Asset Pricing with Heterogeneous Beliefs and Illiquidity
AAsset Pricing with Heterogeneous Beliefsand Illiquidity ∗ Johannes Muhle-Karbe † Marcel Nutz ‡ Xiaowei Tan § March 26, 2020
Abstract
This paper studies the equilibrium price of an asset that is tradedin continuous time between N agents who have heterogeneous be-liefs about the state process underlying the asset’s payoff. We pro-pose a tractable model where agents maximize expected returns underquadratic costs on inventories and trading rates. The unique equi-librium price is characterized by a weakly coupled system of linearparabolic equations which shows that holding and liquidity costs playdual roles. We derive the leading-order asymptotics for small transac-tion and holding costs which give further insight into the equilibriumand the consequences of illiquidity. Keywords
Equilibrium; Liquidity; Heterogeneous Beliefs
AMS 2010 Subject Classification
Heterogeneous beliefs about fundamental values are a key motive for tradein financial markets. Accordingly, a rich literature studies how prices form asthe aggregate of subjective beliefs; see e.g. the survey [44] for numerous ref-erences. This synthesis happens by means of trading: agents with lower indi-vidual valuations sell to agents who are more optimistic about fundamentals.Hence, liquidity —the ease with which trades can be implemented—plays animportant role in determining how beliefs are reflected in prices. ∗ The authors are grateful to the Associate Editor and two anonymous referees for theirhelpful comments. † Department of Mathematics, Imperial College London, [email protected] supported by the CFM-Imperial Institute of Quantitative Finance. ‡ Departments of Statistics and Mathematics, Columbia University,[email protected]. Research supported by an Alfred P. Sloan Fellowship andNSF Grants DMS-1512900 and DMS-1812661. § Department of Mathematics, Columbia University, [email protected]. a r X i v : . [ q -f i n . M F ] M a r n the present study, we propose a tractable model that allows us tostudy the interplay of heterogeneous beliefs and liquidity in determiningasset prices. We consider N (types of) agents who have different beliefsabout the state process determining the payoff of a given asset. They tradethe asset in continuous time to maximize their expected returns, penalizedwith quadratic costs on inventories and trading rates. We show that thismodel admits a unique Markovian equilibrium. The equilibrium price ischaracterized as the solution of a linear system of parabolic equations witha weak coupling (i.e., the equations are coupled only through the zeroth-orderterms). The solution, as well as the necessary estimates on its derivatives, areobtained by combining a fixed-point argument of [8] for reaction–diffusionequations, classical Schauder theory for parabolic equations and a gradientestimate that seems to be novel.This characterization allows us to study the influence of the two costs.The holding costs on inventories, parametrized by a coefficient γ , can beseen as a proxy for risk aversion, whereas the costs on the trading rate,with coefficient λ > , stand in for the liquidity (or transaction) cost causedby market impact. The two costs determine how agents take into accountcurrent and future expected returns when choosing their portfolios: withbigger transaction costs, further weight is placed on future market conditionsto avoid trades likely to be reversed later on. Conversely, larger holding costsmake it less appealing to hold a given position, so that the current tradingopportunities play a bigger role. Accordingly, liquidity and holding costsplay inverse roles in our analysis. Specifically, when the asset is in zero netsupply (a natural assumption for derivative contracts, say) the two costsonly enter through their ratio γ/λ . For a positive supply, the asset priceremains invariant if the inverse of the supply is rescaled in the same manneras transaction and holding costs, so that the larger trading and holding costsof bigger asset positions are offset by reducing both frictions.Explicit asymptotic formulas obtain in the limiting regimes where eithertransaction costs or holding costs are small ( γ/λ ≈ ∞ or γ/λ ≈ , respec-tively). For small transaction costs λ → , a singular perturbation expansionidentifies the leading-order correction term relative to the frictionless equi-librium in which assets are priced by taking conditional expectations under arepresentative agent’s probability measure that averages the agents’ beliefs.The correction term turns out to be proportional to the square root √ λ of the transaction costs. The corresponding constant of proportionality isrelated to the average of the subjective drifts of the agents’ frictionless port-folios. Thus, equilibrium prices increase relative to their frictionless counter-parts if agents on average expect to increase their positions in the future, and2ice versa. The interpretation is that in illiquid markets, agents take intoaccount their future trading needs to reduce transaction costs. Accordingly,expectations of future purchases already lead to increased positions earlieron and equilibrium prices increase according to the excess demand createdby the aggregated adjustments of all agents, and vice versa.The equilibrium for small holding costs γ → can be approximated bya regular perturbation expansion around the risk-neutral equilibrium pricewhich averages all agents’ subjective conditional expectations. Here, theleading-order correction term is determined by γ times the average of theagents’ expectations of their future positions. Other things equal, agentsreduce the magnitude of their positions when holding costs are introduced,thereby reducing the demand of agents who expect to be long on averageand increasing the demand of agents who expect to be short. The resultingsign of the price correction therefore depends on the aggregate expectationsin the market.To illustrate the implications of these results and test the accuracy ofthe expansions, we consider an example where the state process determiningthe asset’s payoff has Ornstein–Uhlenbeck dynamics, a simple model for aforward contract on a mean-reverting underlying such as an FX rate. Agentsagree on the mean-reversion level and volatility, but disagree about the speedof mean-reversion. For these linear state dynamics, the parabolic PDE sys-tem describing the equilibrium price can be reduced to a system of linearODEs by a suitable ansatz, and in turn compared to the explicit formulasthat obtain for our small-cost asymptotics in this case. In this example,we find that the introduction of small transaction costs increases volatil-ity, in line with the asymmetric information model of [19], the risk-sharingmodel studied in [32], numerical results of [1, 14] and empirical studies suchas [29, 33, 47]. By contrast, the introduction of small holding costs decreasesthe equilibrium volatility. The reason is the opposite manner in which thetwo costs influence how agents take into account future trading opportuni-ties. Without transaction costs, agents who believe in faster than averagemean-reversion perceive a mean-reverting price process and therefore sellwhen its value is high, whereas agents who believe in slower mean-reversionperceive a price process that exhibits “momentum” and therefore buy in thiscase. While this frictionless tradeoff only depends on the current dynamicsof the asset, transaction costs force the agents to take into account futuretrading opportunities as well. That makes the current trading opportunitiesless attractive for the agent believing in faster mean-reversion and thereforecreates an excess demand for the asset when its price is high. This in turnfurther increases high prices and conversely decreases low ones, leading to3dditional volatility.Holding costs have the opposite effect, by discounting the importance offuture trading opportunities and therefore reducing volatility relative to therisk-neutral limiting price. In fact, the exact equilibrium volatility smoothlyinterpolates between the risk-neutral volatility (which is highest) and itscounterpart without transaction costs (which is lowest). For model param-eters estimated from time series data for the USD/EUR exchange rate, wefind that the exact equilibrium prices agree with these comparative staticsgleaned from their asymptotic approximations.For models where trading is frictionless, there is an extensive literatureon asset pricing under heterogeneous beliefs; see, e.g., [10, 21, 44, 18] andthe references therein. To obtain tractable results with limited liquidity, wefocus on a model with quadratic holding and trading costs as well as linearpreferences over gains and losses.Similar linear-quadratic liquidity models are used in partial equilibriumcontexts by [2, 5, 6, 36]. Risk-sharing equilibria with homogeneous beliefsare studied in [12, 26, 32, 42]. As the corresponding first-order conditionsare linear, these models are considerably more tractable than equilibriummodels with other preferences or trading costs, where analytical results areonly available if prices or trading strategies are deterministic [38, 48, 49, 50]or agents only trade once [43, 20]. Numerical analyses of equilibrium modelswith heterogeneous beliefs and transaction costs are carried out in [1, 14].Considering a holding cost on risky positions as in [15, 17, 41, 42] furthersimplifies the analysis compared to models where the corresponding riskpenalty is imposed on the variance of the risky positions as in [25, 26, 32].Indeed, in the present model, we can characterize the equilibrium price bya system of linear PDEs, avoiding the nonlinear equations that naturallyappear in models where agents have risk aversion in the form of concaveutility functions. As the present work focuses on equilibrium asset priceswith heterogeneous beliefs about the underlying state process, we abstractfrom heterogeneous holding costs. These are considered in [32] for agentswith homogeneous beliefs and in [7, 16] for partial equilibrium models withheterogeneous beliefs.This paper is organized as follows. Section 2 details the financial marketand the definition of an equilibrium. In Section 3 we derive the optimal port-folio of any agent given an exogenous asset price process. Section 4 providesthe existence, uniqueness and PDE characterization of the equilibrium price.The leading-order asymptotics for small transaction and holding costs arepresented in Sections 5 and 6, respectively. The concluding Section 7 coversthe example with mean-reverting state process.4 otation. As usual, C = C ( R n ) is the space of continuous functions g ( x ) on R n and C k is the space of functions g ∈ C whose partial derivatives upto order k exist and belong to C . Similarly, C ,k is the space of continuousfunctions g ( t, x ) such that g ( t, · ) , ∂ t g ( t, · ) ∈ C k . For any of these spaces, asubscript “ b ” indicates that the functions and all mentioned derivatives arebounded. The dimension n of the underlying domain is often understoodfrom the context. Conditional expectations are denoted E t [ · ] = E [ ·|F t ] forbrevity and when F is a functional of the paths of a process Y , we will oftenwrite E t,y [ F ( Y )] = E [ F ( Y ) | Y t = y ] . In this context, Y will be the solutionof an SDE and E [ F ( Y ) | Y t = y ] can be unambiguously defined as E [ F ( Y t,y )] where ( Y t,ys ) s ≥ t is the unique solution of the corresponding SDE with initialcondition y at time t . Beliefs.
Let X be the coordinate-mapping process on the space Ω = C ([0 , T ] , R d ) of continuous, d -dimensional paths ω with ω = 0 , equippedwith the canonical σ -field F and filtration ( F t ) t ∈ [0 ,T ] generated by X . Weconsider N (types of) agents with heterogeneous views on the distribution ofthe state process X . Specifically, for each ≤ i ≤ N , let Q i be a probabilitymeasure on Ω under which X satisfies dX t = b i ( t, X t ) dt + σ i ( t, X t ) dW it (2.1)where W i is a d (cid:48) -dimensional Brownian motion. We assume that b i : [0 , T ] × R d → R d and σ i : [0 , T ] × R d → R d × d (cid:48) are jointly Lipschitz and bounded. Thisguarantees in particular that (2.1) has a unique strong solution. Moreover,we assume that the matrix σ i := σ i σ (cid:62) i is uniformly parabolic: there is aconstant κ > such that ξ (cid:62) σ i ξ ≥ κ | ξ | for all ξ ∈ R d . The associatedgenerator is denoted by L i = ∂ t + b i ∂ x + 12 Tr σ i ∂ xx . (2.2)We allow the measures Q i to differ in drift as well as volatility. The differ-ences regarding the drifts are more important in practice, and our resultsremain relevant also without disagreement on volatility. We refer to [23] fora detailed discussion of volatility uncertainty. Remark 2.1.
The above assumptions imply that the support of Q i is thewhole space Ω ; cf. [45, Theorem 3.1]. Thus, if F and G are continuous5unctions on Ω , then F ( X ) = G ( X ) Q i -a.s. is equivalent to F = G . Thisfact will be used throughout the paper, often implicitly. Uniform parabolicityis convenient to simplify the exposition, but of course results similar to ourscould be obtained under different assumptions. When the support of X isnot the whole space, the statements involving price functions and PDEs needto be restricted to a suitable domain (as, e.g., in [40]). Market Model.
Let f ∈ C b ( R d ) . We consider N agents that dynamicallytrade an asset with a single payoff f ( X T ) at the time horizon T > . Fixa constant a ≥ , the exogenous supply at time t = 0 , and the initial assetallocation a i ∈ R to each agent, where (cid:80) Ni =1 a i = a . Let L p ( Q i ) denotethe set of progressively measurable processes φ = ( φ t ) ≤ t ≤ T (of appropriatedimension) such that E i [ (cid:82) T φ pt dt ] < ∞ . An (admissible) portfolio for agent i is a scalar process φ ∈ L ( Q i ) which satisfies φ = a i and is absolutelycontinuous with rate ˙ φ ∈ L ( Q i ) . We say that portfolios φ i , ≤ i ≤ N clear the market if N (cid:88) i =1 φ it = a , t ∈ [0 , T ] holds pointwise. A price process (for f ) is a progressively measurable pro-cess S = ( S t ) ≤ t ≤ T which satisfies S T = f ( X T ) and is an Itô process withsufficiently integrable coefficients under each Q i : dS t = µ it dt + ν it dW it with µ i , ν i ∈ L ( Q i ) (2.3)for some Q i -Brownian motion W i , for all ≤ i ≤ N . Equilibrium.
To formulate the agents’ optimization criteria, we fix a hold-ing cost parameter γ > and a transaction cost parameter λ > . (Theboundary cases λ = 0 and γ = 0 will be considered in Sections 5 and 6,respectively.) For a given price process S , agent i maximizes her expectedreturns, penalized for inventories and trading costs, J i ( φ ) = E i (cid:20)(cid:90) T (cid:18) φ t dS t − γ φ t dt − λ φ t dt (cid:19)(cid:21) (2.4)over the set of her admissible portfolios. A portfolio φ i is optimal for agent i if it is a maximizer. If S is a price process such that there exist optimal The precise integrability condition is not crucial; we simply need to ensure that thelocal martingale part of (cid:82) φdS has vanishing expectation when S is defined as in (2.3). Inour main equilibrium result the optimal portfolios are bounded. φ i , ≤ i ≤ N for the agents which clear the market, then S isan equilibrium price process . Finally, v : [0 , T ] × R d → R is an equilibriumprice function if v ( t, X t ) defines an equilibrium price process. We shall beinterested in symmetric equilibria with prices of this Markovian form; how-ever, the associated portfolios are usually path-dependent in the presence oftransaction costs. As is implicit in (2.4), the interest rate is assumed to bezero throughout.In this model, agents have fixed beliefs Q i and agree-to-disagree. Thebeliefs can be inconsistent with one another or with observations of X overtime, especially in the case of disagreement on future volatilities. As in [28],the beliefs are used by the agents to compute expected future profits or lossesand eventually determine the initial price S of the asset. This occurs at theinitial time and without actual observation of the future. Thus, the resultingprice is consistent for all agents even though they disagree.The linear-quadratic criterion (2.4) includes several simplifications to en-hance the tractability of the model. First, as in [31], transaction costs onlydepend on each agent’s individual trading rate and not on the total orderflow in the market. Accordingly, the trading cost should be interpreted asa tax or the fee charged by an exchange, rather than as a price impactcost as in [3]. Partial equilibrium models where the agents interact throughtheir common price impact are studied in [11], for example. Nevertheless,to obtain tractable first-order conditions, we use a quadratic rather thanlinear cost. This simplification is motivated by recent results [27] for risk-sharing equilibria which show that equilibrium prices are robust with respectto the specification of the trading cost. Second, we assume as in [42] thatall investors penalize inventories through a quadratic holding cost on posi-tions. This leads to a system of linear PDEs for equilibrium prices, unlikethe systems of nonlinear PDEs that appear for holding costs on variancesin [32] even with homogeneous beliefs. Concave utility functions over termi-nal wealth or intermediate consumption would further complicate the analy-sis by introducing each agent’s value function as an additional component ofthe nonlinear PDE system. Third, we suppose that the inventory costs arehomogeneous across agents in order to single out the effect of heterogeneity inbeliefs. Heterogeneous holding costs have been studied in [12, 32] for modelswith homogeneous beliefs. Finally, we do not model “where the transactioncosts go” in equilibrium: like in most of the related literature [13, 38, 48],we consider the trading cost as a deadweight loss for the financial marketunder consideration. This seems reasonable for a transaction tax or the feesimposed by an exchange, for example.7
Single-Agent Optimality
As a preparation for the equilibrium result, we first fix agent i and solve herindividual optimization problem in the face of an exogenous price process.Similar linear-quadratic optimization problems have been considered, e.g.,in [5, 6, 12, 15, 25, 26, 36]; we provide a self-contained derivation for theconvenience of the reader. Lemma 3.1.
Let γ > , λ > and G ( t ) = cosh (cid:18)(cid:114) γλ ( T − t ) (cid:19) , t ∈ [0 , T ] . (3.1) Given a price process S as in (2.3) , the dt × Q i -a.e. unique optimal portfoliofor agent i is φ it = G ( t ) G (0) a i + (cid:90) t G ( t ) G ( s ) E is (cid:20)(cid:90) Ts G ( u ) G ( s ) µ iu λ du (cid:21) ds. (3.2) In particular, the optimal trading rate is characterized by the random ODE ˙ φ it = G (cid:48) ( t ) G ( t ) φ it + E it (cid:20)(cid:90) Tt G ( s ) G ( t ) µ is λ ds (cid:21) , φ i = a i . (3.3)As in the previous literature, the optimal trading strategy tracks the aver-age E it [ (cid:82) Tt − γG ( s ) λG (cid:48) ( t ) µ is γ ds ] of the discounted future values of the no-transactioncost portfolio µ it /γ obtained by pointwise maximization of the drift of (2.4).To wit, illiquidity is accounted for by “aiming in front of the moving tar-get” [25]. Both the tracking speed − G (cid:48) ( t ) /G ( t ) and the discount kernel K ( t, s ) = − γG ( s ) /λG (cid:48) ( t ) are determined by the ratio γ/λ of holding andtransaction costs, with relatively lower transaction costs leading to fastertrading and more emphasis on the current returns of the asset. Proof of Lemma 3.1.
Direct differentiation shows that ˙ φ i of (3.3) is indeedthe derivative of φ i in (3.2). Moreover, µ i ∈ L ( Q i ) and Doob’s inequalityimply that φ i ∈ L ( Q i ) and then (3.3) yields that ˙ φ i ∈ L ( Q i ) .Note that J i ( φ ) = E i (cid:20)(cid:90) T (cid:18) φ t µ it − γ φ t − λ φ t (cid:19) dt (cid:21) for any portfolio φ and that portfolios can be parametrized by their trad-ing rates as the initial allocations are fixed and ˙ φ ∈ L ( Q i ) implies φ ∈ ( Q i ) . The strict concavity of J i implies that any optimizer is (a.e.) uniqueand that a trading rate ˙ φ is optimal if and only if the Gâteaux derivative lim ε → ε [ J i ( ˙ φ + ε ˙ ϑ ) − J i ( ˙ φ )] of (2.4) vanishes in all directions ˙ ϑ ∈ L ( Q i ) ;that is, E i (cid:20)(cid:90) T (cid:18) µ it (cid:90) t ˙ ϑ s ds − γφ it (cid:90) t ˙ ϑ s ds − λ ˙ φ it ˙ ϑ t (cid:19) dt (cid:21) = E i (cid:20)(cid:90) T (cid:18)(cid:90) Tt (cid:0) µ is − γφ is (cid:1) ds − λ ˙ φ it (cid:19) ˙ ϑ t dt (cid:21) , ˙ ϑ ∈ L ( Q i ) . As ˙ ϑ is arbitrary, this is equivalent to ˙ φ it = λ E it [ (cid:82) Tt ( µ is − γφ is ) ds ] , which isin turn equivalent to ˙ φ it = M it − λ (cid:90) t ( µ is − γφ is ) ds, ˙ φ iT = 0 for some Q i -martingale M i . Put differently, ˙ φ ∈ L ( Q i ) is optimal if andonly if it solves the linear forward-backward SDE dφ t = ˙ φ t dt, φ = a i , (3.4) d ˙ φ t = γλ (cid:18) φ t − µ it γ (cid:19) dt + dM t , ˙ φ T = 0 . (3.5)Direct computation shows that ( φ i , ˙ φ i ) solves this system: (3.4) is trivial andfor (3.5) we note that d ˙ φ it = (cid:26) (cid:18) G (cid:48)(cid:48) ( t ) G ( t ) − G (cid:48) ( t ) G ( t ) (cid:19) φ it + G (cid:48) ( t ) G ( t ) (cid:18) G (cid:48) ( t ) G ( t ) φ it + E it (cid:20)(cid:90) Tt G ( s ) G ( t ) µ is λ ds (cid:21)(cid:19) − G (cid:48) ( t ) G ( t ) E it (cid:20)(cid:90) Tt G ( s ) µ is λ ds (cid:21) − µ it λ (cid:27) dt + 1 G ( t ) dE it (cid:20)(cid:90) T G ( s ) µ is λ ds (cid:21) = γλ (cid:18) φ it − µ it γ (cid:19) dt + dM t for the Q i -martingale M = (cid:82) · G ( t ) dE it [ (cid:82) T G ( s ) µ is λ ds ] , where G (cid:48)(cid:48) ( t ) = γλ G ( t ) was used. As G (cid:48) ( T ) = 0 , the terminal condition ˙ φ iT = 0 is also satisfied. The following result establishes the existence and uniqueness of an equilib-rium price function v ∈ C , b ([0 , T ] × R d ) and its characterization through aweakly coupled system of linear parabolic equations. Recall that the function G was defined in (3.1) as G ( t ) = cosh( (cid:112) ( γ/λ )( T − t )) .9 heorem 4.1. Let γ > , λ > . The parabolic system ∂ t v i + 12 Tr σ i ∂ xx v i + b i ∂ x v i + G (cid:48) ( t ) G ( t ) ( v i − v ) = 0 , ≤ i ≤ N, (4.1) v := 1 N N (cid:88) i =1 v i + λG (cid:48) ( t ) N G ( t ) a , (4.2) v i ( T, · ) = f, ≤ i ≤ N (4.3) has a unique solution v , . . . , v N ∈ C , b ([0 , T ] × R d ) , and the function v defined via (4.2) is an equilibrium price function. It is unique in the sensethat any equilibrium price function w ∈ C , ([0 , T ] × R d ) with polynomialgrowth must be equal to v . The equilibrium portfolios are given by φ it = G ( t ) G (0) a i + (cid:90) t G ( t ) G ( s ) E is (cid:20)(cid:90) Ts G ( u ) G ( s ) L i v ( u, X u ) λ du (cid:21) ds. For comparison, let us first consider how this result simplifies for homo-geneous beliefs. Then, all drift and volatility coefficients, and in turn thefunctions v i in Theorem 4.1, coincide. Together with the definition of thefunction G , it follows that v ( t, x ) = E t,x [ f ( X T )] − γa N ( T − t ) , where the expectation is taken under the common probability measure. Thisis the same equilibrium price that obtains in the frictionless version of themodel where trading costs λ are zero; cf. Proposition 5.1. Whence, withhomogeneous beliefs and holding costs, equilibrium prices do not depend onliquidity; see [42, 12, 32] for similar results. In contrast, the correspondingequilibrium trading speed depends on both the trading cost λ and the holdingcost λ through their ratio. With homogenous beliefs, it is deterministic andthe same for all agents ≤ i ≤ N , ˙ φ it = (cid:114) γλ tanh (cid:18)(cid:114) γλ ( T − t ) (cid:19) (cid:16) a N − φ it (cid:17) , φ i = a i . To wit, the (identical) agents simply gradually adjust their initial allocationsuntil the total supply of the risk asset is split equally among all of them.Returning to the general case, an immediate consequence of Theorem 4.1is that the holding costs γ and transaction costs λ have dual roles. In par-ticular, in the case of zero net supply a = 0 , the equilibrium price depends10nly on the ratio γ/λ , so that small transaction costs are equivalent to largeholding costs. When a > , the theorem shows that the equilibrium priceis 0-homogeneous in ( γ, λ, /a ) . We discuss this in more detail in Section 6below, after deriving the limiting cases for small costs.If all agents have equivalent beliefs, one could also consider heterogeneousholding costs γ i as previously studied in models with homogeneous beliefs [12,32] or exogenous price dynamics [7, 16]. However, as in those studies, allagents’ current positions would then appear as additional state variables andthe system would become semilinear. (The extra state variables cancel in thepresent setting because the function G ( t ) is the same for all agents.) We thusfocus on homogeneous holding costs to highlight the effect of heterogeneityof beliefs.As a preparation for the proof of Theorem 4.1, we first establish theanalytic properties of the parabolic system. Given α ∈ (0 , , the Hölderspace C α/ , α ([0 , T ] × R d ) consists of the functions w ( t, x ) such that w, ∂ t w, ∂ x w, ∂ xx w exist, are bounded, and uniformly Hölder continuous withexponents α/ in t and α in x . Proposition 4.2.
The system (4.1–4.3) has a unique solution v , . . . , v N ∈ C , b ([0 , T ] × R d ) . In fact, v , . . . , v N ∈ C α/ , α ([0 , T ] × R d ) for all α ∈ (0 , and uniqueness holds in the larger class of functions w , . . . , w N ∈ C , ([0 , T ) × R d ) ∩ C ([0 , T ] × R d ) satisfying | w i ( t, x ) | ≤ c exp( c | x | ) forsome constants c , c ≥ .Proof. The system (4.1–4.3) is a weakly coupled, uniformly parabolic linearsystem; see [24, Chapter 9] for background. Uniqueness in the stated class isa special case of [9, Theorem 1]. An existence result for such linear systemsis contained, e.g., in [24, Theorem 3, p. 256], but this does not yield growthestimates of the type we require here. Our system is also covered by aliterature on reaction–diffusion systems. Specifically, [8, Theorem 2.4] yieldsthat (4.1–4.3) has a unique solution v , . . . , v N ∈ C , ([0 , T ) × R d ) ∩ C b ([0 , T ] × R d ) . The main point (which we have not found provided in the literature)is to prove a useful growth estimate on the derivatives, and for that, the keyelement is to provide a Hölder estimate for v i .(i) In this step we show that v i is globally Lipschitz in x , uniformly in t .Writing u = ( u , . . . , u N ) , our system is of the general form L i u i ( t, x ) + h i ( t, u ( t, x )) = 0 , u i ( T, x ) = f i ( x ) , ≤ i ≤ N (4.4)satisfying the following conditions, for some constant c > : the function h = ( h , . . . , h n ) is jointly Lipschitz with norm Lip( h ) ≤ c (hence h ( t, · )
11s of linear growth, uniformly in t ); the coefficients of L i are bounded andLipschitz; each f i is bounded and Lipschitz with norm Lip( f i ) . According to[8, Theorem 2.4], such a system has a (unique) solution v , . . . , v N ∈ C , ∩ C b .Indeed, define F i ( u )( t, x ) := E i (cid:20) f i ( X t,xT ) + (cid:90) Tt h i ( s, u ( s, X t,xs )) ds (cid:21) , ≤ i ≤ N. It is shown in the proofs of [8, Theorems 2.3 and 2.4] that F = ( F , . . . , F N ) is a contraction on ( C b ) N = C b × · · · × C b for a complete norm which isequivalent to the uniform norm. More precisely, this holds after suitablytruncating h (so that the truncation does not affect the bounded solution).It is shown that if we start at any u ∈ ( C b ) N and iterate F , the sequence u n = ( F ◦ · · · ◦ F )( u ) will converge uniformly to a solution ( v , . . . , v N ) ∈ ( C , ∩ C b ) N of (4.4). We may, in particular, pick u ∈ ( C b ) N such that sup ≤ s ≤ T Lip( u i ( s, · )) < ∞ for all ≤ i ≤ N as our starting point for theiteration.By a standard estimate on SDEs (e.g., [46, Theorem 2.4 (i), p. 8]), E i | X t,xs − X t,ys | ≤ K | x − y | , ≤ s ≤ T for a constant K depending only on the Lipschitz constants of the coefficientsof L i and T . Fix a small time interval [ t, T ] of length τ = T − t > and let L u = L ( t ) u = max ≤ i ≤ N sup t ≤ r ≤ T Lip( u i ( r, · )) . Then for t ≤ s ≤ T we have that | F i ( u )( s, x ) − F i ( u )( s, y ) |≤ E i (cid:20) Lip( f i ) | X s,xT − X s,yT | + (cid:90) Ts c max i Lip( u i ( r, · )) | X s,xr − X s,yr | dr (cid:21) ≤ [ K Lip( f i ) + τ cKL u ] | x − y | . This holds for all i . Choose τ such that ε := τ cK < and set L f =max ≤ i ≤ N K Lip( f i ) , then Lip( F ( u )( s, · )) ≤ L f + εL u , t ≤ s ≤ T, the notation of course meaning that each component F i ( u ) satisfies thisproperty. Iterating yields that u n = ( F ◦ · · · ◦ F )( u ) satisfies the geometricestimate Lip( u n ( s, · )) ≤ L f n − (cid:88) k =0 ε k + ε n L u ( v , . . . , v N ) = lim u n satisfies Lip( v i ( s, · )) ≤ L f (1 − ε ) − for t ≤ s ≤ T .Note that the size τ of the interval in the above argument does not dependon Lip( f ) . Hence we can repeat the argument on the interval [ T − τ, T − τ ] ,replacing the terminal condition f i by ˜ f i := v i ( T − τ, · ) . Continuing finitelymany times, we conclude that sup ≤ s ≤ T Lip( v i ( s, · )) < ∞ .(ii) Next, we show that v i is globally / -Hölder in t , uniformly in x . Asimple SDE estimate shows that E i | X t (cid:48) ,xs − X t,xs | ≤ K | t (cid:48) − t | / , ≤ t ≤ t (cid:48) ≤ s ≤ T where K now depends on the Lipschitz constants and uniform bounds for b i and σ i as well as T . (To see this one may, e.g., go through the proofof [46, Theorem 2.4 (ii), p. 8] and use the uniform bounds in the estimatebelow Equation (2.5) of that reference to avoid a dependence on x in thefinal estimate for E i | X t,xs − X t,ys | .)As mentioned above, the relevant function h in (4.4) is truncated in u ,so that (cid:107) h (cid:107) ∞ < ∞ . This yields the (crude but simple) estimate (cid:90) t (cid:48) t | h ( s, X t,xs , u ( s, X t,xs )) | ds ≤ (cid:107) h (cid:107) ∞ | t (cid:48) − t | ≤ c (cid:48) | t (cid:48) − t | / (4.5)for some constant c (cid:48) , because | t (cid:48) − t | ≤ T . If u ∈ ( C b ) N is Lipschitz in x with constant L u uniformly in t , we then have, similarly as in (i) but usingalso (4.5), | F i ( u )( t (cid:48) , x ) − F i ( u )( t, x ) |≤ E i (cid:20) Lip( f i ) | X t (cid:48) ,xT − X t,xT | + c (cid:48) | t (cid:48) − t | / + (cid:90) Tt (cid:48) cL u | X t (cid:48) ,xs − X t,xs | ds (cid:21) ≤ [ K Lip( f i ) + c (cid:48) + T cL u K ] | t (cid:48) − t | / . Again, we iterate the mapping F to generate u n = ( F ◦ · · · ◦ F )( u ) . By (i) wehave that sup n L u n < ∞ . Hence, the above shows that | u ni ( t (cid:48) , x ) − u ni ( t, x ) | ≤ c (cid:48)(cid:48) | t (cid:48) − t | / for a uniform constant c (cid:48)(cid:48) , and then the same holds for the limit ( v , . . . , v N ) = lim u n .(iii) We have shown above that v is globally Lipschitz in x and 1/2-Hölderin t ; in particular v j ∈ C α/ ,α for all α ∈ (0 , . (See [37, p. 117] for a detaileddefinition of the Hölder spaces.) For fixed i , we can see v i as the solution ofa scalar PDE which contains ( v j ) j as coefficients: ϕ = v i is the solution of ˜ L ϕ ( t, x ) + g ( t, x ) = 0 , ϕ ( T, · ) = f [0 , T ] × R d with terminal value f ∈ C α , parabolic operator ˜ L u := L i u − u and inhomogeneous term g ∈ C α/ ,α defined by g = v i + G (cid:48) ( t ) G ( t ) (cid:32) v i − N N (cid:88) i =1 v i + λG (cid:48) ( t ) N G ( t ) a (cid:33) using the fixed functions ( v j ) ≤ j ≤ N . We can now apply a suitable version ofthe Schauder estimates to conclude that v i = ϕ ∈ C α/ , α ([0 , T ] × R d ) ;cf. [37, Theorem 9.2.3, p. 140]. Remark 4.3.
Suppose that b i , σ i , f ∈ C ∞ b for ≤ i ≤ N . Then we alsohave v i ∈ C ∞ b . Proof. If b i , σ i ∈ C ∞ ([0 , T ) × R d ) , interior regularity for parabolic systems asstated in [24, Theorem 11, p. 265] immediately yields that the solution fromProposition 4.2 is in C ∞ ([0 , T ) × R d ) . We need to show that the partialderivatives of any order are bounded.Fix ≤ i ≤ N and ≤ k ≤ d and consider the function ϕ = ∂ x k v i . Wecan differentiate the system (4.1) with respect to x k and rearrange the termsto find that ϕ is the solution of a scalar parabolic equation L ϕ ( t, x ) + g ( t, x ) = 0 , ϕ ( T, · ) = ∂ x k f on [0 , T ] × R d with terminal value ∂ x k f ∈ C ∞ b ⊆ C α . Here the inho-mogeneity g incorporates all other terms resulting from the differentiatedequation: it is a linear combination, with coefficients in C ∞ ([0 , T ) × R d ) , ofthe functions v j , ≤ j ≤ N as well as their first and second-order spatialderivatives. As v j ∈ C α/ , α by Proposition 4.2, we see in particular that g ∈ C α/ ,α . Thus, we can conclude from [37, Theorem 9.2.3, p. 140] that ∂ x k v i = ϕ ∈ C α/ , α ([0 , T ] × R d ) . In particular, the third-order spatialderivatives of v i are bounded and uniformly Hölder continuous. Moreover, bythe parabolic form of the above equation, the same follows for ∂ t ∂ x k v i = ∂ t ϕ .This argument can be iterated to the higher-order derivatives. Proof of Theorem 4.1.
The formula for the equilibrium portfolios is a directconsequence of Proposition 5.1, so we focus on the price.(i) Let v , . . . , v N ∈ C , b be the solution from Proposition 4.2 and define v by (4.2); we show that v is an equilibrium price function. Itô’s formula showsthat S t = v ( t, X t ) is a price process as defined in (2.3); the coefficients µ i and ν i are even bounded. In view of (4.2), the function w i ( t, x ) := G ( t ) v i ( t, x ) satisfies L i w i = G L i v i + G (cid:48) v i = G (cid:48) v w i ( T, x ) = G ( T ) v i ( T, x ) = f ( x ) . Thus, Itô’s formula and the bounded-ness of ∂ x v i imply that under Q i we have the Feynman–Kac representation w i ( t, x ) = E it,x [ f ( X T )] − (cid:90) Tt G (cid:48) ( u ) E it,x [ v ( u, X u )] du. As a result, v i ( t, x ) = E it,x [ f ( X T )] G ( t ) − (cid:90) Tt G (cid:48) ( u ) G ( t ) E it,x [ v ( u, X u )] du (4.6)for all ( t, x ) ∈ [0 , T ] × R d . Lemma 3.1 shows that given the price S t = v ( t, X t ) ,the portfolio φ it = (cid:90) t G ( t ) G ( s ) E is,X s (cid:104) (cid:90) Ts G ( u ) G ( s ) 1 λ L i v ( u, X u ) du (cid:105) ds + G ( t ) a i G (0) (4.7)is optimal for agent i . It remains to prove that these portfolios clear the mar-ket. Recalling that G ( T ) = 1 , taking expectations in the integration-by-partsformula (cid:82) Ts G ( u ) dS u = G ( T ) S T − G ( s ) S s − (cid:82) Ts G (cid:48) ( u ) S u du and applying (4.6)yield E is,x (cid:20)(cid:90) Ts G ( u ) G ( s ) 1 λ L i v ( u, X u ) du (cid:21) = 1 λG ( s ) (cid:18) E is,x [ f ( X T )] − G ( s ) v ( s, x ) − (cid:90) Ts G (cid:48) ( u ) E is,x [ v ( u, X u )] du (cid:19) = 1 λ [ v i ( s, x ) − v ( s, x )] . (4.8)In view of (4.2), we deduce that N (cid:88) i =1 E is,x (cid:20)(cid:90) Ts G ( u ) G ( s ) 1 λ L i v ( u, X u ) du (cid:21) = − G (cid:48) ( s ) G ( s ) a . Using this in (4.7) and integrating − G (cid:48) ( s ) G ( s ) = ∂ s G ( s ) − , we conclude that N (cid:88) i =1 φ it = − a (cid:90) t G ( t ) G ( s ) G (cid:48) ( s ) G ( s ) ds + G ( t ) a G (0) = a as desired.(ii) Let S t = w ( t, X t ) be an equilibrium price process for some func-tion w ∈ C , ([0 , T ] × R d ) of polynomial growth (or, more generally, w ∈ , ([0 , T ) × R d ) of polynomial growth and locally Hölder continuous on [0 , T ] × R d ). We have w ( T, · ) = f by Remark 2.1. Recall that the coef-ficients µ it = L i w ( t, X t ) and ν it = ∂ x w ( t, X t ) (cid:62) σ it of (2.3) are in L ( Q i ) aspart of our definition of a price process. We define w i by the Feynman–Kacformula (4.6) with w instead of v . In view of the assumptions on b i and σ i , the function w i has polynomial growth like w . Moreover, by a carefulapplication of standard PDE results, w i ∈ C , and w i is a solution of theassociated linear PDE (4.1). Specifically, we can use an approximation withbounded domains as detailed in [30, Theorem 1, Condition (A3’), Lemma 2and the comments above it] under the stated conditions on w .It remains to show (4.2). As a consequence of Lemma 3.1, the agents’equilibrium portfolios φ i are given by (4.7). Because these portfolios clearthe market, (cid:80) i φ is = a and thus ∂ s (cid:80) i φ is G ( s ) = a ∂ s G ( s ) − . Reversing theintegration by parts (4.8), we conclude that N (cid:88) i =1 G ( s ) λ [ w i ( s, x ) − w ( s, x )] = N (cid:88) i =1 G ( s ) E is,x (cid:20)(cid:90) Ts G ( u ) G ( s ) 1 λ L i w ( u, X u ) du (cid:21) = ∂ s N (cid:88) i =1 φ is G ( s ) = a ∂ s G ( s ) − = − a G (cid:48) ( s ) G ( s ) which is equivalent to (4.2). We have thus established that w , . . . , w N ∈ C , are a solution of (4.1) with polynomial growth. The claim now follows bythe uniqueness of the solution as stated in Proposition 4.2. Remark 4.4.
The restriction to Markovian equilibria in Theorem 4.1 (mean-ing that the price is a function of t and x ) is related to our choice ofproof through PDE arguments rather than fundamental. For instance, ifthe volatility σ i is the same for all agents, similar arguments could be car-ried out using backward SDEs (e.g., [35]). In that framework, one wouldobtain uniqueness within a class of possibly non-Markovian equilibria andone could also cover beliefs where (2.1) is replaced by coefficients that maydepend on the path of X . In this section we provide intuition for the equilibrium price from Theo-rem 4.1 by describing its asymptotics for small transaction costs λ → . Forlater comparison, we first consider the model without transaction costs; i.e., λ = 0 . In this case we drop the requirement of absolute continuity in the16efinition of the admissible portfolios and we also do not enforce the initialholdings a i (in any event, agents can instantaneously adjust their positionafter t = 0 without incurring costs). The following result, which is a specialcase of [41, Theorem 2.1 and Remark 3.5], shows that the correspondingequilibrium corresponds to the price of a representative agent with a view ¯ Q defined by the averaged drift and volatility coefficients. Proposition 5.1.
Let λ = 0 and γ > . There exists a unique equilibriumprice function v ∈ C , b , given by v ( t, x ) = ¯ E t,x [ f ( X T )] − ( T − t ) γa N where ¯ E [ · ] is the expectation for the probability ¯ Q under which dX t = ¯ b ( t, X t ) dt + ¯ σ ( t, X t ) dW t , ¯ b = N (cid:80) Ni =1 b i , ¯ σ = N (cid:80) Ni =1 σ i for a Brownian motion W . Equivalently, v is the unique bounded classicalsolution of ∂ t v + 12 Tr ¯ σ ∂ xx v + ¯ b∂ x v − γa N = 0 , v ( T, · ) = f. (5.1) In equilibrium, the dt × Q i -a.e. unique optimal portfolio for agent i is φ i, t = L i v ( t, X t ) γ . (5.2)In the remainder of this section we denote the equilibrium price fromTheorem 4.1 by v λ to emphasize the dependence on λ . Our goal is to computeits leading-order deviation λ − / ( v λ − v ) from the frictionless equilibriumprice v of Proposition 5.1. For simplicity, we focus on the case of a one-dimensional state variable ( d = 1 ) with smooth drift and diffusion coefficientsand terminal condition: b i , σ i , f ∈ C ∞ b for ≤ i ≤ N , and hence v, v i ∈ C ∞ b on the strength of Remark 4.3. Theorem 5.2.
For fixed holding costs γ > and small transaction costs λ → , the equilibrium price function v λ from Theorem 4.1 has the expansion v λ ( t, x ) = v ( t, x ) + √ λv ∗ ( t, x ) + o ( √ λ ) locally uniformly on [0 , T ] × R . (5.3) For the arguments below, C b is sufficient. Some weakening of these regularity assump-tions is certainly possible, for the ease of exposition we have retained the given smoothnessassumptions. ere, v is the frictionless equilibrium price from Proposition 5.1 and v ∗ ( t, x ) = √ γN N (cid:88) i =1 ¯ E t,x (cid:20)(cid:90) Tt L i ˆ φ i, ( s, X s ) ds (cid:21) (5.4) where ˆ φ i, ( s, x ) = L i v ( s, x ) /γ is the feedback function determining agent i ’sfrictionless optimal portfolio (5.2) and the expectation is taken under theprobability measure ¯ Q of the frictionless representative agent for which dX t = ¯ b ( t, X t ) dt + ¯ σ ( t, X t ) dW t , ¯ b = N (cid:80) Ni =1 b i , ¯ σ = N (cid:80) Ni =1 σ i . The singular perturbation expansion (5.3) shows that the leading-orderdeviation of the frictional equilibrium price v λ from its frictionless counter-part v scales with the square root √ λ of the trading cost, as in the risk-sharing equilibrium of [32]. With the heterogeneous beliefs considered in thepresent study, the constant of proportionality (5.4) is determined by the in-tegrated drift rates (cid:82) Tt L i ˆ φ i, ( s, X s ) ds of the agents’ frictionless equilibriumportfolios, averaged with respect to agents and states. Thus, equilibriumprices increase relative to their frictionless counterparts if agents on averageexpect to increase their positions in the future, and vice versa. The in-terpretation is that in illiquid markets, agents take into account their futuretrading needs to save cumulative transaction costs over the whole time inter-val. Accordingly, expectations of future purchases already lead to increasedpositions earlier on, and vice versa. To clear the market, equilibrium pricesincrease or decrease according to the excess demand or supply created bythe aggregated adjustments of all agents.
The first step towards the proof of Theorem 5.2 is to show that the functions v λi from Theorem 4.1 are not just bounded for each λ , but that this bound isin fact uniform for λ ∈ (0 , ∞ ) . In view of the PDEs (4.1) from Theorem 4.1and as λG (cid:48) ( t ) NG ( t ) a is uniformly bounded for all λ > and t ∈ [0 , T ] by thedefinition of G , this is a special case of the following more general result thatwill also allow us to derive estimates for small holding costs in the subsequentsection. Note that while the actual future portfolio changes add up to zero by market clearing,this is not necessarily true for the changes as anticipated by the heterogeneous agents undertheir subjective probability measures, L i ˆ φ i, . In the formula for v ∗ , these anticipatedchanges are averaged across all states under the probability measure ¯ Q corresponding tothe frictionless representative agent. emma 5.3. For i = 1 , . . . , N and an arbitrary parameter ε ∈ E , considerfunctions α i , β i , a εi , b εi , h i ∈ C ∞ b ([0 , T ] × R ) and write L i = ∂ t + 12 β i ∂ xx + α i ∂ x . Suppose that a εi , b εi are bounded uniformly in ε ∈ E and let u i = u i ( ε, λ, γ ) , i = 1 , . . . , N denote the unique classical bounded solution of the system L i u i + (cid:16) G (cid:48) G + a εi (cid:17) u i − G (cid:48) G N N (cid:88) j =1 u j + b εi = 0 , u i ( T, · ) = h i , i = 1 , . . . , N. Then, | u i ( t, x ) | ≤ M for a constant M > independent of ε, λ, γ ∈ (0 , ∞ ) and ( t, x ) ∈ [0 , T ] × R .Proof. Existence and uniqueness of the u i is a special case of [8, Theorem 2.4].Because these functions are bounded, the Feynman–Kac formula as in [34,Theorem 5.7.6] as well as G (cid:48) /G = (log G ) (cid:48) and G ( T ) = 1 give e (cid:82) t a εi dτ u i ( t, x ) = E it,x (cid:34) (cid:90) Tt − G (cid:48) ( s ) G ( t ) 1 N N (cid:88) j =1 e (cid:82) s a εi dτ u j ( s, X s ) ds + (cid:90) Tt e (cid:82) s a εi dτ G ( s ) G ( t ) b εi ds + e (cid:82) T a εi dτ G ( t ) h i ( X T ) (cid:35) where the expectation is taken under the measure for which the state variablehas dynamics dX t = α i ( t, X t ) dt + β i ( t, X t ) dW it . Choose a uniform bound M for (cid:12)(cid:12) e (cid:82) s a εi dτ b εi (cid:12)(cid:12) and (cid:12)(cid:12) e (cid:82) T a εi dτ h i (cid:12)(cid:12) , and define K ( t, λ, γ ) = max (cid:110) | e (cid:82) t a εi ( τ,x τ ) dτ u i ( t, x t ) | (cid:111) < ∞ , where the maximum is taken over i ∈ { , . . . , N } , ε ∈ E , and ( x τ ) τ ∈ [0 ,t ] ∈ C ([0 , t ] , R ) . With this notation, | e (cid:82) t a ρi dτ u i ( t, x ) | ≤ (cid:90) Tt − G (cid:48) ( s ) G ( t ) K ( s, λ, γ ) ds + M (cid:90) Tt G ( s ) G ( t ) ds + MG ( t ) , which in turn leads to G ( t ) K ( t, λ, γ ) ≤ (cid:90) Tt (cid:16) − G (cid:48) ( s ) G ( s ) (cid:17) G ( s ) K ( s, λ, γ ) ds + M (cid:90) Tt G ( s ) ds + M.
19e may read this as an inequality of the form u ( t ) ≤ (cid:82) Tt B ( s ) u ( s ) ds + A ( t ) for u ( t ) = G ( t ) K ( t, λ, γ ) . Using G (cid:48) /G = (log G ) (cid:48) and that G is decreasing, (cid:82) Tt G ( r ) dr ≤ G ( t ) T and Grönwall’s lemma yield G ( t ) K ( t, λ, γ ) ≤ M ( G ( t ) T + 1) − M G ( t ) (cid:90) Tt (cid:16) (cid:90) Ts G ( r ) dr + 1 (cid:17) G (cid:48) ( s ) G ( s ) ds. (5.5)Observe that G satisfies G = λγ G (cid:48)(cid:48) and G (cid:48) ( T ) = 0 and λγ ( G (cid:48) ) G ≤ , so that − (cid:90) Tt (cid:16) (cid:90) Ts G ( r ) dr (cid:17) G (cid:48) ( s ) G ( s ) ds = (cid:90) Tt λγ G (cid:48) ( s ) G ( s ) ds ≤ T − t ≤ T. Together with − (cid:90) Tt G (cid:48) ( s ) G ( s ) ds = 1 − G ( t ) ≤ , it follows from (5.5) and G ( t ) ≥ that K ( t, λ, γ ) ≤ M ( T + 1) . As a ε isuniformly bounded in ε, t, x , the function u i is therefore uniformly boundedin ε, γ, λ, t, x by the definition of K ( t, λ, γ ) . Corollary 5.4.
There exists
M > such that | v λi ( t, x ) | ≤ M for all λ > and ( t, x ) ∈ [0 , T ] × R . Next, we establish an analogous uniform bound for the derivatives of thefunctions v λi and v λ from Theorem 5.2. Lemma 5.5.
Fix k ≥ . There exists M > such that | ∂ kx v λi ( t, x ) | , | ∂ kx v λ ( t, x ) | , | ∂ t ∂ kx v λ ( t, x ) | ≤ M for all λ > and ( t, x ) ∈ [0 , T ] × R .Proof. By Theorem 4.1 and Remark 4.3, the x -derivatives of the functions v λi , i = 1 , . . . , N from Theorem 4.1 satisfy the following PDEs obtained bydifferentiating (4.1) with respect to the spatial variable: ∂ t ∂ x v λi + 12 σ i ∂ xx ∂ x v λi + ( b i + σ i ∂ x σ i ) ∂ x ∂ x v λi (5.6) + (cid:16) ∂ x b i + G (cid:48) G (cid:17) ∂ x v λi − G (cid:48) G N N (cid:88) j =1 ∂ x v λj = 0 , ∂ x v λi ( T, · ) = f (cid:48) . | ∂ x v λi ( t, x ) | , and inturn also for ∂ x v λ ( t, x ) = N (cid:80) Ni =1 ∂ x v λi ( t, x ) . The corresponding bounds forthe higher-order x -derivatives follow by iterating this argument. Finally, theuniform bound for the time derivative of ∂ kx v λ is then direct consequences ofthe parabolic form of the PDEs (4.1), (5.6), etc., and their sums. Lemma 5.6.
For λ > , consider α , β , a λ , h of class C ∞ b and write L = ∂ t + 12 β ∂ xx + α∂ x . Suppose that w λ ∈ C ∞ b satisfies w λ ( T, · ) = h and ∂ t w λ , ∂ x w λ , ∂ xx w λ , a λ arebounded uniformly in λ . Then, the unique bounded classical solution u λ of L u λ + a λ ( t, x ) + G (cid:48) ( t ) G ( t ) ( u λ − w λ ) = 0 , u λ ( T, · ) = h satisfies | u λ ( t, x ) − w λ ( t, x ) |√ λ ≤ M for some M > independent of λ > and ( t, x ) ∈ [0 , T ] × R .Proof. Following the same steps as in the derivation of (4.8) yields that u λ ( t, x ) = w λ ( t, x ) + E t,x (cid:20)(cid:90) Tt G ( u ) G ( t ) (cid:16) a λ + L w λ (cid:17) ( u, X u ) du (cid:21) (5.7)where the expectation is taken under the measure for which the state variablehas dynamics dX t = α ( t, X t ) dt + β ( t, X t ) dW t . With a uniform bound M for a λ + L w λ , the desired bound is | u λ ( t, x ) − w λ ( t, x ) |√ λ ≤ M √ λ (cid:90) Tt G ( u ) G ( t ) du = − M √ λγ G (cid:48) ( t ) G ( t ) ≤ M √ γ where we have once again used G ( u ) = λγ G (cid:48)(cid:48) ( u ) and G (cid:48) ( T ) = 1 in the secondstep and the definition of G for the last inequality. Corollary 5.7.
Fix k ≥ . There exists M > such that | ∂ kx v λi ( t, x ) − ∂ kx v λ ( t, x ) |√ λ ≤ M for all λ > and ( t, x ) ∈ [0 , T ] × R and i ∈ { , . . . , N } . roof. In view of the PDEs (4.1) from Theorem 4.1 and the uniform boundsfrom Lemma 5.5, Lemma 5.6 yields that λ − / | v λi − v λ | ≤ M for some con-stant M . This proves the assertion for k = 0 . The analogous bounds forthe derivatives follow by applying the same argument to the correspondingPDEs obtained by differentiating (4.1) as in the proof of Lemma 5.5.We can now estimate the difference between v λ and the frictionless equi-librium price v of Proposition 5.1. Proposition 5.8.
Fix k ≥ . There exists M > such that | ∂ kx v λ ( t, x ) − ∂ kx v ( t, x ) |√ λ , | ∂ t ∂ kx v λ ( t, x ) − ∂ t ∂ kx v ( t, x ) |√ λ ≤ M for all λ > and ( t, x ) ∈ [0 , T ] × R .Proof. Using (4.2) and then (4.1) for v , subtracting the PDE (5.1) for v ,and using once again G ( u ) = λγ G (cid:48)(cid:48) ( u ) , we obtain ∂ t ( v λ − v ) + 12 ¯ σ ∂ xx ( v λ − v ) + ¯ b∂ x ( v λ − v ) (5.8) + 1 N N (cid:88) i =1 σ i ( ∂ xx v λi − ∂ xx v λ ) + 1 N N (cid:88) i =1 b i ( ∂ x v λi − ∂ x v λ ) = 0 with ( v λ − v )( T, · ) = 0 . Here ¯ b, ¯ σ are as defined in Proposition 5.1. The de-sired uniform bound for λ − / | v λ − v | is now a consequence of the Feynman–Kac formula and Corollary 5.7. The analogous result for λ − / | ∂ kx v λ − ∂ kx v | follows from the same argument because Remark 4.3 shows that these deriva-tives satisfy similar PDEs obtained by differentiating (5.8). The correspond-ing bounds for the time derivatives in turn are a consequence of the parabolicform of the equations.We have the following version of “Laplace’s method” for our function G ( t ) = cosh( (cid:113) γλ ( T − t )) as λ → . Lemma 5.9.
Given t ∈ [0 , T ) and a continuous function F on [ t, T ] , (cid:114) γλ (cid:90) Tt (cid:18) − G (cid:48) ( u ) G ( t ) (cid:18)(cid:90) ut F ( s ) ds (cid:19)(cid:19) du → F ( t ) as λ → . (5.9) Proof.
The left-hand side of (5.9) can be decomposed as (cid:114) γλ (cid:90) Tt − G (cid:48) ( u ) G ( t ) (cid:90) ut (cid:16) F ( s ) − F ( t ) (cid:17) dsdu + (cid:114) γλ (cid:90) Tt − G (cid:48) ( u ) G ( t ) (cid:90) ut F ( t ) dsdu. F on [ t, T ] and observing that (cid:113) γλ − G (cid:48) ( · ) G ( t ) converges to locally uniformly on ( t, T ] , one verifies that the first termvanishes for λ → . Integration by parts and G = λγ G (cid:48)(cid:48) show that the secondterm converges to F ( t ) .Together with the uniform bounds from Proposition 5.8, Lemma 5.9 al-lows us to compute the leading-order expansions of v λi − v λ and its derivatives. Lemma 5.10.
For k = 0 , , and ( t, x ) ∈ [0 , T ) × R , we have lim λ → ∂ kx v λi ( t, x ) − ∂ kx v λ ( t, x ) √ λ = ∂ kx L i v ( t, x ) √ γ . (5.10) Proof.
The proof is similar for k = 0 , , ; we only spell it out in the case k = 2 for which the computations are most involved. By Theorem 4.1 andRemark 4.3, the second-order x -derivatives of the functions v λi , i = 1 , . . . , N from Theorem 4.1 satisfy the following PDEs obtained by differentiating (4.1)twice with respect to the spatial variable: ∂ t ∂ xx v λi + 12 σ i ∂ xx ∂ xx v λi + ( b i + 2 σ i ∂ x σ i ) ∂ x ∂ xx v λi + (cid:16) c i + G (cid:48) G (cid:17) ∂ xx v λi + ∂ xx b i ∂ x v λi − G (cid:48) G N N (cid:88) j =1 ∂ xx v λj = 0 , ∂ xx v λi ( T, · ) = f (cid:48)(cid:48) , where c i = 2 ∂ x b i + ( ∂ x σ i ) + σ i ∂ xx σ i . As all functions appearing here arebounded either by assumption or by Remark 4.3, the Feynman–Kac formulaand G (cid:48) /G = (log G ) (cid:48) yield the stochastic representation ∂ xx v λi ( t, x )= E (cid:48) t,x (cid:34) (cid:90) Tt − G (cid:48) ( u ) G ( t ) (cid:16) e (cid:82) ut c i dτ ∂ xx v λ ( u, X u ) (cid:17) du + e (cid:82) Tt c i dτ f (cid:48)(cid:48) ( X T ) G ( t )+ (cid:90) Tt G ( u ) G ( t ) (cid:16) e (cid:82) ut c i dτ ∂ xx b i ∂ x v λi ( u, X u ) (cid:17) du (cid:35) where the expectation E (cid:48) [ · ] is taken under the measure Q (cid:48) for which thestate variable has dynamics dX t = ( b i + 2 σ i ∂ x σ i )( t, X t ) dt + σ i ( t, X t ) dW it .23ogether with (cid:82) Tt − G (cid:48) ( u ) G ( t ) du = 1 − G ( t ) , this implies ∂ xx v λi − ∂ xx v λ √ λ = E (cid:48) t,x (cid:34) (cid:90) Tt − G (cid:48) ( u ) √ λG ( t ) (cid:16) e (cid:82) ut c i dτ ∂ xx v λ ( u, X u ) − ∂ xx v λ ( t, x ) (cid:17) du + e (cid:82) Tt c i dτ f (cid:48)(cid:48) ( X T ) − ∂ xx v λ ( t, x ) √ λG ( t ) (5.11) + (cid:90) Tt G ( u ) √ λG ( t ) (cid:16) e (cid:82) ut c i dτ ∂ xx b i ∂ x v λi ( u, X u ) (cid:17) du (cid:35) . Recalling that c i , f (cid:48)(cid:48) and (by Lemma 5.5) also ∂ xx v λ are bounded (uniformlyin λ ), dominated convergence and the definition of G show that the expec-tation of the second term on the right-hand side of (5.11) converges to zeroas λ → . In view of lim λ → G ( T ) √ λG ( t ) = 0 , dominated convergence and in-tegration by parts show that the expectation of the third term convergesto E (cid:48) t,x (cid:20) lim λ → (cid:90) Tt − G (cid:48) ( u ) √ λG ( t ) (cid:16) (cid:90) ut e (cid:82) st c i dτ ∂ xx b i ∂ x v λi ( s, X s ) ds (cid:17) du (cid:21) = ∂ xx b i ∂ x v ( t, x ) . (5.12)Here we have used Corollary 5.7 and Proposition 5.8, and Lemma 5.9 forthe equality. Finally, the expectation of the first term on the right-hand sideof (5.11) can be rewritten by applying Itô’s formula to e (cid:82) ut c i dτ ∂ xx v λ ( u, X u ) ,inserting the Q (cid:48) -dynamics of X and taking into account that the correspond-ing local martingale part has expectation zero because all involved func-tions are bounded. Dominated convergence as well as Proposition 5.8 andLemma 5.9 then show that the corresponding limit for λ → is E (cid:48) t,x (cid:20) lim λ → (cid:90) Tt − G (cid:48) ( u ) √ λG ( t ) (cid:90) ut e (cid:82) st c i dτ L (cid:48)(cid:48) i ∂ xx v λ ( s, X s ) dsdu (cid:21) = L (cid:48)(cid:48) i ∂ xx v ( t, x ) , (5.13)where L (cid:48)(cid:48) i = ∂ t + σ i ∂ xx + ( b i + 2 σ i ∂ x σ i ) ∂ x + c i Id . The assertion for k = 2 now follows from (5.11–5.13) by observing that L (cid:48)(cid:48) i ∂ xx + ∂ xx b i ∂ x = ∂ xx L i .We can now prove the expansion of the equilibrium price for small trans-action costs. 24 roof of Theorem 5.2. We first observe that the PDE (5.8) for v λ − v admitsthe Feynman–Kac representation ( v λ − v )( t, x )= 1 N N (cid:88) i =1 ¯ E t,x (cid:20)(cid:90) Tt (cid:16) σ i ( ∂ xx v λi − ∂ xx v λ ) + b i ( ∂ x v λi − ∂ x v λ ) (cid:17) ( s, X s ) ds (cid:21) . Dominated convergence and the limits computed in Lemma 5.10 then yield lim λ → v λ − v √ λ ( t, x ) = (cid:114) γ N N (cid:88) i =1 ¯ E t,x (cid:20)(cid:90) Tt (cid:16) σ i ∂ xx + b i ∂ x (cid:17) L i v ( s, X s ) ds (cid:21) = (cid:114) γ N N (cid:88) i =1 ¯ E t,x (cid:20)(cid:90) Tt (cid:16) L i − ∂ t (cid:17) γ ˆ φ i, ( s, X s ) ds (cid:21) (5.14)where ˆ φ i, = L i v /γ is the frictionless equilibrium portfolio function ofagent i ; cf. (5.2). As these strategies clear the market, the sum of theirtime derivatives is zero and the pointwise limit (5.14) simplifies to (5.4).The family { λ − / ( v λ − v ) } λ> is bounded and equicontinuous by Proposi-tion 5.8; whence, the convergence is in fact locally uniform as a consequenceof the Arzelà–Ascoli theorem. Next, we study the asymptotics of the equilibrium price from Theorem 4.1 forsmall holding costs γ → (and fixed transaction costs λ > ). To emphasizethe dependence on γ , we denote the price v by v γ in this section. We againfocus on the case of a one-dimensional state variable ( d = 1 ) with smoothdrift and diffusion coefficients and terminal condition; see Remark 4.3.To formulate the result, we first note that the risk-neutral version γ = 0 of our model is well posed and essentially covered as a simple special case ofTheorem 4.1 (with the same proof, read with the conventions G ( u ) /G ( s ) = 1 and G (cid:48) ( u ) /G ( s ) = 0 ). The corresponding equilibrium price is the average ofall agents’ conditional expectations, v ( t, x ) = 1 N N (cid:88) i =1 v i ( t, x ) = 1 N N (cid:88) i =1 E it,x [ f ( X T )] , (6.1)and the corresponding portfolios are φ i, t = a i + (cid:90) t E is (cid:20)(cid:90) Ts L i v ( u, X u ) λ du (cid:21) ds. (6.2)25The above notation for the case γ = 0 should not be confused with thenotation for the case λ = 0 in the preceding section.)Lemma 3.1 shows that when γ > and λ > , the optimal portfoliostake into account future expected returns that are discounted with a kerneldetermined by γ/λ . As a limiting case, we have seen that the no-transaction-cost portfolio (5.2) for λ = 0 only takes into account the current (subjective)drift rates; this corresponds to an infinite discount. In the opposite extreme,the no-holding-cost portfolio (6.2) aggregates the future expected returnswithout discounting.Accordingly, we expect small holding costs to play a similar role as largetransaction costs. Indeed, Theorem 4.1 shows that when the supply a van-ishes, the equilibrium price only depends on the ratio γ/λ —the “urgencyparameter” that determines optimal execution trajectories [3] and, more gen-erally, optimal trading strategies with transaction costs in various contexts;cf., e.g., [39] and the references therein. When a > , Theorem 4.1 showsthat the equilibrium price is 0-homogeneous in ( γ, λ, /a ) . This means thatthe asset price remains invariant if the inverse of the supply is rescaled in thesame manner as transaction and holding costs: the larger trading and hold-ing costs of bigger asset positions are offset by reduced friction coefficients.The main result of this section is the following regular perturbation ex-pansion for small holding costs γ → . Theorem 6.1.
For fixed transaction costs λ > and small holding costs γ → , the equilibrium price function from Theorem 4.1 has the expansion v γ ( t, x ) = v ( t, x ) + γv ∗ ( t, x ) + o ( γ ) uniformly on [0 , T ] × R . (6.3) Here v is the equilibrium price (6.1) for γ = 0 and v ∗ ( t, x ) = − N N (cid:88) i =1 E it,x (cid:20)(cid:90) Tt φ i, s ds (cid:21) where φ i, t is the optimal strategy (6.2) of agent i for γ = 0 and the expecta-tion is taken under agent i ’s belief Q i . The reference point for the expansion (6.3) is the risk-neutral price v of (6.1). In this limiting case, agents only consider future expected returns.Other things equal, agents reduce the magnitude of their positions whenholding costs are introduced. The above expression for v ∗ reflects eachagent’s expectation E it,x [ (cid:82) Tt φ i, s ds ] of their average future position. Adding26olding costs reduces the demand by agents who expect to be long on av-erage, and the converse holds for shorts. The resulting sign of the pricecorrection will thus depend on the aggregate expectations in the market. In-deed, the formula for v ∗ shows that at the first order, the arithmetic averageover all agents’ expected average positions is the negative of the correction. Proof of Theorem 6.1. Step 1 . Similarly as for Theorem 5.2, the first steptowards proving this expansion is to establish that the functions v γi fromTheorem 4.1 are uniformly bounded in γ . Indeed, note that the function λG (cid:48) ( t ) NG ( t ) a is bounded locally uniformly in γ . Hence, Lemma 5.3 applied withthe PDEs (4.1–4.2) from Theorem 4.1 yields that given < ¯ γ < ∞ , thereexists M > such that | v γi ( t, x ) | ≤ M for all γ ∈ [0 , ¯ γ ] . (6.4) Step 2 . Next, we show that as γ → , | v γi ( t, x ) − v i ( t, x ) | → uniformly on [0 , T ] × R . (6.5)Indeed, (4.1–4.2) show that ∂ t ( v γi − v i ) + 12 σ i ∂ xx ( v γi − v i ) + b i ∂ x ( v γi − v i )+ G (cid:48) ( t ) G ( t ) v γi − N N (cid:88) j =1 v γj − λG (cid:48) ( t ) N G ( t ) a = 0 , ( v γi − v i )( T, · ) = 0 . Thus, the Feynman–Kac formula yields ( v γi − v i )( t, x )= E it,x (cid:90) Tt G (cid:48) ( s ) G ( s ) (cid:16) v γi ( s, X s ) − N N (cid:88) j =1 v γj ( s, X s ) − λG (cid:48) ( s ) N G ( s ) a (cid:17) ds (6.6)where the expectation is taken under agent i ’s subjective probability mea-sure Q i . Note that G (cid:48) ( t ) → and G ( t ) → as γ → , uniformly on [0 , T ] .In view of (6.4), we conclude (6.5). Step 3 . We can now prove the expansion from Theorem 6.1. By (6.6), ( v γi − v i )( t, x ) γ = E it,x (cid:90) Tt G (cid:48) ( s ) γG ( s ) (cid:16) v γi ( s, X s ) − N N (cid:88) j =1 v γj ( s, X s ) − λG (cid:48) ( s ) N G ( s ) a (cid:17) ds . G and Dini’s theorem, lim γ → G (cid:48) ( t ) G ( t ) = 0 and lim γ → G (cid:48) ( t ) γG ( t ) = − T − tλ , uniformly on [0 , T ] . (6.7)Together with (6.4), dominated convergence, (6.5) and (6.1), this yields lim γ → v γi ( t, x ) − v i ( t, x ) γ = E it,x (cid:20)(cid:90) Tt T − sλ (cid:16) v ( s, X s ) − v i ( s, X s ) (cid:17) ds (cid:21) uniformly on [0 , T ] × R . In view of the definition of v γ in (4.2), and (6.7), itfollows that lim γ → v γ ( t, x ) − v ( t, x ) γ = 1 N N (cid:88) i =1 E it,x (cid:20)(cid:90) Tt T − sλ (cid:16) v ( s, X s ) − v i ( s, X s ) (cid:17) ds (cid:21) − ( T − t ) a N . (6.8)By (6.2) and the first identity of (4.8) in the special case γ = 0 , we have φ i, t − a i = (cid:90) t E is (cid:20) λ (cid:16) f ( X T ) − v ( s, X s ) (cid:17)(cid:21) ds = (cid:90) t λ (cid:16) v i ( s, X s ) − v ( s, X s ) (cid:17) ds. Using this identity to integrate (6.8) by parts and taking into account themarket-clearing condition (cid:80) Ni =1 φ i, = a , the theorem follows. To gain further intuition for the equilibrium of Theorem 4.1, we consider anexample that can be solved explicitly up to a system of linear ODEs. Wewill also use this example to test the numerical accuracy of the expansionsfor small transaction and holding costs relative to the exact solution.Suppose that f ( x ) = x , so that at time T , the state X represents theasset’s payoff. Agents believe that X has mean-reverting dynamics dX t = κ i ( ¯ X − X t ) dt + σdW it . (7.1)That is, agents agree on the volatility σ > and the mean-reversion level ¯ X > , but disagree about the mean-reversion speed κ i > . This can beinterpreted as a simple model for a forward contract on a mean-revertingunderlying such as an FX rate. As is natural in that context, and to simplifythe exposition, we henceforth assume that the net supply of the contract is a = 0 . 28 .1 Equilibrium with Costs We first consider the exact equilibrium price v with transaction costs λ > and holding costs γ > from Theorem 4.1. For the linear state dynam-ics (7.1), the parabolic system (4.1–4.3) can be reduced to a system of linearODEs by the ansatz v λi ( t, x ) = A i ( t ) + B i ( t ) x, i = 1 , . . . , N. Indeed, writing N and I N for the N × N -matrices of ones and the identitymatrix, respectively, the deterministic functions B = ( B , . . . , B N ) (cid:62) and A = ( A , . . . , A N ) (cid:62) satisfy B (cid:48) ( t ) = (cid:20) diag( κ , . . . , κ n ) + G (cid:48) ( t ) G ( t ) (cid:18) N N − I N (cid:19)(cid:21) B ( t ) ,B ( T ) = 1 and A (cid:48) ( t ) = G (cid:48) ( t ) G ( t ) (cid:18) N N − I N (cid:19) A ( t ) − ¯ X diag( κ , . . . , κ N ) B ( t ) ,A ( T ) = 0 . These ODEs have unique, smooth solutions. Moreover, the equilibrium pricethen satisfies v ( t, x ) = 1 N N (cid:88) i =1 ( A i ( t ) + B i ( t ) x ) = ¯ X + ( x − ¯ X ) 1 N N (cid:88) i =1 B i ( t ) , (7.2)where we have used the ODEs for the A i and B i for the second equality. Tobe precise, the unbounded terminal conditions and state dynamics (7.1) donot satisfy the boundedness assumptions of Theorem 4.1. However, with theunique solutions A and B of the above ODEs at hand, the arguments in theproof of Theorem 4.1 show that (7.2) identifies the unique equilibrium pricein the class from smooth functions with linear growth, say. We first study the equilibrium v with vanishing transactions costs λ = 0 and fixed holding costs γ > . As the state variable has the dynamics dX t = ¯ κ ( ¯ X − X t ) dt + σdW t with ¯ κ = 1 N N (cid:88) i =1 κ i (7.3)29nder the aggregate measure ¯ Q , Proposition 5.1 with a = 0 yields that v ( t, x ) = ¯ E t,x [ X T ] = ¯ X + ( x − ¯ X ) e − ¯ κ ( T − t ) . (7.4)As a result, agent i believes that the frictionless equilibrium price has dy-namics dv ( t, X t ) = ( κ i − ¯ κ ) e − ¯ κ ( T − t ) ( ¯ X − X t ) dt + e − ¯ κ ( T − t ) σdW it = ( κ i − ¯ κ ) (cid:0) ¯ X − v ( t, X t ) (cid:1) dt + e − ¯ κ ( T − t ) σdW it . (7.5)This means that agents who believe in faster than average mean-reversion(i.e., κ i > ¯ κ ) observe a mean-reverting process. By contrast, agents whobelieve in slower than average mean reversion conclude that the processexhibits “momentum” in that prices above the mean-reversion level are fol-lowed by further positive drifts. Whence, in equilibrium, the market is en-dogenously populated by both “mean-reversion traders” and “trend-followers”even though all agents believe that the underlying has a mean-reverting fun-damental value.Next, we study the leading-order correction v λ ( t, x ) − v ( t, x ) for λ → .Again, Theorem 5.2 does not apply directly due to the unbounded coeffi-cients, but it is straightforward to carry out the arguments in the proof forthe example at hand. Thus, the leading-order correction is √ λv ∗ ( t, x ) with v ∗ ( t, x ) = 1 √ γN N (cid:88) i =1 ¯ E t,x (cid:20)(cid:90) Tt e − ¯ κ ( T − s ) (¯ κ − κ i ) ( X s − ¯ X ) ds (cid:21) = 1 √ γ (cid:32) N N (cid:88) i =1 (¯ κ − κ i ) (cid:33) (cid:20)(cid:90) Tt ¯ E t,x [ X s − ¯ X ] e − ¯ κ ( T − s ) ds (cid:21) = 1 √ γ (cid:32) N N (cid:88) i =1 (¯ κ − κ i ) (cid:33) (cid:20)(cid:90) Tt e − ¯ κ ( T − t ) ( x − ¯ X ) ds (cid:21) = (cid:114) γ (cid:32) N N (cid:88) i =1 (¯ κ − κ i ) (cid:33) ( T − t ) e − ¯ κ ( T − t ) ( x − ¯ X ) . (7.6)Note that ∂ x v ∗ ≥ , so that the equilibrium volatility is always increasedwhen small transaction costs are added. This is in line with the asymmetricinformation model of [19], the risk-sharing model of [32], numerical resultsof [1, 14], and empirical studies such as [29, 33, 47].In our model, the reason for the increased volatility is that the sign ofthe correction term v ∗ is determined by x − ¯ X , so that transaction costs30mplify the fluctuations of the frictionless equilibrium price (7.4). Let usnow discuss why illiquidity affects price levels in this manner. In view of theabove formula for v ∗ , adding small transaction costs increases equilibriumprices when X t > ¯ X and reduces prices for X t < ¯ X . If X t > ¯ X , agents whobelieve in larger than average mean-reversion speeds predict the friction-less equilibrium price (7.5) to mean-revert downwards towards its long-runmean. Conversely, agents believing in a lower than average mean-reversionspeed expect the positive trend to continue and prices to rise even further.Accordingly, the first group of agents wants to sell the asset and the sec-ond group wants to purchase it. With small transaction costs added, thesetrading motives persist, yet changes in portfolios can only be implementedgradually. Accordingly, agents do not only take into account the differencebetween the current value of the state variable and its long-run mean, butalso their expected differences in the future. Since agents believing in fastermean-reversion expect differences to disappear faster, they have a weakermotive to act on the trading opportunities they observe. For X t > ¯ X , thismeans that sellers have a weaker motive to trade than buyers, so that pricesneed to rise in order to clear the market. For X t < ¯ X , the situation is re-versed and small transaction costs decrease prices relative to their frictionlesscounterparts.In summary, adding small transaction costs increases prices above themean-reversion level and decreases price below it, thereby generating largerprice fluctuations and a larger equilibrium volatility. As optimists and pes-simists are similarly affected by the costs, the effect of illiquidity on pricelevels in our model is ambiguous. Depending on the situation, an increase of λ can lead to an “illiquidity discount” as observed e.g. in [4], or it may increasethe price as in [22] where illiquidity can be an obstruction to shorting. Onecan note that in our example, the correction term v ∗ mean-reverts aroundzero under each agent’s probability measure. In that sense, the average pricelevel remains unchanged. We now turn to the small-holding-cost asymptotics from Section 6. As a firststep, we compute the equilibrium price v with vanishing holding costs γ = 0 and fixed transactions costs γ > . From (6.1), we have v ( t, x ) = 1 N N (cid:88) i =1 v i ( t, x ) , v i ( t, x ) = E it,x [ X T ] = ¯ X + ( x − ¯ X ) e − κ i ( T − t ) . (7.7)By Itô’s formula, agent i believes that this risk-neutral equilibrium price hasdynamics dv ( t, X t )= 1 N N (cid:88) j =1 ( κ j − κ i ) e − κ j ( T − t ) ( X t − ¯ X ) dt + 1 N N (cid:88) j =1 e − κ j ( T − t ) σdW it = (cid:32) κ i − (cid:80) Nj =1 κ j e − κ j ( T − t ) (cid:80) Nj =1 e − κ j ( T − t ) (cid:33) (cid:0) ¯ X − v ( t, X t ) (cid:1) dt + 1 N N (cid:88) j =1 e − κ j ( T − t ) σdW it . The first factor is the difference between κ i and a (time-dependent) weightedaverage of κ , . . . , κ N . Thus, the interpretation is similar as for the equilib-rium (7.5) with λ = 0 : agents who believe in fast mean reversion observea mean-reverting asset price whereas agents believing in slow mean rever-sion perceive momentum. One can also note that the equilibrium volatilitywithout holding costs is always larger than or equal to its counterpart with-out transaction costs. This follows by applying Jensen’s inequality to thegradients of v and (7.4).We now turn to the leading-order correction term for γ → . Again,the boundedness assumptions in Theorem 6.1 are not satisfied in this ex-ample, but the arguments in the proof still apply. Accordingly, using therepresentation (6.8), we have v ∗ ( t, x ) (7.8) = 1 N N (cid:88) i =1 E it,x (cid:20)(cid:90) Tt T − sλ (cid:16) v ( s, X s ) − v i ( s, X s ) (cid:17) ds (cid:21) = 1 N N (cid:88) i =1 (cid:90) Tt T − sλ (cid:16) N N (cid:88) j =1 e − κ j ( T − s ) − e − κ i ( T − s ) (cid:17) e − κ i ( s − t ) ( x − ¯ X ) ds = ( x − ¯ X )( T − t ) λN (cid:88) i (cid:54) = j κ i − κ j e − κ j ( T − t ) − ( T − t )( N − N (cid:88) i =1 e − κ i ( T − t ) where the last equality follows from an elementary but lengthy integration.The Chebychev sum inequality applied to the second representation showsthat the coefficient multiplying x − ¯ X is always negative. Whence, addingsmall holding costs increases the risk-neutral equilibrium price when the32tate process X t is below its mean-reversion level ¯ X and decreases it when X t > ¯ X . Since larger holding costs play the same role as lower transactioncosts in our model for a = 0 , the intuition for this is the converse of theargument for adding small transaction costs in Section 7.2.In particular, in view of (7.7), adding small holding costs dampens thefluctuations of the risk-neutral equilibrium price and accordingly reduces theequilibrium volatility. This negative effect on the equilibrium volatility andthe positive effect of small transaction costs are consistent with the obser-vation made above that the equilibrium volatility without transaction costsalways lies below its counterpart with no holding costs. In fact, the exactequilibrium volatility N (cid:80) Ni =1 B i ( t ) from Section 7.1 smoothly interpolatesbetween these two extreme cases as γ/λ ranges between ∞ and . To assess the accuracy of the small-cost asymptotics from Sections 7.2 and7.3, we now compare the explicit asymptotic formulas (7.6) and (7.8) tothe numerical solutions of the ODEs from Section 7.1 describing the exactequilibrium price. Throughout, we consider a time horizon of T = 3 years.To obtain reasonable values for the other model parameters, we cali-brate the state dynamics (7.3) to USD/EUR exchange rate data from 2009–2019 available from the website of the St. Louis Fed at https://fred.stlouisfed.org/series/DEXUSEU . The model parameters can then be es-timated by matching the first two stationary moments to their empiricalcounterparts and fitting the (linear) log-autocorrelation function to the em-pirical one using linear regression. This leads to σ = 0 . , ¯ X = 1 . , ¯ κ = 0 . . With a zero net supply, this suffices to pin down the equilibrium price withouttransaction costs (7.4), since the latter does not depend on the agents’ hold-ing costs in this case. For the equilibrium prices with transaction costs (7.2),we additionally need to specify each agent’s individual belief as well as thetransaction cost λ and the holding cost γ . Inspired by similar parametervalues used for commodities and equities in [25, 15], respectively, we use λ = 10 − and γ = 10 − . The free parameter κ = 2¯ κ − κ can in turn be chosen arbitrarily to capturethe agents’ disagreement about the mean-reversion speed of the exchangerate. For κ = 3 κ = 0 . , x = 1 , equilibrium asset prices and volatilities are plotted in Figure 1.These numerical examples clearly display the qualitative properties derivedfrom the asymptotic formulas in the previous sections. Indeed, the equilib-rium values with both holding and transaction costs always lie between thelimiting cases where only one of these costs is present. 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