.21), we know that the transport equation ∂ t m a + d X i = V i ∂ q i ( m a ∂ q i u a ( t , q )) = m a (
0, d q ) = m ( d q , d a ) / ¯ m ( a ) has a unique weak solution m a ∈ C ([ T ] ; L ( R )) for a.e a ∈ D , so that m := m a ¯ m solves, in the weak sense, the following Cauchy problem: ∂ t m + d X i = V i ∂ q i ( m∂ q i u a ( t , q )) = m (
0, d q , d a ) = m ( d q , d a ) . In addition, one easily checks that m belongs to C ([ T ] ; L ( R × D )) .By invoking the uniqueness of solutions to (2.16), we have E a ( t ) = Z q q m a ( t , q ) d q for a.e a ∈ D . Thus through the same computations as in (2.11) we obtain µ it = Z ( q , a ) V i ∂ q i u a ( t , q ) m a ( t , q ) ¯ m ( a ) d a d q , i =
1, ..., d , so that ( u a , m ) a ∈ D solves the MFG system (2.4).By virtue of Proposition 2.1, any constructed solution is unique. So to conclude theproof, it remains to show that: Z R × D | q | m ( t , q , a ) d q d a < ∞ . For that purpose, let us set Ψ ( t ) := R R × D | q | m ( t , q , a ) d q d a . After differentiating Ψ andintegrating by parts, we obtain the following ODE that is satisfied by Ψ : Ψ ( t ) = Ψ ( ) + Z t Z a H a ( w ) · E a ( w ) ¯ m ( d a ) d w + Z t Z a ( H a ( w ) q · q ) m ( t , d q , d a ) d w , so that | Ψ ( t ) | | Ψ ( ) | + C (cid:14) sup w T k E a ( w ) k L m + Z t | Ψ ( w ) | d w (cid:15) holds thanks to (2.17)-(2.18). Hence, as m has a finite second order moment, we deducefrom Gr ¨onwall’s Lemma that for any t ∈ [ T ] Z R × D | q | m ( t , q , a ) d q d a < ∞ , which in turn entails the desired result. (cid:3) Stylized Facts & Numerical Simulations.
Let us now comment our results andhighlight several stylized facts of the system. By virtue of (2.13), the optimal tradingspeed v ∗ a is given by: v ∗ a ( t , q ) = VH a ( t ) q + V H a ( t ) (2.22) = VH a ( t ) q + V Z Tt exp (cid:12)Z wt H a ( s ) V d s (cid:13) A µ w d w =: v ∗ a ( t , q ) + v ∗ a ( t ; µ ) . The above expression shows that the optimal trading speed is divided into two distinctparts v ∗ a , v ∗ a . The first part v ∗ a corresponds to the classical Almgren-Chriss solution inthe case of a complexe portfolio (cf. [23]). The second part v ∗ a adjusts the speed basedon the anticipated future average trading on the remainder of the trading window [ t , T ] .Since the matrix H a is negative, note that the strategy gives more weight to the currentexpected average trading. Moreover, the contribution of the corrective term decreases aswe approach the end of the trading horizon. The correction term aims to take advantageof the anticipated market mean field.Let us set(2.23) G a ( t , w ) := exp (cid:12)Z wt H a ( s ) V d s (cid:13) A . Note that the matrix G a is not necessarily symmetric and could have a different structurethan H a . In view of the market price dynamics, the trading speed expression shows thatan action of an individual investor or trader on asset i could have a direct impact on theprice of asset j , at least when the two assets are fundamentally correlated, i.e. Σ i , j = .This phenomenon of cross impact is related to the fact that other traders already anticipatesthe market mean field and aim to take advantage from that information, especially whenasset j is more liquid than asset i (or vice versa). Thus, if an investor is trading as thecrowd is expecting her to trade, then she is more likely to get a “cross-impact” through theaction of the other traders. This fact is empirically addressed in [9, 26].Another expression of the optimal trading speed can also be derived thanks to (2.15).In fact, we have that:(2.24) v ∗ a ( t , q ) = ˙ E a + VH a ( t )( q − E a ) . MEAN FIELD GAME OF PORTFOLIO TRADING 13
The above formulation shows that an individual investor should follow the market meanfield but with a correction term which depends on the situation of her inventory relativeto the population average inventory.In order to simplify the presentation, we ignore from now on investors heterogeneityand assume that market participants have identical preferences. Under this assumption,system (2.16) simply reads:(2.25) (cid:14) ¨ E = − VA ˙ E + γ V Σ EE ( ) = E , ˙ E ( T ) + V AE ( T ) = Given a discretization step δt = N − , the solution of (2.25) is approached by a sequence ( x k , y k ) k N according to the following implicit scheme: x = E x k − x k − − δty k − = k =
1, ..., Ny k − y k − − δt ( γ V Σx k − VA y k ) = k =
1, ..., N V A x N + y N = Hence, computing an approximate solution to system (2.25) reduces to solving a straight-forward linear system. One checks that under conditions of Proposition 2.2, the abovenumerical scheme converges and is stable.Now, we can present some examples by using the above numerical method. We con-sider a portfolio containing three assets (Asset 1, Asset 2, Asset 3) with the followingcharacteristics: • σ = σ = day − / . share − , σ = day − / . share − ; • V =
2, 000, 000 share . day − , V = V =
5, 000, 000 share . day − ; • η = η = share − , η = share − , A = A = day − . share − ; • α = α = × − $. share − , α = × − $. share − .In Figure 1(a)-1(d), we consider a market with the initial average inventories E = , E =
50, 000 , and E = −
25, 000 shares, for Asset 1, Asset 2, and Asset 3 respec-tively. In this example, we suppose that the correlation between the price increments ofAsset 1 and Asset 2 is 80 %, and we set γ = × − $ − except for Figure 1(c).Figure 1(a) shows that changing the permanent market impact prefactors ( α k ) k has a significant influence on the average execution speed. This fact was pointed out in[13], and is essentially related to the fact that the higher the permanent market impactparameter the more the anticipated influence of the other market participants becomeimportant. Namely, when α k is large, traders anticipate a more significant pressure onthe price of Asset k , and adjust their trading speed. On the other hand, dynamics of Asset2 shows that the higher the market liquidity the faster is the execution. This is expectedsince the more liquid the faster assets are traded. Finally, dynamics of Asset 3 shows thattraders accelerate their execution on volatile asset. It corresponds to a natural reactiondue to risk aversion; a trader will try to reduce his exposure to the more risky (hencevolatile) assets in priority. Time -4-2024681012 A v e r age i n v en t o r y Asset 1 ( ref. case )Asset 2 ( ref. case )Asset 3 ( ref. case )Asset 1 ( smaller )Asset 2 ( bigger V )Asset 3 ( bigger ) (a) Market mean field with different parameters Time -4-2024681012 I n v en t o r y Asset 1 ( market trend )Asset 2 ( market trend )Asset 3 ( market trend )Asset 1 ( an individual investor )Asset 2 ( an individual investor )Asset 3 ( an individual investor ) (b) Optimal trading of an individual investor with: q =
40, 000 , q = , and q = Time -4-2024681012 A v e r age i n v en t o r y Asset 1 ( = 5.10 -3 )Asset 2 ( = 5.10 -3 )Asset 3 ( = 5.10 -3 )Asset 1 ( = 5.10 -2 )Asset 2 ( = 5.10 -2 )Asset 3 ( = 5.10 -2 ) (c) Market mean field with high risk aversion Time -4-2024681012 I n v en t o r y Asset 1 ( market trend )Asset 2 ( market trend )Asset 3 ( market trend )Asset 1 ( an individual investor )Asset 2 ( an individual investor )Asset 3 ( an individual investor ) (d) Optimal trading of an individual investor with: q = , and q = q = F IGURE
1. Simulated examples of the dynamics of E and optimal tradingcurves of an individual investor. The dashed lines in Figure 1(a) corre-spond to: α = × − $. share − , V =
7, 000, 000 share . day − , and σ = day − / . share − .Figure 1(c) illustrates the behavior of the crowd of investors with an increasing riskaversion (higher γ ). In the two presented scenarios, one can observe that Asset 2 is liqui-dated very quickly, then a short position is built (around t = for γ = × − $ − )and it is finally progressively unwound. This exhibits the emergence of a Hedging Strat-egy: indeed, since Asset 1 and Asset 2 are highly correlated, investors can slow downthe execution process for the less liquid asset (Asset 1) to reduce the transaction costs, byusing the more liquid asset (Asset 2) to hedge the market risk associated to Asset 1. Thetrader has an incentive to use such a strategy as soon as the cost of the roundtrip in Asset2 is smaller than the corresponding reduction of the risk exposure (seen from its rewardfunction U a ( t , x , s , q ; µ ) defined by equality (2.2)). MEAN FIELD GAME OF PORTFOLIO TRADING 15
Now, we provide examples of individual players’ optimal strategies. We consider twoexamples: an individual investor with initial inventory q =
40, 000 , q = , and q = in Figure 1(b); and an individual investor with initial inventory q = ,and q = q = in Figure 1(d).In Figure 1(d) the considered investor starts from q = E . Hence, by virtue of (2.24)her liquidation curve follows exactly the market mean field. Moreover, the investor takesadvantage of the anticipated evolution of the market by building favorable positions onAsset 2 and Asset 3: building a short (resp. a long) position on Asset 2 (resp. Asset 3), andbuying (resp. selling) back in order to take advantage of price drop (resp. raise) inducedby the massive liquidation (resp. purchase). The trading strategies on Asset 2 and Asset3 are related to the term v ∗ in (2.22). This strategy can be described as a “LiquidityArbitrage Strategy”.Figure 1(b) shows two interesting facts: on the one hand, the individual player buildsa short position on Asset 1 after achieving her goal (complete liquidation) in order to takeadvantage of the market selling pressure; on the other hand, by taking into account themarket buying pressure on Asset 1, the investor slows down her liquidation to reduceexecution costs since she anticipates no sustainable price decline.3. T HE D EPENDENCE S TRUCTURE OF A SSET R ETURNS
The main purpose of this section is to analyze the impact of large transactions on theobserved covariance matrix between asset returns, by using the Mean Field Game frame-work of Section 2. For that purpose, we assume a simple model where a continuum ofplayers trade a portfolio of assets on each day, and where the initial distribution of in-ventories across the investors m changes randomly from one day to another accordingto some given law of probability. We assume that the price dynamics is given by (2.1),and we consider the problem of estimating the covariance matrix of asset returns givena large dataset of intraday observations of the price. For the sake of simplicity, we ignoreinvestors heterogeneity and assume that market participants have identical preferences.Next, we compare our findings with an empirical analysis on a pool of 176 US stockssampled every 5 minutes over year 2014 and calibrate our model to market data.Throughout this section, we denote by (cid:10) X (cid:11) the variance of X , and h X , Y i the covariancebetween X and Y , for any two random variables X , Y . Moreover, we will call a “bin” aslice of 5 minutes. We focused on continuous trading hours because the mechanism ofcall auctions (i.e. opening and closing auctions is specific). Since US markets open from9h30 to 16h, our database has 78 bins per day. They will be numbered from 1 to M andindexed by k .3.1. Estimation using Intraday Data.
We suppose that E is a random variable with agiven realization on each trading period [ T ] , where T = day (trading day); and weconsider the problem of estimating the covariance matrix of asset returns given the fol-lowing observations of the price: (cid:10) (cid:16) S nt , ..., S nt M (cid:17) , (cid:16) S nt , ..., S nt M (cid:17) , ...., (cid:16) S nt N ,1 , ..., S nt N , M (cid:17) (cid:11) , n =
1, ..., d where S nt ℓ , k is the price of asset n in bin k of day ℓ . We suppose that t ℓ ,1 = , t ℓ , M = T , forany ℓ N , and t ℓ , k = t ℓ ′ , k = t k for any k M , ℓ , ℓ ′ N .For simplicity, we suppose that the covariance matrix of asset returns between t k and t k + is estimated form data by using the following “naive” estimator :(3.1) C i , j [ t k , t k + ] := N − N X l = (cid:16) δS i , k , l − δS i , k (cid:17) (cid:16) δS j , k , l − δS j , k (cid:17) , where δS n , k , l = S nt l , k + − S nt l , k and δS n , k = N − P Nl = δS n , k , l , n = i , j . We define thecorrelation matrix as follows:(3.2) R i , j [ t k , t k + ] := C i , j [ t k , t k + ] (cid:16) C i , i [ t k , t k + ] C j , j [ t k , t k + ] (cid:17) / . Suppose that the price dynamics is given by (2.1), then the following proposition pro-vides an exact computation of C i , j [ t k , t k + ] . Proposition 3.1.
Assume that E is independent from the process ( W t ) t ∈ [ T ] , then for any k M − and i , j d , the following hold: (3.3) C i , j [ t k , t k + ] = ( t k + − t k ) Σ i , j + α i α j η i η j V i V j Λ i , jk + ǫ N , where ǫ N → as N → ∞ , Λ i , jk := X ℓ , ℓ ′ d D θ i , ℓk , θ j , ℓ ′ k E + X ℓ , ℓ ′ d D π i , ℓk , θ j , ℓ ′ k E + X ℓ , ℓ ′ d D θ i , ℓk , π j , ℓ ′ k E + X ℓ , ℓ ′ d D π i , ℓk , π j , ℓ ′ k E , and π n , ℓk := Z t k + t k H n , ℓ ( s ) E ℓ ( s ) d s , θ n , ℓk := Z t k + t k Z Ts G n , ℓ ( s , w ) µ ℓ ( w ) d w d s . Proof.
Use the exact expression of the price dynamics (2.1), the law of large numbers, andthe independence between E and ( W t ) t ∈ [ T ] to obtain:(3.4) C i , j [ t k , t k + ] = ǫ N + ( t k + − t k ) Σ i , j + α i α j ( N − ) N X l = Z t k + t k (cid:0) µ i , ls − ¯ µ is (cid:1) d s Z t k + t k (cid:16) µ j , ls ′ − ¯ µ js ′ (cid:17) d s ′ , where ¯ µ nu = N − P Nl = µ n , lu , and µ l , E l are respectively the realizations of µ , E in day l .Now, owing to (2.22)-(2.23), we know that µ lt = VH ( t ) E l ( t ) + V Z Tt G ( t , w ) µ lw d w =: ν l ( t ) + ν l ( t ) . Thus by setting ˜ ν n , lk := Z t k + t k ν n , l ( s ) − N − N X l = ν n , l ( s ) ! d s , n =
1, 2,
MEAN FIELD GAME OF PORTFOLIO TRADING 17 we deduce that Z t k + t k (cid:0) µ i , ℓs − ¯ µ is (cid:1) d s Z t k + t k (cid:16) µ j , ℓs ′ − ¯ µ js ′ (cid:17) d s ′ = (cid:16) ˜ ν l , ik + ˜ ν l , ik (cid:17) (cid:16) ˜ ν l , jk + ˜ ν l , jk (cid:17) . The desired result ensues by noting the existence of estimation noises ǫ N , ǫ N , ǫ N and ǫ N ,such that: ( N − ) − N X l = ˜ ν l , ik ˜ ν l , jk = η i η j V i V j X ℓ , ℓ ′ d D π i , ℓk , π j , ℓ ′ k E + ǫ N ; ( N − ) − N X l = ˜ ν l , ik ˜ ν l , jk = η i η j V i V j X ℓ , ℓ ′ d D θ i , ℓk , θ j , ℓ ′ k E + ǫ N ; ( N − ) − N X l = ˜ ν l , ik ˜ ν l , jk = η i η j V i V j X ℓ , ℓ ′ d D π i , ℓk , θ j , ℓ ′ k E + ǫ N ; ( N − ) − N X l = ˜ ν l , jk ˜ ν l , ik = η i η j V i V j X ℓ , ℓ ′ d D θ i , ℓk , π j , ℓ ′ k E + ǫ N . The proof is complete. (cid:3)
Remark . One can easily derive an analogous result for (cid:16) C i , j [ T ] (cid:17) i , j d . Namely, itholds that:(3.5) C i , j [ T ] = T Σ i , j + α i α j η i η j V i V j Λ i , j + ǫ N , where ǫ N → as N → ∞ , Λ i , j := X ℓ , ℓ ′ d D θ i , ℓ , θ j , ℓ ′ E + X ℓ , ℓ ′ d D π i , ℓ , θ j , ℓ ′ E + X ℓ , ℓ ′ d D θ i , ℓ , π j , ℓ ′ E , + X ℓ , ℓ ′ d D π i , ℓ , π j , ℓ ′ E , and π n , ℓ := Z T H n , ℓ ( s ) E ℓ ( s ) d s , θ n , ℓ := Z T Z Ts G n , ℓ ( s , w ) µ ℓ ( w ) d w d s . Identities (3.3) and (3.5) show that the realized covariance matrix is the sum of thefundamental covariance and an excess realized covariance matrix generated by the impactof the crowd of institutional investors’ trading strategies. Note on the one hand that thediagonal terms C i , i are always deviated from fundamentals because of the contributionof (cid:10) ( π i , i ) (cid:11) and (cid:10) ( θ i , i ) (cid:11) . On the other hand, since H and G inherit a structure similar to Σ , the excess of realized covariance in the off-diagonal terms is non-zero as soon as one –or both – of the conditions below is satisfied: • there exists i = j such that Σ i , j = ; • there exists i = j such that D E i , E j E = . Moreover, (3.3) and (3.5) show that the excess realized covariance can deviate signifi-cantly from fundamentals when: the market impact is large (high crowdedness), the consideredassets are highly liquid (small η i /V i ), the risk aversion coefficient γ is high, or when the standarddeviation of E is large . In addition, since the contribution of θ n , ℓk and π n , ℓk vanishes as weapproach the end of the trading horizon, observe that(3.6) C i , j [ t k , t k + ] ∼ ( t k + − t k ) Σ i , j , as t k + → T , which means that one converges to market fundamentals at the end of the trading period.This is due to the fact that, in our model, all traders have high enough risk aversions sothat their trading speeds go to zero close to the terminal time T .By virtue of (3.3), one can also explain the realized correlation matrix in terms of thefundamental correlations ρ i , j := Σ i , j / ( Σ i , i Σ j , j ) / . Namely, it holds that:(3.7) R i , j [ t k , t k + ] = ρ i , j ( t k + − t k ) Σ i , i Σ j , j C i , i [ t k , t k + ] C j , j [ t k , t k + ] / + α i α j η i η j Λ i , jk V i V j (cid:16) C i , i [ t k , t k + ] C j , j [ t k , t k + ] (cid:17) / + ǫ N =: ρ i , j A i , jk + B i , jk + ǫ N for any i < j d . This expression shows that the deviation of the realized correlationfrom fundamentals is a linear map. The numerator of the multiplicative part A i , jk doesnot depend on the off-diagonal terms of H while it is the case for the additive part B i , jk .3.2. Numerical Simulations.
In this part, we conduct several numerical experiments inorder to illustrate the impact of trading on the structure of the covariance matrix of assetreturns.We consider the example of Section 2.3 by choosing ρ = , ρ = and ρ = . For simplicity, we suppose that E is a centered Gaussian random vector with acovariance matrix Γ that is given by: Γ := λ . − − , where λ =
10, 000 share . We fix a time step δt = − day ( ∼ min ), set t k + − t k = δt ,and estimate (cid:16) C i , j [ t k , t k + δt ] (cid:17) i j k M − and (cid:16) R i , j [ t k , t k + δt ] (cid:17) i 10, 000 observations using the numerical method of Section 2.3.Figures 2(a)-2(d) show that the observed covariance and correlation matrices are sig-nificantly deviated from fundamentals and especially at the beginning of the trading day.Figures 2(b)-2(d) also illustrates the sensitivity of the deviation relative to the change ofthe standard deviation of initial inventories: as λ diminishes, the impact of trading islower and the covariance and correlation matrices converge toward fundamentals.On the other hand, we observe that the beginning of the trading period is dominatedby the dependence structure of initial inventories. This is due to the domination of the MEAN FIELD GAME OF PORTFOLIO TRADING 19 Time -0.200.20.40.60.81 A ss e t r e t u r n s c o rr e l a t i on R [t+ t]1,2 ( ref. case )R [t+ t]1,3 ( ref. case )R [t+ t]2,3 ( ref. case ) (a) Intraday correlation Time V o l a t ili t y C [t+ t]1,1 ( ref. case )C [t+ t]2,2 ( ref. case )C [t+ t]3,3 ( ref. case )C [t+ t]1,1 ( smaller )C [t+ t]2,2 ( smaller )C [t+ t]3,3 ( smaller ) (b) Intraday volatility for λ = and λ = Time -0.200.20.40.60.81 A ss e t r e t u r n s c o rr e l a t i on R [t+ t]1,2 ( smaller )R [t+ t]1,3 ( smaller )R [t+ t]2,3 ( smaller ) (c) Intraday correlation for λ = Time -0.0100.010.020.030.040.050.060.070.08 A ss e t r e t u r n s c o v a r i an c e C [t+ t]1,2 ( ref. case )C [t+ t]1,3 ( ref. case )C [t+ t]2,3 ( ref. case )C [t+ t]1,2 ( smaller )C [t+ t]1,3 ( smaller )C [t+ t]2,3 ( smaller ) (d) Intraday covariance for λ = and λ = F IGURE 2. Simulated examples of intraday covariance and correlationmatrices using (2.1).additive terms ( B i , jk ) i Now, we carry out an empirical analysis on a pool of d = stocks. The data consists of five-minute binned trades ( δt = min) and quotes infor-mation from January 2014 to December 2014, extracted from the primary market of eachstock (NYSE or NASDAQ). We only focus on the beginning of the continuous trading sessionremoving 30 min after the open and the last 90 min before the close, in order to avoid the partic-ularities of trading activity in these periods and target close strategies. Thus, the numberof days is N = and the number of bins per day is M = . Days will be labelled by l = 1, ..., N , bins by k = 1, ..., M , and for simplicity we note C i , jk instead of C i , j [ t k − δt , t k ] forany i , j d . Our goal is to empirically assess the the influence of trading activity on the intradaycovariance matrix of asset returns, and then compare the obtained models with our pre-vious theoretical observations. Given our analysis in Sections 3.1 and 3.2, we expect anexcess in the observed covariance matrix of asset returns and especially at the beginningof the trading period. Moreover, we expect the magnitude of this effect to be an increas-ing function of the typical size of market orders as it is noticed in Figures 2(b) and 2(d).3.3.1. Market Impact. Let us start by assessing the relationship between the intraday vari-ance of asset returns and the intraday variance of net exchanged flows ( F i , ik ) i d , thatis defined by: F i , ik := N − N X l = (cid:0) ν ik , l − ¯ ν ik (cid:1) (cid:0) ν ik , l − ¯ ν ik (cid:1) for any i d and k = 1, ..., M ; where ν ik , l is the net sum of exchanged volumesbetween t k − δt and t k for stock i in day l , and ¯ ν ik = N − P Nl = ν ik , l (i.e. ¯ ν ik is an estimateof the expectation of ν ik , l regardless of the day). As a by-product, we obtain estimatesfor the permanent market impact factors. Though ν does not represent exactly the samequantity as the variable µ of Section 3.1, both variables are an indicator of market orderflows and for simplicity we shall use ν as a proxy for µ . (a) GOOG (b) Histogram of Corr ( C , F ) F IGURE 3. Dependence structure between ( C i , ik ) k M and ( F i , ik ) k M .Figure 3(a) displays the relationship between ( C i , ik ) k M and ( F i , ik ) k M for GOOG . Figure 3(b) exhibits the histogram of corre-lations denoted Corr ( C , F ) .Figure 3(a) shows a strong positive correlation between ( C i , ik ) k M and ( F i , ik ) k M for GOOG . Figure 3(b) shows that this is true for almost all the stocks and reinforcesour findings in Sections 3.1 and 3.2. Furthermore, as (3.4) suggests, we suppose a linearrelationship between ( C i , ik ) k M and ( F i , ik ) k M ; thus for every i d we fit the MEAN FIELD GAME OF PORTFOLIO TRADING 21 following regression:(3.8) C i , i = ǫ + δt · Σ + α · F i , i where ǫ is the error term (assumed normal), the coefficient Σ is related to the “funda-mental” covariance matrix of asset returns and the square root of the coefficient α ishomogeneous to the market impact factor (cf. (3.4)). In Table 1 we show estimates of α , Σ and the correlation between ( C i , ik ) k M and ( F i , ik ) k M (denoted Corr ( C , F ) ) forseveral examples. In particular, we obtain estimates for the permanent market impact b α .AAPL BMRN GOOG INTC b α ( bp ) . . 43 2 . . c α × − × − × − × − std. ( × − ) ( × − ) ( × − ) ( × − ) p-value b Σ . − × − std. ( ) ( ) ( ) ( × − ) p-value ( C , F ) 90% 92% 94% 84% T ABLE 1. Estimates for α , Σ and the realized correlation between ( C i , ik ) k M and ( F i , ik ) k M for Nasdaq stocks. For each estimate thestandard deviation (std.) is shown in parentheses and the p-value is givenin the third row. Numbers in bold are significant at a level of at least 99%.3.3.2. The Typical Intraday Pattern. Next, we are interested in the intraday evolution ofthe diagonal and off-diagonal terms of the covariance matrix of returns, and in the waythis evolution is affected when the typical size of trades diminishes. For that purpose, wecompute the intraday covariance matrix of returns for our pool of US stocks and we nor-malize each term ( C i , jk ) k M by its daily average, then we consider the median value ofdiagonal terms and off-diagonal terms as a way of characterizing the evolution of a typi-cal diagonal term and a typical off-diagonal term respectively. The impact of the relativesize of orders on the intraday patterns is assessed by conditioning our estimations.More exactly, we start by defining the matrix of trade imbalances for each stock n inorder to be able to compare the relative size of trades. Namely, for any n , k , l , we set: w nk , l := ν nk , l mean l N P k | ν nk , l | , where mean n ∈ A { x n } denotes the average of ( x n ) as n varies in A . This mean is an esti-mate of the expectation of the sum of the absolute values of ν nk , l over a day; it can be seenas a renormalizing constant, enabling us to mix different stocks on Figure 4.Next, we define the conditioned intraday covariance matrix (cid:16) C i , jk ( λ ) (cid:17) i , j d k M for every λ > as follows:(3.9) C i , jk ( λ ) := E i , jk ( λ ) − X l ∈ E i , jk ( λ ) (cid:16) δS i , k , l − δS i , j , kλ (cid:17) (cid:16) δS j , k , l − δS j , i , kλ (cid:17) , where: • the set E i , jk ( λ ) corresponds to a conditioning: it contains only days for which this5 min bins (indexed by k ) for this pair of stocks (indexed by ( i , j ) , note that wecan have i = j ) is such that the renormalized net volumes are (in absolute value)below λ . It is strictly defined as follows: E i , jk ( λ ) := (cid:10) l N : | w ik , l | λ and | w jk , l | λ (cid:11) ; • δS n , k , l is the price increment defined as for (3.1) and is computed from the historicstock prices; • δS i , j , kλ is the average price increment over selected days, given by: δS i , j , kλ = (cid:16) P l ∈ E i , jk ( λ ) δS i , k , l (cid:17) / (cid:16) E i , jk ( λ ) (cid:17) ; • E i , jk ( λ ) denotes the number of elements of E i , jk ( λ ) : the number of selected days.Note that the stricter the conditioning (i.e. the smaller λ ), the less days in theselection, and hence the smaller E i , jk ( λ ) .Here (cid:16) C i , jk ( λ ) (cid:17) i , j d k M represents the intraday covariance matrix of returns conditionedon trade imbalances between − λ and λ . In all our examples, the coefficient λ is chosento have enough days in the selection (for obvious statistical significance reasons), i.e. sothat E i , jk ( λ ) ≫ for any i , j d and k M .Now we define the median diagonal pattern C diag ( λ ) := (cid:16) C diagk ( λ ) (cid:17) k M and themedian off-diagonal pattern C off ( λ ) := (cid:0) C offk ( λ ) (cid:1) k M as follows: C diagk ( λ ) := median i d (cid:12) C i , ik ( λ ) / mean k M (cid:10) C i , ik ( ) (cid:11)(cid:13) and C offk ( λ ) := median i 1, ..., M and λ > . Here the notation median n ∈ A { x n } denotes the medianvalue of ( x n ) as n varies in A . One should note that the choice of the normalization con-stant (i.e. the mean over bins of C i , jk ( ) ) will allow us to compare the different curves withrespect to the reference case, i.e. without conditioning. In fact, it turns out that λ = re-moves all conditionings: is above the maximum value of our renormalized flows. More-over, we set E diagk ( λ ) := median i d (cid:10) E i , ik ( λ ) (cid:11) and E offk ( λ ) := median i Time = 1= 6.10 -4 = 6.10 -4 (a) The median diagonal pattern (squared volatility) Time = 1= 10 -3 = 10 -3 (b) The median off-diagonal pattern F IGURE 4. Plots of the median diagonal pattern C diag ( λ ) and the medianoff-diagonal pattern C off ( λ ) for diverse values of λ . The secondary axiscorresponds to the number of observations for each 5 minutes bin afterthe conditioning.We take medians instead of means to have robust estimates of the expectations. We donot want our estimates to be polluted by few days of potential erratic market data, thatcould for instance be due to trading interruptions.Figures 4(a) and 4(b) displays representations of C diag ( λ ) and C off ( λ ) for various val-ues of λ . Observe that C diag ( ) and C off ( ) exhibits a pattern that is very similar to oursimulation in Figures 2(b) and 2(d), especially between the beginning of the trading pe-riod and : . Indeed, the observed quantities are high at the beginning of the tradingperiod, lower as the day progresses until it reaches a minimum around : , followedby a slight increase until market close. The general shape of these curves (left-slantedsmile) is well-known (see e.g. [17] and references therein).Our core observation is that: given the absolute value of the net flows are small, this averagecurve flattens out, even at the beginning of the day . At our knowledge, it is the first time thatthis conditioning is mentioned, and it is perfectly in line with our simulated Figures 2(b)and 2(d). This suggests the slopes of the “averaged volatility curves” comes essentiallyfrom the days during which there is a large positive or negative imbalance of large orders,that are “optimally” executed. We believe that this analysis should be completed byusing a larger data set.3.3.3. A Toy Model Calibration. Now, we use historical data to fit our MFG model to someexamples of traded stocks. For that purpose, we use a very simple approach by reducingas much as possible the number of parameters:( S 1) We suppose that E is a centered Gaussian random vector with a covariance ma-trix Γ . Moreover, as suggested by (2.12) , we use Corr (cid:16) P k ν ik , P k ν jk (cid:17) as a proxyfor Corr (cid:16) E i , E j (cid:17) , and which is in turn estimated from data by using the standard estimator : N − N X l = X k ν ik , l − X k ν ik , l ! X k ν jk , l − X k ν jk , l ! , where P k ν ik , l = N − P l P k ν ik , l .( S 2) As suggested by Figures 4(a)-4(b) we choose δtΣ i , j = × mean k M (cid:10) C i , jk ( ) (cid:11) , and we shift upward the simulated curves by δ = × mean k M (cid:10) C i , jk ( ) (cid:11) ;( S 3) Finally, we fix the penalization parameters A i = A = , and choose k i := V i /η i , γ , and Γ i , i by minimizing the L -error between the simulated curves and realcurves. Time Realized variance for GOOGSimulated variance for GOOG (a) GOOG Time Realized covariance for GOOG/AAPLSimulated covariance for GOOG/AAPL (b) GOOG/AAPL F IGURE 5. Comparison between the simulated curves and the real curvesfor two examples. Figure 5(a) corresponds to i ≡ j ≡ GOOG and Figure5(b) corresponds to ( i , j ) ≡ ( GOOG , AAPL ) .Figures 5(a)-5(b) show illustrative examples by considering the two-stocks portfolio:Asset 1 ≡ GOOG ; Asset 2 ≡ AAPL . For that example, the parameters of our model arepresented in Table 2.Here Γ , α , α , σ , σ , ρ are estimated from data (cf. Table 1 and Figures 4(a)-4(b)),while Γ , Γ , γ , k , k are computed by minimizing the L -error between the simulatedcurves and real curves. Following this approach, one requires d + parameters to fit aportfolio of d stocks (i.e. d ( d + ) / curves). MEAN FIELD GAME OF PORTFOLIO TRADING 25 Estimated using the regression (3.8) of Section 3.3 and ( S S Corr (cid:0) E , E (cid:1) = , α = × − , σ = , ρ = , α = × − , σ = , Calibrated on curves of Figure 5(a) and 5(b) Γ = × , Γ = × , γ = − , k = × , k = × .T ABLE 2. The MFG model parameters for the two-stocks portfolio: Asset1 ≡ GOOG ; Asset 2 ≡ AAPL . R EFERENCES [1] A. Alfonsi, A. Fruth, and A. Schied, Optimal execution strategies in limit order books with general shapefunctions , Quantitative Finance (2010), no. 2, 143–157.[2] A. Alfonsi and A. 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