Insider Trading with Temporary Price Impact
IInsider Trading with Temporary Price Impact
Weston Barger a , Ryan Donnelly b a University of Washington, Seattle, WA, United States b King’s College London, London, UK
Abstract
We model an informed agent with information about the future value of an asset trying to maxi-mize profits when subjected to a transaction cost as well as a market maker tasked with setting fairtransaction prices. In a single auction model, equilibrium is characterized by the unique root of aparticular polynomial. Analysis of this polynomial with small levels of risk-aversion and transactioncosts reveal a dimensionless parameter which captures several orders of asymptotic accuracy of theequilibrium behaviour. In a continuous time analogue of the single auction model, incorporation ofa transaction costs allows the informed agent’s optimal trading strategy to be obtained in feedbackform. Linear equilibrium is characterized by the unique solution to a system of two ordinary dif-ferential equations, of which one is forward in time and one is backward. When transaction costsare in effect, the price set by the market maker in equilibrium is not fully revealing of the informedagent’s private signal, leaving an information gap at the end of the trading interval. When consider-ing vanishing transaction costs, the equilibrium trading strategy and pricing rules converge to theirfrictionless counterparts.
Keywords: market microstructure, asymmetric information, price impact, transaction cost
1. Introduction
When traders place orders on a securities exchange, they face transaction frictions. Direct frictionsinclude exchange and brokerage fees, but consumers of liquidity also experience indirect costs. Liq-uidity providers adjust their limit orders to reflect the information contained in incoming marketorders by moving their price quotes in the direction of order flow. This adversely affects traders whotake liquidity as their subsequent orders will be transacted at a less favourable price. Additionally,if an aggressive order is large enough then it consumes all of the liquidity at the best available priceand the remainder of the order is executed at sequentially worse prices. This can be thought of asa transaction cost which is dependent on the size of the aggressive order and the state of the orderbook.
Email addresses: [email protected] (Weston Barger), [email protected] (Ryan Donnelly) a r X i v : . [ q -f i n . T R ] J u l isk-aversion and transaction costs have been previously studied in the insider trading literature.The authors Holden and Subrahmanyam (1994) extend the discrete-time model of Kyle (1985) toinclude an exponentially risk-averse insider. Furthermore, their model allows for multiple insiderswith the same level of risk-aversion who all receive identical information. Subrahmanyam (1998)further extends the model of Holden and Subrahmanyam (1994) to include a quadratic transactioncost for a risk-averse insider in discrete-time. Baruch (2002) extends the continuous-time modelwhich was first given by Kyle (1985) and generalized by Back (1992) by including risk-aversion.In this work, we model an exponentially risk-averse (or risk-neutral) insider who faces a transactioncost per share that is proportional to the size of the order. We first present a single-auction modeland classify the unique linear equilibrium. We show that the market maker’s equilibrium pricingrule corresponds to the unique positive root of a particular polynomial determined by the modelparameters. An asymptotic expansion of the roots of the aforementioned polynomial is performedfor small transaction cost and small risk-aversion, which allows us to examine the effects of thetransaction cost relative to frictionless models.We then formulate an analogous model in continuous-time and present a linear equilibrium classifiedby the solution to a forward-backward ordinary differential equation (FBODE). We show that theresulting FBODE has a unique solution and is explicitly solvable when the insider is risk-neutral. Wethen analyze the effects of varying the model parameters on equilibrium using numerical solutionsof the associated FBODE. Although we cannot solve for equilibrium explicitly unless the insider isrisk-neutral, we are able to make conclusions about the nature of equilibrium in certain limiting casesof the transaction cost parameter. In particular, when the transaction costs vanish the equilibriumtrading and pricing rules converge to their frictionless counterparts. This result could be used tosimplify the analysis of other similar asymmetric information models because it provides a family offeedback controls which converge to the equilibrium control in the frictionless case.The models formulated in this paper are also related to those of optimal execution literature. Often inthat literature, the pressure on the asset price exerted by order flow is referred to as permanent priceimpact, while the immediate cost associated with market microstructure is referred to as temporaryprice impact. In their seminal work, the authors of Almgren and Chriss (2001) model permanent andtemporary price impact by defining two distinct price processes: the midprice and the transactionprice. The midprice is the midpoint between the best quoted bid and ask prices set by liquidityproviders, and the transaction price is the average price per unit of asset at which the trader collectsproceeds from trades. The authors of Almgren and Chriss (2001) model permanent impact by lettingthe drift of the midprice be an exogenous function of the trader’s order flow. Temporary impact ismodelled by defining the transaction price of trades to be equal to the midprice plus an exogenousfunction of the trader’s order volume.Our model also includes midprice and transaction price processes. As distinct from Almgren andChriss (2001), we directly model price setting market makers which allows for the permanent priceimpact to be endogenous. However, we define a transaction price process that is analogous themodel of Almgren and Chriss (2001) by explicitly introducing an exogenous transaction cost. To beconsistent with the insider trading literature, we refer the permanent price impact effect simply asprice impact and temporary price impact effect as transaction cost.2he continuous-time version of our model is a direct generalization of the continuous-time modelsof the aforementioned papers Kyle (1985), Back (1992) (when the insider’s signal is Gaussian),and Baruch (2002) (with constant volatility of noise trading). As such, some qualitative features ofequilibrium in these works also arise in the present paper, but there are also some notable differences.In particular, a key feature of many other models with asymmetric information is that the asset priceis always fully revealing of the insider’s signal at the end of the trading horizon (the insider alwayshas incentive to exploit her informational advantage). This is not the case in our model whenthe transaction cost is non-zero. As a consequence, revelation of the insider’s signal does containinformation not already incorporated in the publicly available price.The rest of the paper is organized as follows. In Section 2, we develop a single-auction model,present the unique linear equilibrium, and analyze the effects of the transaction cost by performingan asymptotic expansion for small transaction cost. We shift our focus to continuous-time in Section3 and begin by presenting a continuous-time model in Section 3.1. We begin Section 3.2 by developingthe mathematical machinery necessary the subsequent presentation of the linear equilibrium, and wefinish the section by presenting a linear continuous-time equilibrium. In Section 4, we demonstratethe effects on the equilibrium of the previous section of varying the model parameters. In Section 4.1,we pay special attention to the transaction cost parameter by analyzing the limit of the equilibriumas this parameter tends to zero and infinity. Some concluding remarks are offered in Section 5.
2. Single-Auction
In this section, we consider a single-auction market where the transaction price of the insider’s tradesincurs an additional cost per share which is linear with respect to the trade volume. After presentingthe model which describes the dynamics of trade and the objective of the insider and market maker,we prove the existence of a unique equilibrium in this setting. We then investigate the effects of thetransaction cost on the associated equilibrium.The single-auction model in this section is similar to work contained in Subrahmanyam (1998) forthe case of a single agent. Though structurally similar, this previous work never considers a modelin which risk-averse insiders interact with unpredictable noise traders. In that paper either theinsider is risk-neutral or the volume traded by the noise traders is directly observed by the insiderbefore submitting their own trade (in that case the stochasticity in the model comes from a randomendowment to the insiders). Some of our results are analogous to Subrahmanyam (1998), but thereare some distinctions, for example that our model guarantees equilibrium whereas the lack of noisetraders can give rise to situations with no equilibrium (see Lemma 2 of Subrahmanyam (1998)). Weinclude the single-auction results so that we may perform a more in-depth analysis of the equilibriumthrough an asymptotic expansion, and for the sake of completeness before investigating a continuous-time version of the model. 3 .1. Model
In the spirit of Kyle (1985), we consider a single-auction on a market with one risky asset that istraded on an exchange with three types of traders: market makers who set the asset’s midprice, aninsider who has information about the future value of the asset, and noise traders. We let v denotethe ex-post liquidation value of the asset, and we assume that v ∼ N ( v , Σ v ).The insider receives the realization of v before the auction takes place, but this information isunavailable to the public. Thus, she wishes to utilize her informational advantage by submitting anorder of size ∆ x in an auction. We assume that the number of noise traders on the exchange is large,and we denote the aggregate order of the noise traders by ∆ z , which we assume to be distributed as∆ z ∼ N (0 , σ ). The aggregate order of the insider and noise traders are submitted to the marketmaker who observes on the total quantity ∆ y , where∆ y = ∆ x + ∆ z . (1)The auction takes place in two phases. First, the insider and the noise traders submit orders to theexchange, and second, the market maker observes the aggregate order ∆ y and sets the midprice p .We assume that the insider pays a transaction cost proportional to the size of her order so that theeffective transaction price is (cid:98) p = p + c ∆ x , (2)where c > p , and the effective transaction price, (cid:98) p , could arise fromone of many sources. The interpretation given in Subrahmanyam (1998) is that of a transactiontax, possibly invoked upon the market by a regulator. A different interpretation could be that theeffective transaction price is due to different preferences between a large number of market makers. Inprevious works, it is assumed that there is a very large number of perfectly competitive risk-neutralmarket makers which drives all of them to quote the same price. In reality, liquidity providers mayset different prices than each other, giving rise to a demand structure depending on their aggregatequotes. We do not explicitly model this behaviour or interaction between market makers and insteadcapture this effect through the linear dependence of transaction price on trade volume.Without a loss of generality, we assume that the insider holds no shares of the asset before theauction. The wealth of the insider after the trades are executed is thus w = ( v − (cid:98) p ) ∆ x . (3)The insider would like to maximize the utility of her expected wealth w . That is, the insider chooses4 x to achieve max ∆ x E [ U ( w ) | v ] , (4)where U is the insider’s utility function. The market maker is tasked with setting prices efficiently.That is, the market maker chooses the price p such that p = E [ v | ∆ y ] . (5) We begin this section by defining what it means for a pricing rule and a trading strategy to forman equilibrium. We then classify the unique linear equilibrium for an exponentially risk-averse (orrisk-neutral) insider.We assume that the market maker and insider choose pricing rules and trading strategies, respectively,as functions of information available to them during the auction. That is, the market maker choosesthe price p as a function of ∆ y , and the insider chooses her order size ∆ x as a function of v . Let P and X be functions such that p = P (∆ y ) and ∆ x = X ( v ). Definition 1. A single-auction equilibrium ( P, X ) consists of a pricing rule P and tradingstrategy X such that • given the pricing rule P , the order ∆ x given by the trading strategy X ( v ) achieves the maximumin (4) , and • given the trading strategy X , the price p given by the pricing rule P (∆ y ) satisfies the efficiencycondition (5) .The pair ( P, X ) is a linear single-auction equilibrium if both P and X are linear functions oftheir arguments. Suppose that P is the set of pricing rules P ≡ P (∆ y ) and let P ⊂ P be the set of pricing rulesfor which there exists a corresponding trading strategy X that satisfies (4). Similarly, let X bethe set of trading strategies X ≡ X ( v ) and let X ⊂ X be the set of trading strategies for whichthere exists a pricing rule P that satisfies (5). The sets P and X induce mappings ρ : P → X and ξ : X → P . A pricing rule P and trading strategy X form a single-auction equilibrium if ρ ( P ) ∈ X , ξ ( X ) ∈ P and ( P, X ) = ( ξ ( X ) , ρ ( P )).5e will restrict our focus to the exponentially risk-averse (or risk-neutral) insider. For any constant A ≥
0, we define the utility function U ( w ) = (cid:26) w, A = 0 − e − Aw , A > , (6)and we refer to A as the risk aversion parameter.Before presenting the classification of linear equilibrium it is helpful to introduce the constants λ K = 12 (cid:114) Σ v σ , β K = 12 λ K . (7)The constants λ K and β K correspond to the single-auction pricing rule and trading strategy, respec-tively, of Kyle (1985). In that work, the market maker’s single-auction equilibrium pricing rule fora risk-neutral insider is P K (∆ y ) = v + λ K ∆ y , and the corresponding optimal trading strategy is X K ( v ) = β K ( v − v ). We will write the equilibrium pricing rule and trading strategy of our modelin terms of λ K which helps to illuminate the effect of the added transaction cost. Theorem 2 (Single Auction Equilibrium).
Choose A ≥ and let U , defined in (6) , be theinsider’s utility function. Then the unique linear single-auction equilibrium is given by P (∆ y ) = v + λ ∆ y , (8) X ( v ) = β ( v − v ) (9) where β = 12 ( λ + c ) + A σ λ , (10) and λ is the unique positive root of the polynomial r ( x ) = A σ x + 4 A σ x + 4 (1 + A c σ ) x + 4 (2 c − A σ λ K ) x + 4 ( c − λ K ) x − c λ K . (11) Proof
For a proof see Section 6.1 in the appendix.Intuitively, a risk-averse trader prefers to submit a smaller order compared to her risk-neutral coun-terpart because the inventory holdings are exposed to the randomness associated with the noisetraders. For this reason one might expect that the insider submits smaller orders with as the value6f A is increased. Correspondingly, less information about the true value of the asset would becontained in the order signal ∆ y received by the market maker, and she thus reduces the severityof her price adjustment. One might also expect that as the transaction cost parameter c increasesit becomes less worthwhile for the insider to submit large orders, regardless of her risk preference,thus leading to a smaller β and λ . Both of these statements are indeed true as summarized by thefollowing proposition. Proposition 3 (Single Auction Parameter Dependence).
Let P (∆ y ) = v + λ ∆ y and X ( v ) = β ( v − v ) be the pricing rule and trading strategy forming the unique single-auction equilibrium givenby Theorem 2. If A > then ∂β∂c < , ∂β∂A < ,∂λ∂c < , ∂λ∂A < . Proof
For a proof see Section 6.2 in the appendix.The classification of equilibrium given by Theorem 2 is in terms of a root of a fifth degree polynomial.In general the quantities λ and β involved in the equilibrium will not have closed form expressions interms of model parameters. However, we are able to find approximations to these quantities whichhold when certain model parameters are small. These approximations are given in the followingproposition. Proposition 4 (Single Auction Approximation).
Let P (∆ y ) = v + λ ∆ y and X ( v ) = β ( v − v ) be the pricing rule and trading strategy forming the unique single-auction equilibrium given byTheorem 2, and define the dimensionless parameter ν := λ K σ A cλ K . (12) Then the quantities λ and β admit the following approximations: λ = λ K (cid:18) − ν + ν (cid:19) + o ( ν ) , (13) β = 12 λ K (cid:18) − ν + 32 ν (cid:19) + o ( ν ) . (14) Proof
For a proof see Section 6.3 in the appendix.7s should be expected, the approximations to λ and β both converge to their counterparts presentin Kyle (1985) as A, c →
0, namely λ K and 1 / (2 λ K ). In addition, there are some other interestingobservations to be made about this result.The first and most significant observation is that both approximations depend on the dimensionlessparameter ν as defined in (12), and not on the individual values of A or c . A priori, there is no reasonwhy this dimensionless parameter ν fully determines the approximations, and not the individualvalues of A and c . The authors have verified that higher order approximations with respect to A and c fully depend on their individual values, and thus expanding with respect to the dimensionlessparameter ν alone is not possible to higher order that what appears in the Proposition 4.The second observation stems again from the dimensionless parameter ν , which means that bothparameters have the same qualitative effect on equilibrium locally around A = c = 0. The magnitudeof the effects, however, depend on other model parameters. For example, if λ K is large, then increasing A will have more effect on equilibrium than increasing c . This can be made sense of intuitively fromthe perspective of the insider. If Kyle’s model is taken as a reference, then λ K σ represents amagnitude of price risk faced by the insider due to the random noise traders, and λ K on its ownrepresents a cost of trading to the insider. If λ K is large, then the insider faces a large amount of riskand we would expect any increase in risk-aversion from zero to have a significant effect on equilibrium.Similarly, when λ K is large, increasing the value of c from zero represents a relatively small changein the total trading cost to the insider and should have insignificant effect on equilibrium.The third observation is that the approximation of λ does not have any first order correction terms.This is an indication that locally around A = c = 0, the equilibrium value of λ is relatively robustwith respect to changes in A and c because it is affected only at second order and higher. Theexpansion for β however does have first order corrections with respect to both parameters, so theinsider’s trading strategy does not retain the same level of robustness to small changes of A and c from 0.To demonstrate the accuracy of these approximations, we plot both the exact equilibrium quantitiesand the approximations for various sets of parameters. For the sake of visual clarity we plot thiscomparison with respect to a single scale parameter denoted θ which acts as follows: we fix valuesof A and c and find the equilibrium quantities and the corresponding approximations after makingthe replacements A (cid:55)→ θ A , c (cid:55)→ θ c . We then plot the results as a function of θ in order to observe convergence as θ →
0. The results areshown in Figure 1 for three pairs of A and c . Due to the scaling introduced by θ , the actual valuesof A and c are of less significance than their ratio. The curves which are shown would not be alteredby changing A and c such that their ratio is fixed after a reparameterization of θ . Recall that λ K depends on the other market parameters Σ v and σ , so this remark depends on changing λ K withoutchanging σ Figure 1: Dotted curves show the exact equilibrium quantities for risk-aversion level θA and transaction cost θc . Solidcurves show the approximation of equilibrium quantities given by Proposition 4. Other parameters are σ = 1 andΣ v = 1.
3. Continuous-Time Auction
In this section, we present a continuous-time model which incorporates an analogous transaction costto the previous section. After introducing the model in Section 3.1, we give the notion of marketequilibrium which we consider. Some mathematical machinery is then established which is necessaryfor classifying equilibrium in continuous-time. When the insider is risk-neutral, we can solve for linearequilibrium in closed form. When they are risk-averse, the equilibrium is classified by a solution toan ODE, of which we prove there is always a solution. Finally, we analyze the dependence on themodel parameters of the equilibrium solution, and give closed form expressions for the limits of theequilibrium as the transaction cost parameter tends to extreme values.
We consider a similar setting to the previous section with three types of market participants, exceptorders are submitted and prices are set in continuous-time. Let the processes P = ( P t ) ≤ t ≤ T be themidprice of the asset, X = ( X t ) ≤ t ≤ T be the insider’s inventory, and Z = ( Z t ) ≤ t ≤ T be a standardBrownian motion. The cumulative orders submitted by the noise traders up to time t is equal to σZ t , where σ > Y = ( Y t ) ≤ t ≤ T be the total number of shares submitted to themarket up to time t . That is, Y t = X t − X + σ Z t . (15)As in the single-auction model, we assume that the market maker observes the aggregate order flow Y but neither of the components X or Z . We let the time T value of the asset v be normallydistributed with mean v and variance Σ v , independent of the Brownian motion Z . The insider isprivy to the realization of v at the initial time t = 0.It is helpful to explicitly denote the sets of information available to the various market participants9t a given time. We define the filtrations F Yt and F Zt to be the filtrations generated by the processes Y and Z , respectively. As the market maker observes only Y , we define F Mt = F Yt to be the marketmaker’s information. The insider is aware of her trades, so she can back out the path of liquiditytrades by observing historical prices (see Back (1992)), and the insider has received the realizationof v before trading begins. So we set the insider’s information to be F It = σ ( F Zt ∪ σ ( v )).We assume that insider employs trading strategies that yield an absolutely continuous inventoryprocess almost surely. To this end, we write dX t = θ t dt , (16)and analogously to the single-auction model, we define the insider’s transaction price (cid:98) P = ( (cid:98) P t ) ≤ t ≤ T to be (cid:98) P t = P t + c θ t , (17)where c is a positive constant.The assumption that the insider’s inventory path is absolutely continuous may seem restrictive. Inmost other model formulations with asymmetric information, the insider may trade according toany process which is adapted to the filtration F It , including processes with discontinuities or diffusivecomponents. However, it is generally the case that in equilibrium the insider’s inventory is absolutelycontinuous. We do not attempt to model the transaction cost for trading strategies which are notabsolutely continuous. There are many works in the context of portfolio optimization which includetransaction costs of various forms (among many others, see Magill and Constantinides (1976), Davisand Norman (1990), and Muhle-Karbe et al. (2017)), but it is unlikely that the optimal strategieswhich result in those models would be consistent with the notion of equilibrium that we consider.At the end of the trading horizon, the insider’s wealth is X T v minus the cost of trading throughoutthe period. Thus, the terminal wealth W T is given by W T = X T v − (cid:90) T (cid:98) P s θ s ds . (18)The insider chooses a strategy to maximize her expected utility of terminal wealth. That is, θ ischosen to achieve sup θ ∈A E [ U ( W T ) | F I ] , (19)where U is an increasing, concave function and the set of admissible strategies is A = { θ | θ is F I -predicable and E (cid:20)(cid:90) T θ t dt (cid:21) < ∞} . (20)The market maker is tasked with setting the price of the asset efficiently at all times 0 ≤ t ≤ T and10o should choose the price according to P t = E [ v | F Mt ] . (21)We now define the concept of equilibrium in continuous-time. This concept is analogous to thesingle-auction equilibrium concept of Section 2.2. Definition 5. A continuous-time equilibrium ( P, X ) consists of a price process P and an in-ventory process X such that • given a price process P , the inventory process X achieves the supremum in (19) , and • given an inventory process X , the price process P is efficient, i.e. P satisfies (21) .3.2. Continuous-Time Equilibrium The goal of this section is to classify a linear equilibrium in continuous-time for an exponentiallyrisk-averse or risk-neutral insider. We will restrict our consideration to pricing rules and tradingstrategies that have a form which are analogous to those of the single-auction equilibrium presentedin Theorem 2. Namely, increments of the price set by the market maker will be linear with respect toincrements of trade volume, and the insider’s trading strategy will be linear with respect to v − P t .Before stating the classification of equilibrium in continuous-time we need to develop some additionalmathematical machinery. These are contained in Lemmas 6, 7, and 9.Using expressions (16), (17), and (18), we rewrite the expression for terminal wealth W T as W T = (cid:90) T ( v − P s − c θ s ) θ s ds , (22)where we have taken X = 0 for simplicity. The computations that follow can be carried out with X (cid:54) = 0, but this choice will not affect the insider’s optimal trading strategy (it will have only a minoreffect on the insider’s value function).We will develop optimal insider trading strategies by first fixing the market maker’s pricing rule andthen solving the Hamilton-Jacobi-Bellman (HJB) equation associated with the optimization problem(19). We will then subsequently verify that the solution is optimal. This motivates us to introducethe dynamic version of 19 as H ( t, P ) = sup θ ∈A E (cid:20) U (cid:18)(cid:90) Tt ( v − P s − c θ s ) θ s ds (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) F It (cid:21) . (23)11or a given time t ∈ [0 , T ] and midprice P t , the quantity H ( t, P t ) gives the optimal expected utilitythat the insider can achieve by trading during the time interval [ t, T ]. We refer to H as the insider’svalue function. Lemma 6 (Insider’s Value Function and Optimal Strategy).
Let A ≥ and let the utilityfunction U be as defined in (6) . Let λ be a positive, bounded, deterministic function such that theRiccati differential equation dh ( t ) dt = − (1 − A c σ ) λ ( t ) c h ( t ) + λ ( t ) c h ( t ) − c , h ( T ) = 0 , (24) has a global solution h : [0 , T ] → R . Suppose the midprice process P is given by P t = v + (cid:90) t λ ( s ) dY s , (25) where Y is given in (15) . Then the insider’s value function (23) is given by H ( t, P ) = − exp (cid:26) − A (cid:18) ( v − P ) h ( t ) + σ (cid:82) Tt λ ( s ) h ( s ) ds (cid:19)(cid:27) , A > , ( v − P ) h ( t ) + σ (cid:82) Tt λ ( s ) h ( s ) ds , A = 0 , (26) and the optimal strategy in feedback form is given by θ ∗ t = β ( t ) ( v − P t ) , β ( t ) = 1 − λ ( t ) h ( t )2 c . (27) Proof
For a proof see Section 6.4 in the appendix.In the statement of Lemma 6 we have eliminated consideration of many arbitrary pricing rules byconsidering only functions λ for which there is a solution to the ODE (24). This is necessary inestablishing this lemma in generality, as it is easy to find functions λ for which this ODE does nothave a solution defined on all of [0 , T ]. This is because for some pricing rules there may be incentivefor the insider to acquire arbitrarily large inventory positions which will push prices in her favour, andthen liquidating the acquired position when price impact is smaller, thereby making an unboundedprofit. We make more comments on how we avoid this issue after we present the continuous-timeequilibrium. If λ is relatively large and decreases linearly to λ T which is relatively small, then the ODE will only have asolution on an interval of the form ( T ∗ , T ] where T ∗ >
12e use an HJB approach to prove this lemma, and the elimination of many pricing rules fromconsideration is analogous to a similar result in the standard Kyle model. If the HJB approach istaken in the Kyle model, then it is straightforward to show that there is no solution to the HJBequation if λ is not constant in the risk-neutral case, or if λ doesn’t have very specific dynamics inthe risk-averse case. Thus, acquiring an optimal trading strategy which is admissible requires thediscarding of many pricing rules. On the other hand, using the HJB approach in our setting withtransaction cost does have a significant difference compared to the standard Kyle model. Due to thequadratic nature of the performance criteria, we are immediately provided with the feedback form ofthe optimal trading strategy. In addition, the linear performance criteria in the Kyle model meansthat the HJB equation reduces to a system of PDE’s rather than a single equation as in our model.The addition of the transaction cost makes many of the mathematical components of our problemmore straightforward.In the next Lemma we provide the details of the market maker’s task of setting efficient prices. Recallthat in the proof of Theorem 2 the pricing rule was shown to be efficient by applying the projectiontheorem for normal random variables directly to the expectation E [ v | ∆ y ], where we had assumedthat the insider’s trading strategy was linear in v . Our approach in continuous-time is analogous,but we need a generalization of the projection theorem to continuous-time. In following Lemma, weassume the insider follows a linear trading strategy and apply optimal filtering theory to compute E [ v | F Mt ]. Lemma 7 (Market Maker’s Efficient Pricing).
Suppose that insider’s inventory process is spec-ified by dX t = β ( t ) ( v − P t ) dt , (28) for a deterministic function β . Then the process P specified by the dynamics dP t = λ ( t ) dY t , P = v , (29) where Y is given by (15) and λ ( t ) = β ( t )Σ( t ) σ , d Σ( t ) dt = − σ λ ( t ) , Σ(0) = Σ v , (30) satisfies the efficiency condition (21) . Furthermore, the function Σ is equal to the posterior varianceof v given the market maker’s information: Σ( t ) = E [( v − P t ) |F Mt ] . (31)13 roof The result is a direct application of (Liptser and Shiryaev, 2001, Theorem 12.1). (cid:3)
The previous two lemmas are analogous to the steps in the single-auction case. The reader willrecall that the proof of Theorem 2 was done in two steps. The first step was the computation ofthe insider’s optimal strategy for a fixed, linear pricing rule. In discrete time, this came down tosolving an algebraic equation, of which Lemma 6 is the continuous-time analogue. The second stepwas the computation of an efficient pricing rule for a fixed, linear trading strategy, and Lemma 7 isthe corresponding result in continuous-time.We reduced the computation of the linear single-auction equilibrium to coupled algebraic equations,which were subsequently reduced to a single algebraic equation. In continuous-time, we reduce thecomputation of an equilibrium to two coupled, nonlinear ODEs for which one ODE is prescribedan initial condition and the other is prescribed a terminal condition. This motivates the followingdefinition.
Definition 8.
Let
T, ξ , ξ T ∈ R be constants with T > . Let x : R → R and F : R → R . A forward-backward ordinary differential equation (FBODE) is a system of the form dx ( t ) dt = F ( x ( t )) , x (0) = ξ x ( T ) = ξ T . (32)In the following lemma, we state the FBODE that will appear in our continuous-time equilibriumand prove the existence and uniqueness of a solution. Lemma 9 (Solution to FBODE).
Let A ≥ , c ≥ , and Σ v > . For x ∈ R , let us define thefunction F : R → R as F ( x ) = F ( x ) F ( x ) , F ( x ) = − σ x c σ + x ) , F ( x ) = − σ x ( c σ + x − A x )4 ( c σ + x ) . (33) Then the FBODE dx ( t ) dt = F ( x ( t )) , x (0) = Σ v x ( T ) = 0 , (34) has a solution. If c > then this solution is unique and it satisfies x ( t ) > for all t ∈ [0 , T ] and ( t ) > for all t ∈ [0 , T ) . If c = 0 then the solution is unique if we impose x ( t ) > for all t ∈ [0 , T ) . In this case the solution satisfies x ( t ) > for all t ∈ [0 , T ) and x ( T ) = 0 .In addition, when A = 0 the solution is given by x ( t ) = 2 c λ σ + λ σ ( T − t ) , x ( t ) = T − t λ ( T − t ) + 4 c , λ = (cid:114) Σ v σ T + c T − cT , (35) and when c = 0 the solution which satisfies x ( t ) > for t ∈ [0 , T ) is given by x ( t ) = 4 σ Λ K ( T − t )( A Σ v + 2 S ) ( A Σ v t − TT + 2 S ) , x ( t ) = σ Λ K ( T − t ) A Σ v + 2 S , Λ K = (cid:114) Σ v σ T , S = (cid:115)(cid:18) A Σ v (cid:19) +Λ K . (36) Proof
For a proof see Section 6.5 in the appendix.Note that the results of Lemma 9 include the possibility of c = 0 even though our model specifiesthat c is positive. The fact that the FBODE in Lemma 9 has a desired solution for c = 0 will beuseful in proving the nature of equilibrium in the limit c →
0, and so we include these results.It is also useful to mention which equilibrium quantities are represented by the functions x and x as solutions to the FBODE. Taking x and x the solution when c >
0, we will construct equilibriumby letting Σ( t ) = x ( t ) and h ( t ) = x ( t ) /x ( t ), which is well defined due to the result of Lemma 9that x ( t ) > t ∈ [0 , T ].Before giving a continuous-time equilibrium it is useful to define the constantΛ K = (cid:114) Σ v σ T . (37)The constant Λ K is the continuous-time pricing rule of Kyle (1985). That is, when a risk-neutralinsider faces a risk-neutral market maker on a frictionless exchange, the processes specified by thedynamics dP t = Λ K dY t , P = v , (38a) dX t = 1Λ K ( T − t ) ( v − P t ) dt , X = 0 , (38b)form an equilibrium. Writing the risk-neutral continuous-time equilibrium in terms of Λ K will helpilluminate the effect that transaction costs have on the market participant’s respective strategies.15 heorem 10 (Continuous-Time Equilibrium). Let U be as given in (6) for some A ≥ , andlet c > . Let x = ( x , x ) be the unique solution to the FBODE (34) , and let Σ( t ) = x ( t ) and h ( t ) = x ( t ) /x ( t ) . Then the midprice process P and inventory process X specified by dP t = λ ( t ) dY t , P = v , (39) dX t = β ( t ) ( v − P t ) dt , X = 0 , (40) where Y is given in (15) and where β ( t ) = σ c σ + Σ( t ) h ( t )) , (41a) λ ( t ) = Σ( t )2 ( c σ + Σ( t ) h ( t )) , (41b) form a continuous-time equilibrium. Furthermore, Σ( t ) = E [( v − P t ) | F Mt ] and the value function H is given by (26) . When A = 0 , λ ( t ) ≡ λ is a constant and β ( t ) = 1 λ ( T − t ) + 2 c , (42a) λ = (cid:114) Λ K + c T − cT . (42b) Proof
For a proof see Section 6.6 in the appendix.Recall that in Lemma 6 we removed from consideration many pricing rules in which an associatedODE did not have a solution, as failure to do so would result in the ODE solutions blowing up withinthe interval [0 , T ]. Our method of classifying equilibrium in Theorem 10 avoids this issue by takingthe function h as given (in terms of the unique solution to an FBODE which does not blow up), andthen specifying the pricing rule λ which yields h as the solution to the ODE (24).Definition 5 defines what it means for processes ( P, X ) to form a continuous-time equilibrium. InTheorem 10, we gave deterministic functions β and λ such that the processes defined by dP t = λ ( t ) dY t and dX t = β ( t ) ( v − P t ) dt with ( P , X ) = ( v ,
0) form a continuous-time equilibrium. While thefunctions ( β, λ ) do not themselves form an equilibrium, the functions ( β, λ ) correspond directly to alinear equilibrium (
P, X ). In the sequel, we will refer to linear equilibria (
P, X ) using ( β, λ ), and werefer to β as an equilibrium trading rule and λ as an equilibrium pricing rule .We will now discuss and interpret some characteristics of the continuous-time equilibrium of Theorem10. 16 roposition 11 (Trading and Pricing Rule Monotonicity). Let β and λ be the trading andpricing rules, respectively, corresponding to the continuous-time equilibrium of Theorem 10. Then,1. β is an increasing function of t ,2. if A > then λ is a decreasing function of t . Proof
For a proof see Section 6.7 in the appendix.In equilibrium, β is increasing in time and β ( T ) = 1 / c , so0 < β ( t ) < c , t ∈ [0 , T ) . (43)To gain some intuition as to why the trading rule is bounded by 1 / c , let b ( t ) be an arbitrary tradingrule and consider the expression for terminal wealth W T = (cid:90) T b ( t ) (1 − c b ( t )) ( v − P t ) dt . (44)Thus, at time t the insider gains wealth at a rate equal to b ( t ) (1 − c b ( t )) ( v − P t ) . This expressionis positive for 0 < b ( t ) < /c and maximized at b ( t ) = 1 / c . However, in choosing b ( t ) = 1 / c priceimpact effects are also maximized and this lowers the potential gain of wealth at future times. Theoptimal balance of instantaneous and future gains is therefore achieved for trading rules which arebounded by 1 / c .In the absence of transaction costs, the authors of Kyle (1985) and Baruch (2002) show that inequilibrium both the risk-neutral and exponentially risk-averse insider, respectively, take tradingrules β that blow up as t approaches T so as to force the midprice to v , resulting in Σ( T ) = 0. Fora non-zero transaction cost c , the discussion above shows that an insider of any risk tolerance neverwishes to choose β > / c . In this setting, the insider does not reveal the true value of the asset tothe market maker by the end of the trading period i.e. Σ( T ) >
4. Parameter Dependence
In this section, we discuss the dependence of the equilibrium of Theorem 10 on model parameters.First we summarize the effects of varying model parameters on the continuous-time equilibrium bynumerically solving the associated FBODE. Then we study the limits of the equilibrium rules as thetransaction cost parameters are taken to extreme values.In analyzing the effects of varying the model parameters on the resulting equilibria, we considerthe solution, x , to the FBODE (34) to be a function of the underlying parameters x ( t ) ≡ x ( t ; Θ),17 Figure 2: Shown are the functions β , λ , and Σ in equilibrium for various values of the transaction cost parameter, c .Other parameter values are A = 1, σ = 1, and Σ v = 0 . where Θ is a collection of parameters of interest. An ideal approach would then be to differentiatethe FBODE (34) with respect to Θ i to get dynamics of ∂x/∂ Θ i . Unfortunately the non-linearitiesmake this approach highly intractable, and so we resort to demonstrating these dependencies bynumerically solving the FBODE for various sets of the underlying parameters.In Figure 2 we show the effect of varying the transaction cost parameter c . As would be expectedof the trading rule β , when there are larger transaction costs the insider trades less aggressively asdemonstrated in the left panel. Consequently, the price impact λ shown in the middle panel decreasesfor larger transaction costs because net order flow contains less information. In addition the varianceof the market maker’s estimate decreases more slowly and price discovery takes comparatively moretime, as seen in the right panel.In Figure 3 we show the effect of varying the risk-aversion parameter A . These results are qualitativelysimilar to those found in Baruch (2002). In particular, larger values of A mean that the insider tradesmore aggressively at the beginning of the trading interval, as seen in the left panel by larger values ofthe trading rule β , because this is when they are exposed to the greatest amount of risk posed by thenoise traders. At the end of the trading interval, the trading rule converges to the same finite valueregardless of risk-aversion level because the remaining risk exposure converges to zero as t → T .The finite limit is a consequence of the positive transaction cost c , whereas in Baruch (2002) thislimit would be infinite. The more aggressive trading at the beginning of the trading interval meansthere is more information contained in the net order-flow, and thus we see in the middle panel thatprice impact is larger at earlier times compared to later times for any fixed value of risk-aversion A >
0. The right panel demonstrates that price discovery also occurs faster when there is increasedrisk-aversion.In both Figure 2 and Figure 3 we note that the terminal conditional variance of the asset value isstrictly positive. This is an immediate consequence of Lemma 9 where it is shown that the solutionto the FBODE (34) satisfies x ( t ) > t ∈ [0 , T ]. This means that the price process is not fullyrevealing of the asset’s value as t → T , unlike other frictionless models where equilibrium results inlim t → T P t = v almost surely. 18 Figure 3: Shown are the functions β , λ , and Σ in equilibrium for various values of the risk-aversion parameter, A .Other parameter values are c = 0 . σ = 1, and Σ v = 0 . In this section, we study the limiting behaviour of the equilibrium rules β and λ of Theorem 10 withrespect to the transaction cost parameter c . The following proposition summarizes this behaviour asthe transaction cost is taken towards its extremes. Proposition 12 (Limiting Transaction Cost Dependence).
For fixed A ≥ , lim c →∞ β ( t ; c ) = 0 , (45)lim c →∞ λ ( t ; c ) = 0 , (46)lim c → β ( t ; c ) = A Σ v (cid:115)(cid:18) A Σ v (cid:19) + Λ K K ( T − t ) , (47)lim c → λ ( t ; c ) = Λ KA Σ v · t − TT + (cid:114)(cid:16) A Σ v (cid:17) + Λ K . (48) The limits (45) and (46) hold uniformly for t ∈ [0 , T ] , and the limits (47) and (48) hold uniformlyon any compact subinterval of [0 , T ) . Proof
For a proof see Section 6.8 in the appendix.The two limits in (45) and (46) indicate that for sufficiently large transaction cost, the equilibriumessentially consists of the insider doing nothing and therefore there being no informational contentto order flow (as it would be comprised of only noise trades). This result is expected given that any19rofits earned by the insider would be more than canceled out by the losses suffered from significantcosts.The two limits in (47) and (48) correspond to the equilibrium trading rule and pricing rule of ina setting where there is no transaction cost as in Kyle (1985), Back (1992), and Baruch (2002).This is significant because in our model the existence of a feedback form for the trading strategyrelies on strict positivity of the transaction cost parameter. Typically in this style of model ofasymmetric information, the HJB approach does not result in a feedback form for the optimaltrading strategy. Proposition 12 shows that in this case the correct equilibrium trading and pricingrules for a frictionless model can be obtained by adding an appropriate friction and then taking alimit as this friction vanishes. With a feedback form of the insider’s trading strategy, some of theanalysis becomes more straightforward, so this limiting technique could be applied in other settingsof asymmetric information where the equilibrium rules are not as straightforward to classify.An important step in obtaining this result is in identifying appropriate processes which will not havea discontinuity at c = 0. When considering a vanishing friction term, some quantities can becomediscontinuous at t = T . In our model, this will happen with the function h which appears in theinsider’s value function (26). To illustrate, in our model with c > h ( T ) = 0. Butin frictionless equilibrium models it is typically the case that lim t → T h ( t ) > h is apositive constant). Thus it is impossible to have a well behaved limit as c →
0. However, our modelalso results in Σ( T ) > c >
0, which means that the product Σ( t ) h ( t ) > t ∈ [0 , T ) andΣ( T ) h ( T ) = 0. But these relations also hold in frictionless models, and so one of the appropriatefunctions to use in the classification of equilibrium is the product Σ( t ) h ( t ), which is represented by x ( t ) in Lemma 9 and Theorem 10.
5. Conclusion
In this paper, we extend the results of Kyle (1985), Holden and Subrahmanyam (1994) and Baruch(2002) by adding friction to the market in the form of a transaction cost which is linear in theinsider’s order size. We consider an exponentially risk-averse insider, allowing for the possibility ofrisk-neutrality. We begin by modeling a single-auction exchange and providing the correspondingunique market equilibrium in the form of an algebraic equation. We then demonstrate the effectof both the transaction cost and risk-aversion on the equilibrium by asymptotically expanding thealgebraic equation about the frictionless risk-neutral equilibrium for small transaction cost and smallrisk-aversion. This procedure results in a single dimensionless quantity which determines an accurateexpansion of the trading strategy to second order and the pricing rule to third order.We then formulate an analogous market model in continuous-time. The mathematical machineryneeded for the presentation of a linear market equilibrium in continuous-time is more sophisticatedthan that of the single-auction equilibrium. We develop an optimal trading strategy for an exponen-tially risk-averse insider by explicitly solving the HJB equation associated with optimization problem(19). We then give the filtering equations which provide an efficient price process for a fixed insiderstrategy. The main result is Theorem 10, which gives a linear, continuous-time market equilibriumin terms of the solution to the FBODE (34), for which we show there exists a unique solution.20here are several qualitative similarities between the equilibria of the frictionless case and ours. First,linear equilibria exist in both cases and are given in terms of the solutions to differential equations.Additionally, the equilibrium insider trading rule β is increasing in time, and the equilibrium marketmaker’s pricing rule λ is constant in the risk-neutral setting and decreasing in time in the risk-averse setting. However our model also has some properties which are not typical in many modelsof asymmetric information. The market maker’s conditional variance Σ has the property Σ( T ) > T ) = 0 meaning that the true value of the asset v isnot reveled to the market maker by time T .The effects of varying model parameters is investigated by numerically computing the equilibriumstrategies and by also explicitly computing some limits of equilibrium processes β and λ as transactioncost parameter c tends to the values zero and infinity. We show that β and λ converge to thefrictionless strategies of Baruch (2002) as c → c → ∞ . Since the additionof friction to the continuous time model provides a feedback control, and the equilibrium processes β and λ converge to their frictionless counterparts when c →
0, this could be used as a techniqueto study other frictionless models of asymmetry in which a feedback control is not directly obtainedfrom the HJB approach.
6. Appendix A
First we show that r has a unique positive root. Define the function s ( λ ) = 2 ( λ + c ) + A σ λ . (49)We will see later that s ( λ ) > r can be written as r ( λ ) = λ ( s ( λ ) + 4 λ K ) − λ K s ( λ ) , (50)and we note that r ( λ ) = 0 if and only if λ ( s ( λ ) + 4 λ K ) = 4 λ K s ( λ ) . (51)The left and right sides of (51) are both polynomials in λ with positive coefficients, and therefore areboth strictly increasing functions of λ ≥
0. Furthermore, for any A ≥ λ = 0the left hand side of (51) is zero and the right hand side is positive. Therefore, (51) has exactly onepositive solution, and hence r has exactly one positive root.The rest of the proof is divided into the two cases A = 0 and A >
0. Some related expressions aredifferent between the two cases, but the structure of the proof in both cases is identical. First, wesuppose that the insider’s trading strategy is a linear function of v and show the efficient pricingrule is linear. Then, we suppose the market maker’s pricing rule is linear and show that the insider’s21ptimal trading strategy is linear. Matching coefficients gives the result.Case A = 0: First, we suppose that the market maker chooses p as a linear function of ∆ y . Namely,we let P (∆ y ) = µ + λ ∆ y , where µ and λ are constants. When the market maker follows the pricingrule P , the insider’s transaction price is given by (cid:98) p = µ + λ ∆ y + c ∆ x = µ + ( λ + c ) ∆ x + λ ∆ z . (52)Thus, E [ U ( w ) | v ] = E [( v − (cid:98) p ) ∆ x | v ] = E [( v − µ ) ∆ x − ( λ + c ) ∆ x − λ ∆ x ∆ z | v ] (53)= ( v − µ ) ∆ x − ( λ + c ) ∆ x . (54)The second order condition for optimality is λ + c >
0, which is equivalent to s ( λ ) >
0. Assumingthis is satisfied, the choice of ∆ x ≡ X ( v ) = v − µ λ + c ) , (55)maximizes (54) for any v . We note that for linear pricing rules P linear trading strategies X areoptimal even if we allow X to be a nonlinear function.Now we assume that the insider chooses ∆ x to be a linear function X of the ex-post price v . Specif-ically, the insider chooses X ( v ) = α + β v , where α and β are constants. Then, by the projectiontheorem for normal random variables we have E [ v | ∆ y ] = E [ v | α + β v + ∆ z ] = E [ v ] + E [( v − E [ v ])(∆ y − E [∆ y ])] E [(∆ y − E [∆ y ]) ] (∆ y − E [∆ y ])= v + 4 λ K β λ K β (∆ y − α − β v )= v − λ K α β λ K β + 4 λ K β λ K β ∆ y . (56)By examining (55) and (56), we see that for ( P, X ) ≡ ( P (∆ y ) , X ( v )) = ( µ + λ ∆ y, α + β v ) to be anequilibrium, it must be that α = − β µ, µ = v − λ K α β λ K β , (57a) β = 12 ( λ + c ) , λ = 4 λ K β λ K β , (57b)subject to the constraint s ( λ ) > µ = v and α = − β v , which gives the insider’strading strategy X ( v ) = β ( v − v ). Recall that the insider’s second order condition is satisfied if22 ( λ ) >
0. Inserting β into the expression for λ in (57b), we see that λ satisfies (51). If λ ≤ s ( λ ) ≤
0, contradicting optimality of ∆ x . Therefore, λ is the unique positiveroot of the polynomial r .Case A >
0: First, suppose that the market maker chooses p as a linear function of ∆ y so that P (∆ y ) = µ + λ ∆ y , where µ and λ are constants. When the market maker follows the pricingstrategy P , then the insider’s transaction price (cid:98) p is given by (52). Thus, E [ U ( w ) | v ] = E [ − exp {− A ( v − (cid:98) p )∆ x }| v ]= − exp {− A ( v − µ )∆ x + A ( λ + c )∆ x + A λ σ ∆ x } . (58)The second order condition for optimality is 2( λ + c ) + Aσ λ > s ( λ ) > x ≡ X ( v ) = v − µ λ + c ) + A σ λ (59)maximizes (58) for any v . We note that for linear pricing rules P linear trading strategies X areoptimal even if we allow X to be a nonlinear function.Now, suppose that the insider chooses the linear trading strategy X ( v ) = α + β v . Then E [ v | ∆ y ] isgiven by (56).By examining (56) and (59), we see that for ( P, X ) ≡ ( P (∆ y ) , X ( v )) = ( µ + λ ∆ y, α + β v ) to be anequilibrium we must have α = − β µ, µ = v − λ K α β λ K β , (60a) β = 12 ( λ + c ) + A σ λ , λ = 4 λ K β λ K β , (60b)where s ( λ ) >
0. The rest of the proof is identical to the case A = 0. (cid:3) In the proof of Theorem 2, we saw that when
A > β and λ satisfy (60b). Inserting λ into β in (60b) we get that in equilibrium β satisfies q ( β ) := 32 c λ K β + 16 λ K β + 16 λ K ( A λ K σ + c ) β + 2 c β − , (61)with β >
0. We see that for fixed x > ∂q ( x ) /∂c > ∂q ( x ) /∂A >
0. Since q ( x )is a strictly increasing function of x , we therefore must have ∂β/∂c < ∂β/∂A <
0. Taking23erivatives of λ in (60b) with respect to c and A yields ∂λ∂c = 4 λ K (1 − λ K β )(1 + 4 λ K β ) ∂β∂c , (62) ∂λ∂A = 4 λ K (1 − λ K β )(1 + 4 λ K β ) ∂β∂A . (63)Therefore, ∂λ∂c < , ∂λ∂A < ⇐⇒ β < λ K = β K . (64)We have already shown that β is strictly decreasing with respect to c and A , and in particular wehave β ≤ β K with equality only when c = A = 0. Thus, λ is strictly decreasing with respect to both c and A . (cid:3) The proof of this proposition relies on the result that the roots of a polynomial depend on thecoefficients analytically in a neighbourhood of a given root (see Brillinger (1966)). With this in mindwe write λ and β as a power series in the quantities A and c as follows: λ = ∞ (cid:88) i =0 ∞ (cid:88) j =0 ˜ λ i,j A i c j , (65) β = ∞ (cid:88) i =0 ∞ (cid:88) j =0 ˜ β i,j A i c j . (66)Our approximation corresponds to computing each ˜ λ i,j for i + j ≤ β i,j for i + j ≤
2, theremainder of the higher order terms being either o ( A + c ) (for λ ) or o ( A + c ) (for β ).We first substitute the expansion for λ into the polynomial r ( x ) given in (11) and recall that λ isthe unique positive root of this polynomial. We then collect terms according to the powers of A and c and set each term equal to zero individually. This results in a system of 10 equations which doesnot have a unique solution. However, inspection shows that the coupling of the equations is arrangedin such a way that the first appearance of each quantity to be solved for is linear, with the exceptionof ˜ λ , . Thus, given ˜ λ , , every other ˜ λ i,j is solved for uniquely. Because we require λ to be positive,we must use the solution which corresponds to a positive value of ˜ λ , , of which there is only one The equations are large and tedious, so are displayed in a subsequent appendix λ = λ K , ˜ λ = 0 , ˜ λ = 0 , ˜ λ = − σ λ K , ˜ λ = − σ λ K , ˜ λ = −
12 1 λ K , ˜ λ = 18 σ λ K , ˜ λ = 34 σ λ K , ˜ λ = 32 σ , ˜ λ = 1 λ K . By letting ν = 12 λ K σ A + cλ K we are able to write the truncated sum corresponding to the abovecoefficients and perform some elementary factoring which yields (cid:88) i + j ≤ ˜ λ i,j A i c j = λ K (cid:18) − ν + ν (cid:19) . This is the desired form as given in the statement of the Proposition.Solving for ˜ β i,j for i + j ≤ q ( x ) defined in (61). By collecting powers of A and c andsetting each to zero individually, this once again yields a system of 6 equations . The resultingsystem of equations does not have a unique solution, but given ˜ β , the remainder of the equationsto be solved are linear. There is only one value of ˜ β , which ensures β is positive (the other rootsbeing negative or imaginary). Solving the remaining linear equations yields˜ β = 12 λ K , ˜ β = − σ , ˜ β = − λ K , ˜ β = 316 σ λ K , ˜ β = 34 σ λ K , ˜ β = 34 λ K . The truncated sum corresponding to these coefficients can again be factored to give (cid:88) i + j ≤ ˜ β i,j A i c j = 12 λ K (cid:18) − ν + 32 ν (cid:19) . This is the desired form as given in the statement of the Proposition. (cid:3)
We consider the cases A = 0 and A > Similarly, shown in a subsequent appendix A = 0: When A = 0, U ( w ) = w , and the value function (23) becomes H ( t, P ) = sup θ ∈A E (cid:20)(cid:90) Tt ( v − P s − c θ s ) θ s ds (cid:12)(cid:12)(cid:12)(cid:12) F It (cid:21) . (67)Associated with this stochastic control problem is the HJB partial differential equation ∂ t H + sup θ (cid:26) σ λ ( t ) ∂ P P H + θ λ ( t ) ∂ P H + ( v − P − c θ ) θ (cid:27) = 0 , H ( T, · ) = 0 . (68)It can be checked by direct substitution that the solution of this equation is given by (26) when h satisfies the ODE (24). The supremum in (68) is achieved at θ ∗ ( t, P ) = β ( t ) ( v − P ) , β ( t ) = 1 − λ ( t ) h ( t )2 c . (69)All that remains to be shown is that this feedback form of θ ∗ yields an admissible trading strategy.Optimality of θ ∗ then follows from a standard verification argument (see Pham (2009)). To this end,define an auxilliary process Q = ( Q t ) ≤ t ≤ T by Q t = P t − v . Then under the control θ ∗ , we have the dynamics dQ t = − λ ( t ) β ( t ) Q t dt + λ ( t ) σ dZ t , Q = v − v . This stochastic differential equation is linear and therefore has a unique strong solution. Further,since v − v is Gaussian, the resulting solution is a Gaussian process (see Karatzas and Shreve (2012)Section 5.6). This gives E (cid:20)(cid:90) T ( θ ∗ s ) ds (cid:21) = E (cid:20)(cid:90) T β ( s ) Q s ds (cid:21) < ∞ . In addition, Q has continuous paths and thus the trading strategy is predictable, and thereforeadmissible.Case A >
0: When
A > U ( w ) = − exp( − Aw ), and the value function (23) becomes H ( t, P ) = sup θ ∈A E (cid:20) − exp (cid:26) − A (cid:90) Tt ( v − P s − c θ s ) θ s ds (cid:27) (cid:12)(cid:12)(cid:12)(cid:12) F It (cid:21) . (70)This stochastic control problem has the associated HJB partial differential equation ∂ t H + sup θ (cid:26) σ λ ( t ) ∂ P P H + θ λ ( t ) ∂ P H − A ( v − P − c θ ) θ H (cid:27) = 0 , H ( T, · ) = − . (71)Once again, it can be checked by direct substitution that this equation has solution given by (26). Theresulting feedback form of the control is again linear with respect to v − P , and thus the remainderof the proof is identical to the risk-neutral case. (cid:3) .5. Proof of Lemma 9 In this section, we present a proof of Lemma 9 for the case
A >
0. The proof for A = 0 isessentially the same but more straightforward, and the solution given by (35) can be checked bydirect substitution. Proof of Lemma 9
We first consider c >
0. Inspection of equation (34) shows that any solutionmust have x a decreasing function, so we look for a solution in which x ( t ) = ρ ( x ( t )) for somefunction ρ . This gives dρdx = dx /dtdx /dt = c σ x + ρx − A ρ x . (72)This ODE has general solution ρ ( x ) = ( γ + 1) x γ − ( γ − k A ( x γ + k ) , (73) γ = √ A c σ , (74)for arbitrary k ∈ R . The value of k must be chosen to match the boundary condition x ( T ) = ρ ( x ( T )) = 0. We can immediately rule out k = 0 because then x ( t ) = ρ ( x ( t )) becomes a non-zeroconstant contradicting the boundary condition x ( T ) = 0. Therefore, we search for non-zero k suchthat g ( k ) = 0, where g ( k ) = ( γ + 1) x γ ( T ; k ) − ( γ − k , and where by substituting (73) into (33) we have that x satisfies dx dt ( t ; k ) = f ( x ( t ; k ); k ) , x (0; k ) = Σ v , (75) f ( x ; k ) = − A σ x ( x γ + k ) (cid:18) ( γ + 1) x γ + ( γ − k (cid:19) , (76)where we have made the dependence of x on the parameter k explicit. Inspection of (75) showsthat x ( t ; k ) must be positive for all t ∈ [0 , T ], and so we must have k > g ( k ) = 0.We now show that there exists exactly one value of k > k > x ( T ; k ) is decreasing with respect to k . As f is continuouslydifferentiable with respect to k , we have that x ( T ; k ) is as well (see Hartman (2002)). By setting z ( t ; k ) = ∂x ( t ; k ) /∂k , we have ∂z ( t ; k ) ∂t = ∂f ( x ( t ; k ); k ) ∂x z ( t ; k ) + ∂f ( x ( t ; k ); k ) ∂k , z (0; k ) = 0 , (77)27here ∂f ( x ; k ) ∂x = − A σ x ( k + x γ ) (cid:18) ( γ + 1) x γ − ( γ − k (cid:19) (cid:18) ( γ + 1) x γ + ( γ − k (cid:19) , (78) ∂f ( x ; k ) ∂k = − A γ σ x γ +21 ( x γ + k ) (cid:18) ( γ + 1) x γ + ( γ − k (cid:19) . (79)Since x ( t ; k ) > t ∈ [0 , T ] and k >
0, we have ∂f ( x ( t ; k ); , k ) ∂x ≤ , ∂f ( x ( t ; k ); k ) ∂k < . (80)Thus, by examining (77), we see that ∂x ( t ; k ) /∂k < t ∈ (0 , T ] and k >
0. This establishesthat x ( T ; k ) is decreasing with respect to k , and therefore any solution to g ( k ) = 0 must be unique.Let k r and k l be given by k r = γ + 1 γ − v ) γ , k l = γ + 1 γ − x γ ( T ; k r ) . Then we see 0 < k l < k r and g ( k l ) > > g ( k r ). Thus we have a unique k > x ( T ; k ) = 0. This establishes existence and uniqueness of the solution to (34) along with theproperty that x ( t ) > t ∈ [0 , T ]. The fact that x ( t ) > t ∈ [0 , T ) then followsimmediately from (73) together with the boundary condition x ( T ) = 0 and that x ( t ) is decreasing.We now consider c = 0 and proceed similarly by looking for a function ρ such that x ( t ) = ρ ( x ( t )).In this case the ODE satisfied by ρ is dρdx = ρx − A ρ x , which has solution ρ ( x ) = x A ( x + k ) , for arbitrary k ∈ R . We immediately eliminate k = 0 for the same reason as before. However, wecannot eliminate k < x now takes the form dx dt ( t ; k ) = − A σ ( x ( t ; k ) + k ) , x (0; k ) = Σ v , which has the unique solution x ( t ; k ) = Σ v − k (Σ v + k ) A σ t v + k ) A σ t . (81)28nforcing the boundary condition x ( T ; k ) = 0 yields two possible values of k : k = − Σ v (cid:18) ± (cid:115) A σ T Σ (cid:19) . (82)This results in one positive and one negative value of k , both of which provide solutions to theFBODE (34). However, the negative value of k results in x ( t ) < t ∈ [0 , T ). By enforcing x ( t ) > t ∈ [0 , T ) we must discard the negative value of k and are left with a unique solution.From (81) we have immediately that x ( t ; , k ) > t ∈ [0 , T ) and x ( T ) = 0. (cid:3) First we fix ( x , x ) to be the unique solution to the FBODE (34) with x (0) = Σ v . Suppose theprice dynamics are given by dP t = λ ( t ) dY t , P t = v , with λ ( t ) given by λ ( t ) = Σ( t )2 ( c σ + Σ( t ) h ( t )) , where Σ( t ) = x ( t ) and h ( t ) = x ( t ) /x ( t ) as in the statement of the Theorem. A straightforwardcomputation shows that dh ( t ) dt = 2 A σ Σ ( t ) h ( t ) − c σ c σ + Σ( t ) h ( t )) , and in addition, that − (1 − A c σ ) λ ( t ) c h ( t ) + λ ( t ) c h ( t ) − c = 2 A σ Σ ( t ) h ( t ) − c σ c σ + Σ( t ) h ( t )) . Thus we have that h satisfies the ODE (24). Therefore, by Lemma 6, the optimal trading strategyis given by θ ∗ t = β ( t ) ( v − P t ) dt , with β ( t ) = 1 − λ ( t ) h ( t )2 c , and the insider’s value function is given by (26).Now, again with ( x , x ) the unique solution to the FBODE, suppose that the insider’s trading29trategy is given by θ t = β ( t ) ( v − P t ) dt , (83)with β ( t ) = σ c σ + Σ( t ) h ( t )) , where Σ( t ) = x ( t ) and h ( t ) = x ( t ) /x ( t ). From (34) we see that Σ satisfies d Σ( t ) dt = − σ Σ ( t )4 ( c σ + Σ( t ) h ( t )) = − σ λ ( t ) . (84)In addition we have by their definitions in the statement of the Theorem that λ ( t ) = β ( t ) Σ( t ) σ . (85)By Lemma 7, the relations in (83), (84), and (85) imply that the efficient pricing rule is given by dY t = λ ( t ) dY t , P = v , and that the function Σ also yields the market maker’s conditional variance:Σ( t ) = E [( v − P t ) |F Mt ] . Finally, when A = 0, applying Lemma 9 gives the closed form expressions β ( t ) = 1 λ ( T − t ) + 2 c ,λ = (cid:114) Λ K + c T − cT . as desired. (cid:3) As in the proof of Theorem 10, let ( x , x ) be the unique solution to the FBODE (34), and letΣ ( t ) = x ( t ) and h ( t ) = x ( t ) /x ( t ). Then writing β ( t ) = σ c σ + Σ( t ) h ( t )) , λ ( t ) = Σ( t )2 ( c σ + Σ( t ) h ( t )) ,
30e have that β is increasing if and only if x is decreasing. Recalling that x ( t ) dt = − σ x ( c σ + x − A x )4 ( c σ + x ) , we have that if x ( t ) > √ A c σ / (4 A ), the x (cid:48) ( t ) >
0. However, satisfying these inequalitiesfor any t would violate the boundary condition x ( T ) = 0, so we must have that x is decreasing and β is increasing.A straightforward computation shows that dλ ( t ) dt = − A σ Σ ( t ) h ( t )4 ( c σ + Σ( t ) h ( t )) , and so if A > λ is decreasing. (cid:3) From Theorem 10 and Proposition 11, the function β is positive, increasing, and satisfies β ( T ) =1 / c , therefore β is uniformly bounded by 1 / c . The uniform limitlim c →∞ β ( t ; c ) = 0 , (86)is then immediate. In equilibrium, the pricing rule is given by λ ( t ; c ) = β ( t ; c ) Σ( t ; c ) σ , (87)and from (30) we have that Σ is a decreasing function of t and therefore bounded by Σ(0; c ) = Σ v .Thus, we also have the limit lim c →∞ λ ( t ; c ) = 0 , (88)uniformly in t .For the remainder of the proof we only consider A >
0, as the proof for A = 0 is similar and morestraightforward. The limits which correspond to c → dx dt ( t ; c, k ) = f ( x ( t ; c, k ); c, k ) , x (0; c, k ) = Σ v ,f ( x ; c, k ) = − A σ x (cid:16) x γ ( c )1 + k (cid:17) (cid:18) ( γ ( c ) + 1) x γ ( c )1 + ( γ ( c ) − k (cid:19) , γ ( c ) = √ A c σ , ans where k is chosen so that( γ ( c ) + 1) x γ ( c )1 ( T ; c, k ) − ( γ ( c ) − k = 0 , (89)and we have made all dependences on c explicit. Note that all of these expressions above apply toboth cases c > c = 0, with the provision that the choice of k when c = 0 is the positive root in(82). Since the function f is continuously differentiable with respect to c , so is the solution x ( t ; c, k )(see Hartman (2002)). Additionally, γ is continuously differentiable with respect to c , and so by theimplicit function theorem we may take k = k ( c ) in view of (89). We then have x given by x ( t ; c, k ( c )) = ρ ( x ( t ; c, k ( c ))) = ( γ ( c ) + 1) x γ ( c )1 ( t ; c, k ( c )) − ( γ ( c ) − k ( c )4 A ( x γ ( c )1 ( t ; c, k ( c )) + k ( c )) , which is also seen to be continuously differentiable with respect to c . Recall that in equilibrium thetrading and pricing rules can be written as β ( t ; c ) = σ c σ + x ( t ; c, k ( c )) , λ ( t ; c ) = x ( t ; c, k ( c ))2 ( c σ + x ( t ; c, k ( c )) . Both of these expressions are continuous functions of t and c except at ( t, c ) = ( T,
0) where bothdenominators are equal to 0. Thus, if we restrict t to a compact subinterval on [0 , T ) and c toan interval of the form [0 , C ], then both β and λ are uniformly continuous with respect to ( t, c ),and the expressions given in (47) and (48) are obtained by direct substitution of c = 0 using thecorresponding solutions of x and x given in Lemma 9. (cid:3) . Appendix B This appendix contains the full systems of equations that are solved in the proof of Proposition 4. ˜ λ i,j (cid:18) λ , − λ , λ K (cid:19) = 0 , (cid:18) λ , σ − λ , λ K σ + 12 ˜ λ , ˜ λ , − λ , λ K (cid:19) = 0 , (cid:18)
12 ˜ λ , ˜ λ , − λ , λ K + 8 ˜ λ , − λ K (cid:19) = 0 , (cid:18) ˜ λ , σ + 16 ˜ λ , ˜ λ , σ + 12 ˜ λ , ˜ λ , + 12 ˜ λ , ˜ λ , − λ , ˜ λ , λ K σ − λ , λ K (cid:19) = 0 , (cid:18)
16 ˜ λ , ˜ λ , + 12 ˜ λ , ˜ λ , − λ , λ K + 4 ˜ λ , σ +16 ˜ λ , ˜ λ , σ + 24 ˜ λ , ˜ λ , ˜ λ , − λ , ˜ λ , λ K σ (cid:19) = 0 , (cid:18)
12 ˜ λ , ˜ λ , + 12 ˜ λ , ˜ λ , + 16 ˜ λ , ˜ λ , + 4 ˜ λ , − λ , λ K (cid:19) = 0 , (cid:18) λ , ˜ λ , σ + 16 ˜ λ , ˜ λ , σ + 24 ˜ λ , ˜ λ , σ + 12 ˜ λ , ˜ λ , + 24 ˜ λ , ˜ λ , ˜ λ , − λ , ˜ λ , λ K σ + 4 ˜ λ , − λ , λ K σ − λ , λ K (cid:19) = 0 , (cid:18)
16 ˜ λ , ˜ λ , + 12 ˜ λ , ˜ λ , + 12 ˜ λ , ˜ λ , − λ , λ K + 8 ˜ λ , + 5 ˜ λ , ˜ λ , σ +12 ˜ λ , ˜ λ , σ + 16 ˜ λ , ˜ λ , σ + 24 ˜ λ , ˜ λ , ˜ λ , + 24 ˜ λ , ˜ λ , ˜ λ , +48 ˜ λ , ˜ λ , ˜ λ , σ − λ , ˜ λ , λ K σ − λ , ˜ λ , λ K σ (cid:19) = 0 , (cid:18) λ , + 16 ˜ λ , ˜ λ , + 16 ˜ λ , ˜ λ , + 12 ˜ λ , ˜ λ , + 12 ˜ λ , ˜ λ , − λ , λ K +12 ˜ λ , ˜ λ , σ + 16 ˜ λ , ˜ λ , σ + 24 ˜ λ , ˜ λ , ˜ λ , + 24 ˜ λ , ˜ λ , ˜ λ , +24 ˜ λ , ˜ λ , σ − λ , λ K σ − λ , ˜ λ , λ K σ (cid:19) = 0 , (cid:18) λ , λ K − (cid:19) = 0 . .2. System for ˜ β i,j (cid:18)
16 ˜ β , λ K − (cid:19) = 0 , (cid:18)
16 ˜ β , λ K σ + 64 ˜ β , ˜ β , λ K (cid:19) = 0 , (cid:18)
32 ˜ β , λ K + 64 ˜ β , ˜ β , λ K + 16 ˜ β , λ K + 2 ˜ β , (cid:19) = 0 , (cid:18)
64 ˜ β , ˜ β , λ K + 96 ˜ β , ˜ β , λ K + 48 ˜ β , ˜ β , λ K σ (cid:19) = 0 , (cid:18)
160 ˜ β , ˜ β , λ K + 64 ˜ β , ˜ β , λ K + 48 ˜ β , ˜ β , λ K σ +192 ˜ β , ˜ β , ˜ β , λ K + 48 ˜ β , ˜ β , λ K + 2 ˜ β , (cid:19) = 0 , (cid:18)