On regularized optimal execution problems and their singular limits
OOn regularized optimal execution problems and their singular limits
Max O. Souza a , Y. Thamsten a a Instituto de Matem´atica e Estat´ıstica, Universidade Federal Fluminense, Niter´oi, RJ, Brasil
Abstract
We investigate the portfolio execution problem under a framework in which volatility and liquidity are bothuncertain. In our model, we assume that a multidimensional Markovian stochastic factor drives both ofthem. Moreover, we model indirect liquidity costs as temporary price impact, stipulating a power law torelate it to the agent’s turnover rate. We first analyze the regularized setting, in which the admissiblestrategies do not ensure complete execution of the initial inventory. We prove the existence and uniquenessof a continuous and bounded viscosity solution of the Hamilton-Jacobi-Bellman equation, whence we obtaina characterization of the optimal trading rate. As a byproduct of our proof, we obtain a numerical algorithm.Then, we analyze the constrained problem, in which admissible strategies must guarantee complete executionto the trader. We solve it through a monotonicity argument, obtaining the optimal strategy as a singularlimit of the regularized counterparts.
Keywords:
Optimal Execution; Illiquid Markets; Stochastic Volatility; Stochastic Liquidity; ViscositySolutions.
1. Introduction
We refer to the situation in which a financial agent must execute a large trade as the execution problem.It constitutes a significant part of algorithmic trading. A paradigm that we can use to find a solution is toset up a trading schedule: stipulate a time horizon to complete the program, and fractionate the originallylarge order into smaller ones. In this setting, to identify an adequate rate of trading, we must take intoaccount the trade-off among two major financial complexities: transaction costs; exposure to price risk. Wedescribe how these effects influence trading decisions as follows. On the one hand, if we send too sizeableorders, then we will undergo considerable price impact. For instance, in this case, our behavior may demandthe consumption of many layers of liquidity of the Limit Order Book (LOB) at each time we execute a trade.On the other hand, if we take too long to complete execution, then we will be excessively exposed to theinherent uncertainty in the price movements.Systematic research on optimal execution began with the works of Almgren and Chriss (AC) [2, 3]— see also [8]. The AC model has as among its premises the following core assumptions: (i) Bachelierprice dynamics; (ii) price impacts are of two types, viz., a temporary and a permanent one, depending ontheir persistence in time; (iii) the trader only sends market orders. This model proved itself to be veryuseful as a framework for the investigation of the trade execution problem. Some efforts reevaluating (ii)include [5, 13, 19]. Regarding (iii), papers [9, 24] consider the use of limit orders is execution algorithms.A geometric Brownian motion models the stock price in [17], in contradistinction to assumption (i). Forextensions envisaging to account for a non-constant market volume, see [10, 23].In the standard continuous-time model of AC, the temporary and permanent price impacts are functionsof the speed of trading of the agent. In this direction, the monographs [11, 29] contain statistical analyses
Email addresses: [email protected] (Max O. Souza), [email protected] (Y. Thamsten)
Preprint submitted to TBA a r X i v : . [ q -f i n . M F ] J a n f micro-structural aspects that are relevant to the study of algorithmic trading. We remark that theycorroborate, to an extent, the adequacy of the AC setting with reality. A popular choice for the functionalform of the impacts is the linear model: ones assumes that the impacts per share are proportional to the sizeof the orders sent to the market by the trader (usually with different slopes). Under suitable hypotheses,Huberman and Stanzl proved in [27] that the absence of quasi-arbitrage implies that the permanent priceimpact must indeed manifest in this way. This assumption also rules out price manipulation, as well asdynamic arbitrage, see [18, 23].From the empirical literature in price impact, e.g., [4, 6, 29], we see that the linearity assumption onthe temporary price impact is typically reasonable. Moreover, it is rather convenient, because it usuallyleads to analytical tractability. However, in many circumstances, assuming a nonlinear temporary impactinstead of the linear model leads to a more realistic shape of the impact curves. In particular, in settings inwhich explicit solutions are not available in the linear case, such as the one we will investigate in this work,it is pertinent to consider this more general assumption, resorting to numerical methods to compute thesolutions. The references [11, 23] contain a few works dealing with nonlinear temporary price impact. Seealso [5] for an early development of the modeling in this direction, and the recent work [30] for an applicationof neural networks to the execution problem with a power law assumption on the nonlinearity of the costunder discussion.In the AC setting, we find the optimal execution strategy through an optimization problem in a mean-variance framework. The design of the performance criteria is done in a way to represent a measure ofexecution quality. More precisely, it consists of minimizing the mean and the variance of the differencebetween the realized revenue and a benchmark price. A common choice for the latter is the pre-trade price,leading to a criterion known as implementation shortfall (IS), which we will consider henceforth. Alternativesare volume-weighted average price (VWAP), target close (TC), and percentage of volume (POV); we referto [11, 23] for in-depth discussions about these criteria.It is a stylized fact that a flat or known deterministic profile for the volatility and liquidity is a suitableassumption for highly liquid assets, such as large-capitalization US stocks. It leads, in turn, to deterministicclosed-form formulas for the corresponding optimal strategies. An advantage of this fact is that we canpre-compute the trading curve. However, as it is described in [1], small-cap stocks are more difficult totrade. Throughout the trading day, there are moments in which trading is cheap, and others in which it isexpensive. Furthermore, these moments alternate randomly, making the problem more difficult. Thus, whenperforming execution of portfolios containing less liquid assets, considering uncertainty in their volatility andliquidity becomes an important feature. In effect, an illiquid scenario ought to lead to higher price impacts;this, together with lower volatility levels, incentivizes the trader to slow down the execution program.Analogously, a situation of higher liquidity and volatility urges her to speed up. In these circumstances, theneed for strategies that adapt to the current market state arises. We stress that there are also many studieson intra-day volatility estimation, of which we mention [16]. Thus, implementing these adaptive strategiesin practice seems to be a feasible endeavor.In the present work, we consider a risk averse agent which assesses the performance of the admissiblestrategies by utilizing criteria of the IS kind. However, two distinguishing modelling features here arethat we allow the temporary impact to be of a general power law type, and that, at a first moment, wetolerate hitting the terminal target only approximately. Then, we consider the constrained problem, i.e., theframework in which the agent must necessarily finish with zero inventory. Moreover, we consider volatilityand the temporary price impact parameter as stochastic processes, with a multidimensional Markov diffusionas their driver. In the literature, a paper considering stochastic price impacts, but constant volatility, is [7].There is a model for treating the problem of slicing a VWAP order under stochastic volatility and marketvolume, but with no form of execution costs, in [28]. The framework of [20] considers constant temporaryand permanent price impacts, but stochastic resilience and volatility (the latter through the uncertain riskaversion parameter). Some advances in the execution problem when simultaneously considering stochasticliquidity and volatility are [1, 21, 22, 25, 26], for the single agent case, [12], in a discrete time setting, and[14, 15], in game-theoretic models.We prove the existence and uniqueness of viscosity solutions of the Hamilton-Jacobi-Bellman (HJB)Partial Differential Equation (PDE) for the penalized problem, in an appropriate class, via an iterative2onotonicity fixed-point method. We obtain appropriate bounds for the solution of the main HJB, and thisallows us to derive bounds for the terminal inventory in terms of the penalization parameter. In particular, wecan identify a range of values of the penalization parameter which guarantee that following the correspondingoptimal trading rate leads to the execution of any prescribed portion of the trader’s initial inventory. Wealso show that this strategy does not lead to speculative trading. Moreover, our method naturally yields anumerical method for computing the solution, which we implement to provide illustrations. Then, we useanother monotonicity argument to establish that the singular limit, as the terminal penalization parametergoes to infinity, is the solution of the constrained problem, i.e., the one in which we require strict execution.The works that are closer to the present one in their modeling aspects are [1, 22, 25]. In [1, 25], theauthors only investigate the case of a linear temporary price impact per share, leading to quadratic objectivefunctionals. The work [22] is based on criteria stemming from the power law hypothesis, as in the presentwork. Our results are in the line of those of [22, 25], but our proofs rely on techniques that are distinctto the ones employed there. In contradistinction to those papers, we analyze the singular problem as thelimit of the regularized counterpart. Our approach has the benefit that we only require continuity of thecoefficients to prove our existence and uniqueness result. Also, the method we employ yields a numericalalgorithm, cf. [25, Remark 2.9].We organize the remainder of this paper as follows. In Section 2 we describe the model in detail, anddefine the value functions for both problems, viz., the regularized and the constrained ones. In Section3, we make an ansatz leading to a semilinear HJB PDE for the corresponding value function. We proveexistence and uniqueness of a viscosity solution of it, in a suitable class. We also prove some properties ofthe optimal turnover rate, and present some numerical experiments. In Section 4 we obtain the solution ofthe constrained problem as a singular limit of the penalized optimal strategies via a monotonicity argument.In Section 5 we present our conclusions.We finish this introductory Section by fixing some notations. Throughout the present work, we fix T > , F , F , P ) a complete filtered probability space,where F = {F t } (cid:54) t (cid:54) T is a filtration satisfying the usual conditions, and F T = F . We also assume that F supports a one-dimensional Brownian motion B, as well as an m − dimensional one W , for a fixed positiveinteger m. All stochastic processes figuring throughout this work will be F − adapted. We interpret P asthe historical (or statistical) measure. For 0 (cid:54) t (cid:54) T, the set U t comprises the F − progressively measurablestochastic processes { ν u } t (cid:54) u (cid:54) T satisfying E (cid:34)(cid:90) Tt ν u du (cid:35) < ∞ . Moreover, given a Markovian multidimensional process { x u } t (cid:54) u (cid:54) T , we write E t,x [ · ] = E [ ·| x t = x ] . The letter C denotes a generic positive constant that may change from line to line. It will depend on allmodel parameters, unless we explicitly state otherwise.
2. The model
For t ∈ [0 , T ] , let us consider an agent trading shares of a certain asset, during the time horizon [ t, T ] , with turnover rate { ν u } t (cid:54) u (cid:54) T , i.e., ν u is the instantaneous rate at which this trader negotiates at time u. Her inventory process { Q u } t (cid:54) u (cid:54) T has dynamics (cid:40) dQ νu = ν u du,Q νt = q. (2.1)3e consider a stock whose price process { S νu } t (cid:54) u (cid:54) T evolves according to (cid:40) dS νu = σ u dB u ,S νt = S. (2.2)Above, the process { σ u } t (cid:54) u (cid:54) T is the absolute volatility of the asset price. The agent’s cash process hasdynamics (cid:40) dX νu = − S νu ν u du − κ u | ν u | φ du,X νt = x, (2.3)where the parameter φ ∈ ]0 ,
1] is exogenously given. Thus, we are accounting for a temporary impact pershare which is not necessarily linear on the agent’s turnover rate. Instead, this indirect cost determined byan appropriate power of its absolute value. The book value of the agent’s cash plus inventory at time t, which we refer to as her wealth, is w νu := X νu + Q νu S νu ( t (cid:54) u (cid:54) T ) . (2.4)Using Itˆo’s formula, it follows that w νT = w νt + (cid:90) Tt (cid:8) − ( S νu ν u + κ u | ν u | φ ) du + S νu ν u du + Q νu σ u dB u (cid:9) = w νt − (cid:90) Tt κ u | ν u | φ du + (cid:90) Tt σ u Q νu dB u . (2.5)From here on, we assume (with slight abuse of notations) κ u = κ ( y u ) and σ u = σ ( y u ) , where κ, σ : R d → [0 , ∞ [ , for a positive integer d, and that { y u } t (cid:54) u (cid:54) T is a d − dimensional Markov diffusion.More precisely, we suppose that there are functions α : R d → R d , β : R d → R d × m , such that (cid:40) d y u = α ( y u ) dt + β ( y u ) d W u , y t = y . (2.6)Henceforth, we make the subsequent hypotheses on the functions introduced above: (H1) The functions α and β are Lipschitz continuous. (H2) Both κ and σ are continuous functions and there are κ, κ, σ > , σ (cid:62) , such that κ (cid:62) κ (cid:62) κ and σ (cid:62) σ (cid:62) σ. In Section 3, we consider the regularized problem, i.e., the circumstance in which we do not require strictexecution but penalize non-vanishing terminal inventory holdings. We intend to work under the dynamicprogramming paradigm of stochastic optimal control, leading us to the following definition.
Definition 2.1.
Given t ∈ [0 , T ] , our performance assessment of a strategy ν ∈ U t is made via the criterion J ν ( t, x, S, q, y ) := E t,x,S,q, y [ w νT − ( x + qS )] − E t,x,S,q, y (cid:34) A | Q νT | φ + γ (cid:90) Tt σ φu | Q νu | φ du (cid:35) = E t,q, y (cid:34)(cid:90) Tt (cid:110) − κ ( y u ) | ν u | φ − γσ φ ( y u ) | Q νu | φ (cid:111) du − A | Q νT | φ (cid:35) , (2.7) where A > is a constant. emark 2.2. The last equality in (2.7) implies J ν ( t, x, S, q, y ) = J ν ( t, q, y ) . Remark 2.3.
The assumption φ ∈ ]0 , ensures that J ν is well-defined, for each t ∈ [0 , T [ and ν ∈ U t . For a given ν ∈ U t , the criterion J ν defined in (2.7) includes two parts. The first one comprises theexpectation of the difference between the agent’s terminal wealth, w T , and her initial cash plus the pre-trade price, x + qS. Therefore, we take an IS viewpoint. Two penalization terms constitute the remainingpart of J ν : (i) The term proportional to | Q νT | φ , for ending up with terminal inventory; (ii) The integral (cid:82) Tt σ φ ( y u ) | Q νu | φ du, which represents a sense of urgency of the trader. We observe from the identity(2.5) that the addition of the latter term is a natural risk management tool to control (cid:8)(cid:82) st σ u Q u dB u (cid:9) t (cid:54) s (cid:54) T , which is a source of uncertainty in the terminal wealth w T collected by the agent via following her strategy.In particular, when φ = 1 , this recovers the popular mean-variance framework of [2]. Furthermore, in viewof the form of the expectation of w T − w t , it seems appropriate to consider the power 1 + φ as we do here(for both terms in (i) and (ii) we described above). We will show that these modeling choices do lead us toa dimensionality reduction, viz., they allow us to drop the dependence on the variable q, in a precise sensethat we will discuss briefly. See the works [22, 30] for a similar approach to related problems.The parameter A in (2.7) makes the trader tolerant for finishing the schedule with a nonzero inventory.Mathematically, it has the effect of regularizing the problem. In Section 4, we will establish that, as A tends to infinity, the optimal strategy of the regularized problem converges, in an appropriate sense, to thesolution of the one in which complete execution is required. However, prior to taking limits, we do obtainestimates on the remaining terminal inventory in terms of A. Thus, we indicate how large a trader shouldchoose A to guarantee the execution of a given percentage of her initial inventory. The interpretation of theparameter γ in (2.7) is that it represents the risk aversion of the agent. In the linear temporary price impactcase, in which case φ = 1 , then we identify 2 γ as the risk aversion parameter for a constant absolute riskaversion model, see [11, 23]. We notice that for the same level of risk aversion γ, the trader is more (less)urgent for higher (lower) variance levels. We remark that we can treat other forms of stochastic urgencyparameters using the techniques of the present work, under mild assumptions. For concreteness, we proceedwith the model we presented above.Alongside ( H1 ) and ( H2 ), we make the following hypothesis on A : (H3) The terminal penalization parameter satisfies
A > (cid:18) γσ φ κ φ φ (cid:19) φφ +1 . We remark that ( H3 ) is convenient (mainly for notational purposes), but it is not a necessary assumption.Moreover, since our main interest is the regime where A is large, it is not restrictive.We subtract the quantity x + qS in (2.7) envisaging to attain a dimensionality reduction. In analogy towhat is exposed in [23], this can be interpreted as a comparison between our revenue from following strategy ν, during the time window [ t, T ] , and the book value of initial inventory, x + qS. Therefore, we follow theIS paradigm by considering these performance criteria. In the sequel, we introduce our value function.
Definition 2.4.
The value function J is given by J ( t, q, y ) := sup ν ∈U t J ν ( t, q, y ) (cid:0) ( t, q, y ) ∈ [0 , T ] × R × R d (cid:1) . (2.8) In Section 4, we will be concerned with the analysis of the constrained problem:sup ν ∈U c (cid:32) J ν ∞ ( q, y ) := E , q, y (cid:34) − (cid:90) T (cid:110) κ ( y t ) | ν t | φ + γσ φ ( y t ) | Q νt | φ (cid:111) dt (cid:35)(cid:33) . (2.9)We call the problem “constrained” because we define the set of admissible controls U c , figuring above, as U c := (cid:40) { ν t } (cid:54) t (cid:54) T ∈ L φ : (cid:90) T ν t dt = − q, P − a.s. (cid:41) , (2.10)5here L φ := (cid:40) { ν t } (cid:54) t (cid:54) T : { ν t } t is F − progressively measurable, and E (cid:34)(cid:90) T | ν t | φ dt (cid:35) < ∞ (cid:41) . That is, we stipulate in (2.10) an execution constraint. The performance criteria for the current problemare the functionals J ν ∞ we defined in (2.9). We remark that J ν ∞ ( q, y ) = J ν (0 , q, y ) , for ν ∈ U c . Moreover,we notice that, properly identifying processes of L φ which agree dt × d P − a.e.a.s., we can render this setinto a Banach space by endowing it with the norm (cid:107) ν (cid:107) φ := E (cid:34)(cid:90) T | ν t | φ dt (cid:35) φ . In view of the form of our performance criteria, the membership in L φ provides the natural integrabilitycondition for a solution of the problem (2.9). The other constraint we placed in the definition of U c in (2.10)means precisely that we are only interested in strategies guaranteeing the complete execution of the initialinventory.
3. Analysis of the regularized problem
From [33, Theorem 4.3.1], we know that the value function J is a viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation ∂ t J + L J + sup ν (cid:110) − κ | ν | φ + ν∂ q J (cid:111) − γσ φ | q | φ = 0 , with J ( T, q, y ) = − A | q | φ , where L is the infinitesimal generator of { y t } t , (cid:40) L := tr (cid:0) ββ (cid:124) ( y ) D y (cid:1) + α ( y ) · D y , for ( D y ) i := ∂ y i and ( D y ) ij := ∂ y i ∂ y j , i, j ∈ { , . . . , d } . We have sup ν (cid:110) − κ | ν | φ + ν∂ q J (cid:111) = κ (cid:101) H (cid:18) ∂ q Jκ (cid:19) , where (cid:101) H ( p ) = φ (cid:18) | p | φ (cid:19) φ , and the optimal control in feedback form is ν ∗ ( t, q, y ) := (cid:101) H (cid:48) (cid:18) ∂ q J ( t, q, y ) κ ( y ) (cid:19) . (3.1)In this way, the HJB reads ∂ t J + L J + κ (cid:101) H (cid:18) ∂ q Jκ (cid:19) − γσ φ | q | φ = 0 . (3.2)We propose the ansatz J ( t, q, y ) = z ( t, y ) | q | φ , (3.3)which leads to the PDE (cid:40) ∂ t z + L z + κH (cid:0) zκ (cid:1) − γσ φ = 0 ,z ( T, y ) = − A, (3.4)6here H ( p ) = (1 + φ ) φ (cid:101) H ( p ) = φ | p | φ . Arguing as in [22, Lemma 2.7], we can show that z solves (3.4)if, and only if, ( t, q, y ) (cid:55)→ z ( t, y ) | q | φ solves (3.2). In the next Section, we turn to the analysis of (3.4).In particular, we will show a verification result, guaranteeing that (3.3) holds.The most fundamental aspect of the analysis of the regularized problem is the investigation of the PDE(3.4). Thus, we refer to this equation as our main PDE. The present Section is devoted to establishing theexistence and uniqueness of a continuous and bounded viscosity solution of it. The subsequent theorem will be key in the remainder of this section. It is a particular case of [32,Theorem 3.42].
Theorem 3.1.
Under (H1) and (H2) , let c, f : [0 , T ] × R d → R and g : R d → R be three continuous andbounded functions. Define h ( t, y ) := E t, y (cid:34)(cid:90) Tt e (cid:82) ut c ( τ, y τ ) dτ f ( u, y u ) du + e (cid:82) Tt c ( τ, y τ ) dτ g ( y T ) (cid:35) . (3.5) Then h is continuous, and it is the unique viscosity solution of the PDE (cid:40) ∂ t h + L h + ch + f = 0 in ]0 , T [ × R d ,h | t = T = g in R d , (3.6) within the class G consisting of continuous functions satisfying lim | y |→∞ h ( t, y ) e − δ [log | y | ] = 0 , for some δ > . In effect, from the representation (3.5), the property of comparison holds. We state it in Corollary 3.2.
Corollary 3.2.
Let c, f, (cid:101) f : [0 , T ] × R d → R , g, (cid:101) g : R d → R be five bounded continuous functions. Define h as in (3.5), and likewise (cid:101) h, the latter having (cid:101) f and (cid:101) g in place of f and g, respectively. If f (cid:62) (cid:101) f and g (cid:62) (cid:101) g, then h (cid:62) (cid:101) h. Let us introduce the operators A ( h ) := ∂ t h + L h + κH ( h/κ ) − γσ φ = ∂ t h + L h + φκ − φ | h | φ − γσ φ , A ( h ) := ∂ t h + L h + φκ − φ | h | φ − γσ φ , A ( h ) := ∂ t h + L h + φκ − φ | h | φ − γσ φ . We observe that A ( h ) (cid:62) A ( h ) (cid:62) A ( h ) . (3.7)We will use the above operators to build a subsolution and a supersolution to (3.4), which will help usto prove the well-posedness of the latter PDE. These constructions will rely upon the following result on aclass of Ordinary Differential Equations (ODEs).We observe that, for each a, b > r > bA r − a > , the scalar initial value problem (cid:40) y (cid:48) = a − b | y | r ,y ( T ) = − A, y on [0 , T ] . Moreover, y is monotone decreasing and − A (cid:54) y < − ( a/b ) /r . (3.8)In effect, y is given by y ( t ) = F − ( T − t ) , for the bijective differentiable mapping F : ξ ∈ (cid:104) − A, − ( a/b ) /r (cid:104) (cid:55)→ − (cid:90) ξ − A dua − b | u | r ∈ [0 , ∞ [ , which satisfies F (cid:48) > . Thus, choosing a and b suitably, and r := 1 + 1 /φ, we infer that there are twodifferentiable deterministic functions z, z : [0 , T ] → [ − A,
0[ (independent of the state variable y ∈ R d )solving (cid:40) A ( z ) = 0 ,z ( T ) = − A, (3.9)and (cid:40) A ( z ) = 0 ,z ( T ) = − A. (3.10)Furthermore, they are subject to the bounds − A (cid:54) z (cid:54) z (cid:54) − (cid:32) γσ φ κ φ φ (cid:33) φφ +1 (cid:54) . (3.11)The inequalities in the extremes of (3.11) are straightforward to derive from (3.8), and we can show theone in the middle by standard ODE comparison arguments. We can also show that there exists a positiveconstant C, independent of A, such that1 C ( A − φ + T − t ) φ (cid:54) | z ( t ) | (cid:54) C ( A − φ + T − t ) φ and 1 C ( A − φ + T − t ) φ (cid:54) | z ( t ) | (cid:54) C ( A − φ + T − t ) φ , (3.12)for 0 (cid:54) t (cid:54) T, see Appendix A. In the next Subsection, we will find the solution z of (3.4), in an appropriatesense, subject to z (cid:54) z (cid:54) z. Intuitively, this is coherent with a comparison principle, cf. (3.7).
We obtain existence and uniqueness results for (3.4) through an iterative monotonicity method. For adescription of this approach in other contexts, we refer to [31, Chapter 7] and [36, Chapter 12]. Here, weapply this technique in the setting of viscosity solutions with milder hypotheses on model coefficients.The first step is to define the bounded continuous coefficient c,c := − ( φ + 1) (cid:18) | z | κ (cid:19) φ = − ( φ + 1) (cid:16) − zκ (cid:17) φ , and designate by A c the operator A c h := ∂ t h + L h + ch. Instead of solving (3.4), we will solve the equivalent problem (cid:40) A c z + κH ( z/κ ) − γσ φ − cz = 0 in ]0 , T [ × R d ,z | t = T = − A in R d . (3.13)8 emma 3.3. Let z (1) , z (1) ∈ G be the viscosity solutions of the PDEs (cid:40) A c z (1) + f (1) = 0 in ]0 , T [ × R d ,z (1) | t = T = − A in R d , (cid:40) A c z (1) + f (1) = 0 in ]0 , T [ × R d ,z (1) | t = T = − A in R d , where f (1) := − γσ φ + φκ − φ | z | φ − ch. and f (1) := − γσ φ + φκ − φ | z | φ − ch Then, z (cid:54) z (1) (cid:54) z (1) (cid:54) z. (3.14) Proof.
We notice that the functions h and h solve A c z + f = 0 , A c z + f = 0 ,z | t = T = − A = z | t = T , where (cid:40) f := − γσ φ + φκ − φ | z | φ − cz,f := − γσ φ + φκ − φ | z | φ − cz. From the relations f (1) − f = γ ( σ φ − σ φ ) + φ (cid:16) κ − φ − κ − φ (cid:17) | z | φ (cid:62) , and z (1) | t = T = z (1) | t = T , we conclude through Corollary 3.2 that z (cid:54) z (1) . Likewise, from the inequalities f (1) − f = γ ( σ φ − σ φ ) + φ (cid:16) κ − φ − κ − φ (cid:17) | z | φ (cid:54) f (1) − f (1) = φκ − φ (cid:16) | z | φ − | z | φ (cid:17) − c ( z − z ) (cid:62) (cid:104) − ( φ + 1) κ − φ | z | φ − c (cid:105) ( z − z )= 0 , alongside the fact that z | t = T = z | t = T = z | t = T = z | t = T , we deduce that we can apply Corollary 3.2 to deducethe other two inequalities in (3.14). Lemma 3.4.
We set z (0) := z and z (0) := z. For some k (cid:62) , we assume that there are functions (cid:8) z ( l ) , z ( l ) (cid:9) kl =0 ⊆ G solving the PDEs A c z ( l ) + f ( l ) = 0 in ]0 , T [ × R d , where f ( l ) := − γσ φ + φκ − φ (cid:12)(cid:12) z ( l − (cid:12)(cid:12) φ − cz ( l − ,z ( l ) = − A in R d , A c z ( l ) + f ( l ) = 0 in ]0 , T [ × R d , where f ( l ) := − γσ φ + φκ − φ (cid:12)(cid:12) z ( l − (cid:12)(cid:12) φ − cz ( l − ,z ( l ) = − A in R d , in the viscosity sense, for (cid:54) l (cid:54) k, and satisfying z = z (0) (cid:54) · · · (cid:54) z ( k − (cid:54) z ( k ) (cid:54) z ( k ) (cid:54) z ( k − (cid:54) · · · (cid:54) z (0) = z. Then, considering the viscosity solutions of z ( k +1) , z ( k +1) ∈ G of A c z ( k +1) + f ( k +1) = 0 , in ]0 , T [ × R d , where f ( k +1) := − γσ φ + φκ − φ (cid:12)(cid:12) z ( k ) (cid:12)(cid:12) φ − cz ( k ) ,z ( k +1) | t = T = − A in R d , and A c z ( k +1) + f ( k +1) = 0 in ]0 , T [ × R d , where f ( k +1) := − γσ φ + φκ − φ (cid:12)(cid:12) z ( k ) (cid:12)(cid:12) φ − cz ( k ) ,z ( k +1) | t = T = − A in R d , we have z ( k ) (cid:54) z ( k +1) (cid:54) z ( k +1) (cid:54) z ( k ) . Proof.
Under the present assumptions, we have z ( l ) (cid:62) z and z ( l ) (cid:62) z, for all 0 (cid:54) l (cid:54) k. Hence, we canestimate f ( k +1) − f ( k ) = φκ − φ (cid:18)(cid:12)(cid:12)(cid:12) z ( k ) (cid:12)(cid:12)(cid:12) φ − (cid:12)(cid:12)(cid:12) z ( k − (cid:12)(cid:12)(cid:12) φ (cid:19) − c ( z ( k ) − z ( k − ) (cid:62) (cid:20) − ( φ + 1) κ − φ (cid:12)(cid:12)(cid:12) z ( k − (cid:12)(cid:12)(cid:12) φ − c (cid:21) ( z ( k ) − z ( k − ) (cid:62) . Likewise, we show f ( k +1) − f ( k +1) (cid:62) , as well as f ( k +1) − f ( k ) (cid:54) . Since z ( k +1) | t = T = z ( k ) | t = T = z ( k ) | t = T = z ( k +1) | t = T , we conclude the desired result from Corollary 3.2.From Lemmas 3.3 and 3.4, we conclude that the sequences { z ( k ) } k (cid:62) and { z ( k ) } k (cid:62) such that z (0) := z and z (0) := z, whereas for k (cid:62) , they are viscosity solutions in the class G of the PDEs (cid:40) A c z ( k ) = γσ φ − φκ − φ (cid:12)(cid:12) z ( k − (cid:12)(cid:12) φ + cz ( k − in ]0 , T [ × R d ,z ( k ) = − A in R d , (cid:40) A c z ( k ) = γσ φ − φκ − φ (cid:12)(cid:12) z ( k − (cid:12)(cid:12) φ + cz ( k − in ]0 , T [ × R d ,z ( k ) = − A in R d , are well-defined. We emphasize that, in particular, the membership in the class G ensures their continuity.Furthermore, they satisfy z (cid:54) z ( k ) (cid:54) z ( k +1) (cid:54) z ( k +1) (cid:54) z ( k ) (cid:54) z, k (cid:62) . Therefore, it is licit to define the following pointwise limits z ∗ ( t, y ) := lim k →∞ z ( k ) ( t, y ) and z ∗ ( t, y ) := lim k →∞ z ( k ) ( t, y ) (( t, y ) ∈ [0 , T ] × R d ) . We observe that they satisfy z (cid:54) z ∗ (cid:54) z ∗ (cid:54) z. Theorem 3.5.
The PDE (3.13) has a unique bounded continuous viscosity solution z. Moreover, it is givenby z = z ∗ = z ∗ . Proof.
Firstly, we note that Theorem 3.1 implies z ( k ) ( t, y ) = E t, y (cid:34) (cid:90) Tt e (cid:82) ut c ( τ, y τ ) dτ (cid:40) − γσ φ ( y u ) + φκ ( y u ) − φ (cid:12)(cid:12)(cid:12) z ( k − ( u, y u ) (cid:12)(cid:12)(cid:12) φ − c ( u, y u ) z ( k − ( u, y u ) (cid:41) du − Ae (cid:82) Tt c ( u, y u ) du (cid:35) . Next, we can let k → ∞ and use the Dominated Convergence Theorem to deduce that z ∗ solves z ∗ ( t, y ) = E t, y (cid:34) (cid:90) Tt e (cid:82) ut c ( τ, y τ ) dτ (cid:40) − γσ φ ( y u ) + φκ ( y u ) − φ | z ∗ ( u, y u ) | φ − c ( u, y u ) z ∗ ( u, y u ) (cid:41) du − Ae (cid:82) Tt c ( u, y u ) du (cid:35) . (3.15)From the representation (3.15), we can show that z ∗ is continuous, see Appendix B. Therefore, accordingto Theorem 3.1, the function on the right-hand side of (3.15), which we proved to be equal to z ∗ , is alsocontinuous and solves the PDE (cid:40) A c ( z ∗ ) = γσ φ − φκ − φ | z ∗ | φ + cz ∗ in ]0 , T [ × R d ,z ∗ | t = T = − A in R d . in the viscosity sense. In other words, z ∗ is a viscosity solution of (3.13) or, equivalently, this function solves(3.4). We can make the same argument to show that z ∗ enjoys this same property. This proves the existencepart of the Theorem. The fact that z ∗ = z ∗ will follow from the proof of the uniqueness of continuous andbounded viscosity solutions of (3.4), which we now turn to show.Let us assume (cid:101) z i , i = 1 , , are two bounded continuous viscosity solutions of (3.4). Applying Theorem3.1, we infer (cid:101) z i ( t, y ) = E t, y (cid:34)(cid:90) Tt (cid:110) − γσ φ ( y u ) + φκ ( y u ) − φ | (cid:101) z i ( u, y u ) | φ (cid:111) du (cid:35) − A. Setting δ := (cid:101) z − (cid:101) z , we obtain δ ( t, y ) = E t, y (cid:34)(cid:90) Tt g ( u, y u ) δ ( u, y u ) du (cid:35) , (3.16)where g ( t, y ) := φκ ( y ) − φ (cid:18) | (cid:101) z ( u, y ) |
1+ 1 φ −| (cid:101) z ( u, y ) |
1+ 1 φ (cid:101) z ( u, y ) − (cid:101) z ( u, y ) (cid:19) if (cid:101) z ( t, y ) (cid:54) = (cid:101) z ( t, y ) , ( φ + 1) κ ( y ) − φ | (cid:101) z ( t, y ) | φ sgn ( (cid:101) z ( t, y )) otherwise.We notice that g is bounded. Let C > | g | (cid:54) C. We set∆( t ) := sup ( a,η ) | δ ( t, a, η ) | . t ) (cid:54) C (cid:90) Tt ∆( u ) du (0 (cid:54) t (cid:54) T ) . (3.17)An application of Gronwall’s Lemma gives ∆ ≡ , whence (cid:101) z ≡ (cid:101) z . This finishes the proof of the Theorem.
Corollary 3.6.
The convergences lim k →∞ z ( k ) = lim k →∞ z ( k ) = z are uniform over compact subsets of [0 , T [ × R d . Proof.
From Theorem 3.5, we know that the limiting functions z ∗ and z ∗ indeed coincide and are continuous.An application of Dini’s Theorem, see [34, Theorem 7.13], gives the result we stated. The first result we expose in this subsection are representations of the optimal speed of trading andinventory in terms of the solution of (3.4).
Theorem 3.7.
The value function is indeed given by (3.3) . Thus, the optimal speed of trading { ν ∗ t } (cid:54) t (cid:54) T and the corresponding optimal inventory holdings (cid:8) Q ∗ t := Q ν ∗ t (cid:9) (cid:54) t (cid:54) T are given by ν ∗ t = − q (cid:18) − z ( t, y t ) κ ( y t ) (cid:19) φ exp (cid:32) − (cid:90) t (cid:18) − z ( u, y u ) κ ( y u ) (cid:19) φ du (cid:33) , (3.18) and Q ∗ t = q exp (cid:32) − (cid:90) t (cid:18) − z ( u, y u ) κ ( y u ) (cid:19) φ du (cid:33) . (3.19) Proof.
We can prove the verification result, namely, that J ( t, q, y ) = z ( t, y ) | q | φ , just as in [22, Proposition2.10] or [25, Proposition 2.8]. The proof here is even easier because we are considering the regularizedproblem, and there is no jump component on the trader’s inventory process; for completeness, we include itin Appendix C.From (3.1) and (3.3), and recalling that z (cid:54) z (cid:54) , we derive ν ∗ ( t, q, y ) := sgn ( ∂ q J ( t, q, y )) (cid:18) | ∂ q J ( t, q, y ) | (1 + φ ) κ ( y ) (cid:19) φ = − (cid:18) − z ( t, y ) κ ( y ) (cid:19) φ q. (3.20)Therefore, from dQ ∗ t = ν ∗ ( t, Q ∗ t , y t ) dt = − (cid:18) − z ( t, y t ) κ ( y t ) (cid:19) φ Q ∗ t dt we deduce Q ∗ t = q exp (cid:32) − (cid:90) t (cid:18) − z ( u, y u ) κ ( y u ) (cid:19) φ du (cid:33) . This proves (3.19). Using (3.19) in (3.20), we show (3.18).
Corollary 3.8.
The optimal terminal inventory holdings satisfies | Q ∗ t | (cid:54) | q | (cid:18) T − t + A − /φ T + A − /φ (cid:19) ( (cid:96)κ ) /φ , (3.21) for each (cid:54) t (cid:54) T, where we have written (cid:96) := inf s ∈ [0 ,T ] (cid:104) | z ( s ) | ( T − s + A − /φ ) φ (cid:105) . (3.22)12 emark 3.9. In view of (3.11) , we observe that the constant (cid:96) we defined in (3.22) is strictly positive, andthat it is independent of A. Proof.
From (3.19), we deduce | Q ∗ t | (cid:54) | q | exp (cid:18) − κ /φ (cid:90) t | z ( u, y u ) | /φ du (cid:19) = | q | exp (cid:20) − κ /φ (cid:90) t | z ( u, y u ) | /φ ( T − u + A − /φ )( T − u + A − /φ ) du (cid:21) (cid:54) | q | exp (cid:20) − (cid:96) /φ κ /φ (cid:90) t T − u + A − /φ ) du (cid:21) (cid:54) | q | exp (cid:20) − (cid:96) /φ κ /φ log (cid:18) T + A − /φ T − t + A − /φ (cid:19)(cid:21) , (3.23)from where the result we stated immediately follows.Since lim A →∞ (cid:18) A − /φ T + A − /φ (cid:19) ( (cid:96)κ ) /φ = 0 , we deduce from (3.21) that, for any 0 (cid:54) θ < , we can choose A sufficiently large so as to have | Q ∗ T | < (1 − θ ) | q | . More precisely, as long as (cid:18) A − /φ T + A − /φ (cid:19) ( (cid:96)κ ) /φ < − θ, or equivalently, A > (cid:40) T exp (cid:34)(cid:18) (cid:96)κ (cid:19) − /φ log (cid:18) − θ (cid:19)(cid:35)(cid:41) φ we guarantee the execution of the fraction 1 − θ of the initial inventory. In practice, we can choose θ in sucha way that (1 − θ ) | q | is less than one lot size of the asset, resulting in a full execution. It is also remarkablethat the bound (3.21) is model-free in the sense that, to derive it, we need only assume that the traderfollows strategy ν ∗ of (3.18).From Theorem 3.7, it promptly follows that our optimal strategy does not lead the agent to engage inspeculative trading. This is the content of the next result. Corollary 3.10.
The optimal strategy { ν ∗ t } (cid:54) t (cid:54) T does not practice price manipulation, i.e., for (cid:54) t (cid:54) T,q ν ∗ t (cid:54) and q Q ∗ t (cid:62) , P − almost surely. (3.24)The first inequality in (3.24) means that the trader does not buy (respectively, sell) in the context of aliquidation (respectively, acquisition) program. The second one describes that such an agent will not oversell(respectively, overbuy) when executing a portfolio liquidation (respectively, acquisition). Hence, this resultdoes indeed guarantee the absence of price manipulation in the current model.13 .6. Some numerical experiments Our proof of the existence of the solution z of (3.4) also establishes the convergence of the followingnumerical algorithm: Algorithm 1:
Iterative numerical algorithm for solving the PDE (3.4).
Result:
Numerical solution of (3.4).Initialize z (0) := z or z (0) := z, k = 0 , the error variable (cid:15), and stipulate the tolerance (cid:15) ; while (cid:15) (cid:62) (cid:15) do . (cid:40) ∂ t z ( k +1) + L z ( k +1) + cz ( k +1) = γσ φ − φκ − φ (cid:12)(cid:12) z ( k ) (cid:12)(cid:12) φ + cz ( k ) in ]0 , T [ × R d ,z ( k +1) | t = T = − A in R d ;2 . Update (cid:15) ;3 . k ← k + 1 . end At a first step, we notice that the initial iterate z (0) must itself be numerically computed, by using aproper ODE integrator, cf. (3.9) and (3.10). There is an easier case, namely when φ = 1 , corresponding toa linear temporary price impact setting. In this case, it is straightforward to derive closed-form formulas forboth z and z. Furthermore, each iteration we make in Algorithm 1, involves solving a linear parabolic PDE.In the numerical experiments that follow, we used a Crank-Nicolson scheme to solve the linear PDE ateach iteration step, determining the boundary conditions in the computation domain by linear extrapolation.Here, we make the simplifying assumption of coordinated variation, see [1, Subsection 1.3] – hence, we have d = m = 1 . We show in Table 1 the parameters that we kept fixed in the numerical experiments that follow. Wewill describe the remaining ones in each of the corresponding plots. Also, the spatial domain we chose tocompute the solution in each of the experiments is (cid:2) y , y (cid:3) = [ − , .T α ( y ) β ( y ) κ ( y ) σ ( y ) Q ∗ = q − y κ ∨ [( κ e y ) ∧ κ ] (cid:16) κ κ ( y ) (cid:17) − / Table 1: Some fixed model parameters we use throughout all of the present simulations. Above, we fix κ := 0 . κ := κ / , and κ := κ × . In Figure 1, we showcase the particular realization of the stock price corresponding to a volatility and atemporary impact parameter paths that we will use to illustrate the behavior of the strategies. We carry outsome comparative statics, varying A in Figure 2, φ in Figure 3, and γ in Figure 4, ceteris paribus . We carryout a Monte Carlo simulation of 10 such paths, and demonstrate in Figures 5 and 6 some histograms toillustrate the behavior of the optimal strategies corresponding to each of the values of φ and γ we consideredpreviously. Of course, the same innovations were used for the different parameter values. For A, we findmore insightful to understand how the terminal inventory Q ∗ T changes with this parameter, whence we plotin Figure 7 the histogram of the values of Q ∗ T , resulting from this same 10 simulations, for A ∈ { , } .γ E [ X ∗ T ] E [ Q ∗ T ] E [ w ∗ T ]5 × − .
446 (17 . .
118 (0 . .
184 (17 . × − .
122 (15 . .
080 (0 . .
335 (15 . × − .
211 (9 . .
004 (0 . .
366 (9 . Table 2: Average values of X ∗ T , Q ∗ T and w ∗ T = X ∗ T + Q ∗ T S T , computed over all 10 paths we simulated, for some values of γ. Here, we fixed A = 3 and φ = 0 . . We have put the corresponding standard deviations within parentheses. igure 1: The paths of the stock price, volatility, and temporary impact parameter we used to illustrate the behavior of thestrategies in the comparative staticsFigure 2: Variation of the strategy and corresponding cash and inventory processes with respect to A. We fixed φ = 0 .
75 and γ = 0 . . Figure 3: Variation of the strategy and corresponding cash and inventory processes with respect to φ. We fixed A = 3 and γ = 0 . . Figure 4: Variation of the strategy and corresponding cash and inventory processes with respect to γ. We fixed A = 3 and φ = 0 . . igure 5: Histograms of X ∗ T , Q ∗ T and w ∗ T = X ∗ T + Q ∗ T S T for three different values of γ, ceteris paribus , resulting from the 10 simulations. We fixed A = 3 and φ = 0 . . We present the averages and corresponding standard deviations that we computedfrom our simulations in Table 2.Figure 6: Histograms of X ∗ T , Q ∗ T and w ∗ T for three different values of φ, ceteris paribus , resulting from the 10 simulations. Wefixed A = 3 and γ = 0 . . We notice that, if we increase φ and maintain all else equal, the behavior of the trader is to slowdown (cf. Figure 3). Therefore, she actually takes less impact over the trading schedule, accumulating slightly higher revenues,but has the downside of reaching terminal time holding a larger inventory position. We present the average values resultingfrom these simulations, as well as their corresponding standard deviations, in Table 3. φ E [ X ∗ T ] E [ Q ∗ T ] E [ w ∗ T ]5 × − .
683 (13 . .
052 (0 . .
751 (13 . . × − .
122 (15 . .
080 (0 . .
335 (15 . .
215 (18 . .
137 (0 . .
694 (18 . Table 3: Average values of X ∗ T , Q ∗ T and w ∗ T , computed over all 10 paths we simulated, for some values of φ. We fixed A = 3and γ = 0 . . We have put the corresponding standard deviations within parentheses.Figure 7: Histograms of the terminal optimal inventory holdings Q ∗ T for the values of A ∈ { , } . We fixed φ = 0 .
75 and γ = 0 . . The dashed lines represent, from the left to the right within each panel, the 5% , ,
75% and 95% quantiles. Eachsolid vertical line lies on the respective average value, taken over all paths we simulated. . Analysis of the constrained problem From now on, corresponding to each A satisfying ( H3 ), let us denote the solution of (3.4) by z A . We also write (cid:8) ν ∗ At (cid:9) (cid:54) t (cid:54) T ∈ U and (cid:8) Q ∗ At (cid:9) (cid:54) t (cid:54) T to represent the optimal strategy and inventory holdings,respectively, corresponding to z = z A . Finally, we set J νA ( q, y ) := z A (0 , y ) | q | φ , i.e., J νA ( q, y ) is the objectivecriteria (2.7) associated to the parameter A, t = 0 , and the strategy ν ∈ U . We observe that J νA ( q, y ) = J ν ( q, y ) ( ν ∈ U c ) . We refer to Subsection 2.3 for the definitions of the performance criteria J ν ∞ , as well as the set of admissiblecontrols for the constrained problem U c . We define the value function J ∞ := sup ν ∈U c J ν ∞ . In the subsequent result, we will use another monotonicity argument to derive asymptotic properties of thesolution of the PDE (3.4), as A → ∞ . Lemma 4.1.
Given ( t, y ) ∈ [0 , T ] × R d , the mapping A (cid:55)→ z A ( t, y ) is strictly decreasing.Proof. Let us consider
A < A (cid:48) (both constrained to ( H3 )). We set δ A := ( A − A (cid:48) )( z A − z A (cid:48) ) . We introducethe function g A,A (cid:48) ( t, y ) := κ ( y ) (cid:18) H ( z A ( t, y ) /κ ( y ) ) − H (cid:16) z A (cid:48) ( t, y ) /κ ( y ) (cid:17) z A ( t, y ) − z A (cid:48) ( t, y )( t, y ) (cid:19) if z A ( t, y ) (cid:54) = z A (cid:48) ( t, y ) ,H (cid:48) (cid:0) z A ( t, y ) /κ ( y ) (cid:1) otherwise . We observe that the function g A,A (cid:48) is continuous and bounded; hence, it is straightforward to check that δ A the unique bounded and continuous viscosity solution of the PDE (cid:40) ∂ t δ A + L δ A + g A,A (cid:48) δ A = 0 in ]0 , T [ × R d ,δ A | t = T = − ( A − A (cid:48) ) in R d . We fix ( t, y ) ∈ [0 , T ] × R d arbitrarily. We can apply Theorem 3.1 to represent δ A in the form δ A ( t, y ) = E t, y (cid:104) − ( A − A (cid:48) ) e (cid:82) Tt g A,A (cid:48) ( τ, y τ ) dτ (cid:105) < , which is clearly equivalent to z A ( t, y ) > z A (cid:48) ( t, y ) . This proves the Lemma.As a consequence of Lemma 4.1, we can prove that the limit z ∞ of the sequence of functions (cid:8) z A (cid:9) A isa viscosity solution of the singular problem. Corollary 4.2.
The function z ∞ : [0 , T [ × R d → R defined as z ∞ ( t, y ) := lim A →∞ z A ( t, y ) (cid:0) ( t, y ) ∈ [0 , T [ × R d (cid:1) (4.1) is subject to C ( T − t ) φ (cid:54) | z ∞ ( t, y ) | (cid:54) C ( T − t ) φ (cid:0) ( t, y ) ∈ [0 , T [ × R d (cid:1) , (4.2) and it is a viscosity solution of (cid:40) ∂ t z ∞ + L z ∞ + κH ( z ∞ /κ ) − γσ φ = 0 in [0 , T [ × R d ,z ∞ ( t, y ) → −∞ , as t ↑ T. (4.3) Moreover, if z ∞ is continuous, then the limit (4.1) is uniform over compact subsets of [0 , T [ × R d . roof. The function z ∞ is well-defined by Lemma 4.1, and the relations (4.2) follow from (3.12). We canshow that it is a (possibly discontinuous) viscosity solution of (4.3) using standard stability arguments, suchas in [35, Theorem 6.8]. If z ∞ is continuous, then Dini’s Theorem implies that the convergence (4.1) is infact uniform over compact subsets of [0 , T [ × R d . We write { ν ∞ t } (cid:54) t (cid:54) T to denote the control given in feedback form by ν ∞ ( t, q, y ) = − (cid:18) − z ∞ ( t, y ) κ ( y ) (cid:19) φ q, and by { Q ∞ t } (cid:54) t (cid:54) T its corresponding inventory process. Remark 4.3.
Proceeding as in Theorem 3.7, we derive Q ∞ t = q exp (cid:32) − (cid:90) t (cid:18) − z ∞ ( u, y u ) κ ( y u ) (cid:19) φ du (cid:33) , as well as ν ∞ t = − q (cid:18) − z ∞ ( t, y t ) κ ( y t ) (cid:19) φ exp (cid:32) − (cid:90) t (cid:18) − z ∞ ( u, y u ) κ ( y u ) (cid:19) φ du (cid:33) . We emphasize that, from the definition of z ∞ in Corollary 4.2, together with the fact that z A (cid:54) , weinfer that z ∞ (cid:54) . Therefore, the limiting strategy ν ∞ does not lead to price manipulation: q ν ∞ t (cid:54) and q Q ∞ t (cid:62) , for (cid:54) t (cid:54) T. Next, we proceed to prove some convergence results.
Lemma 4.4.
The following limits hold P − a.s., as A → ∞ : (cid:40) qQ ∗ At ↓ qQ ∞ t , for (cid:54) t (cid:54) T ; ν ∗ At → ν ∞ t , for (cid:54) t (cid:54) T. Proof.
By Theorem 3.7, the Monotone Convergence Theorem and Remark 4.3, we have qQ ∗ At = q exp (cid:32) − (cid:90) t (cid:18) − z A ( u, y u ) κ ( y u ) (cid:19) φ du (cid:33) ↓ q exp (cid:32) − (cid:90) t (cid:18) − z ∞ ( u, y u ) κ ( y u ) (cid:19) φ du (cid:33) = qQ ∞ t , for 0 (cid:54) t (cid:54) T. Similarly, ν ∗ At = − (cid:18) − z A ( t, y t ) κ ( y t ) (cid:19) φ Q ∗ At A →∞ −−−−→ − (cid:18) − z ∞ ( t, y t ) κ ( y t ) (cid:19) φ Q ∞ t . We finish this section by proving that { ν ∞ t } (cid:54) t (cid:54) T is the optimal trading rate for the constrained controlproblem. Theorem 4.5.
The process { ν ∞ t } (cid:54) t (cid:54) T belongs to U c , and it is the optimal control for the constrainedproblem. roof. We observe that U c (cid:54) = ∅ . In effect, an element in it is the TWAP strategy ν TWAP : ν TWAP t := qT (0 (cid:54) t (cid:54) T ) . We have J ν TWAP ∞ ( q, y ) = E , y (cid:34)(cid:90) T (cid:40) − κ ( y t ) (cid:18) | q | T (cid:19) φ − γσ φ ( y t ) | q | φ (cid:18) − tT (cid:19) φ (cid:41)(cid:35) =: m > −∞ . We emphasize that m is independent of A. Given { ν t } (cid:54) t (cid:54) T ∈ U c ⊆ U arbitrarily, it follows that J ν ∗ A A ( q, y ) (cid:62) J νA ( q, y ) = J ν ∞ ( q, y ) . (4.4)In particular, J ν ∗ A A ( q, y ) (cid:62) m, whence − lim sup A →∞ E (cid:34)(cid:90) T (cid:12)(cid:12) ν ∗ At (cid:12)(cid:12) φ dt (cid:35) = lim inf A →∞ E (cid:34) − (cid:90) T (cid:12)(cid:12) ν ∗ At (cid:12)(cid:12) φ dt (cid:35) (cid:62) κ lim inf A →∞ E (cid:34) − (cid:90) T κ ( y t ) (cid:12)(cid:12) ν ∗ At (cid:12)(cid:12) φ dt (cid:35) (cid:62) κ lim inf A →∞ J ν ∗ A A (0 , q, y ) (cid:62) mκ . Therefore, we can employ Lemma 4.4 and Fatou’s Lemma to infer that E (cid:34)(cid:90) T | ν ∞ t | φ dt (cid:35) (cid:54) lim inf A →∞ E (cid:34)(cid:90) T (cid:12)(cid:12) ν ∗ At (cid:12)(cid:12) φ dt (cid:35) (cid:54) − mκ < ∞ . This proves that { ν ∞ t } (cid:54) t (cid:54) T ∈ L φ . We now turn to the proof of the fact that { ν ∞ t } t ∈ U c . To do this, we notice that A E ,q, y (cid:104)(cid:12)(cid:12) Q AT (cid:12)(cid:12) φ (cid:105) (cid:54) − J ν ∗ A (0 , q, y ) (cid:54) − m, whence, arguing as above, we obtain E ,q, y (cid:104) | Q ∞ T | φ (cid:105) (cid:54) lim inf A →∞ E ,q, y (cid:104)(cid:12)(cid:12) Q AT (cid:12)(cid:12) φ (cid:105) = 0 . In this way, we deduce that Q ∞ T = 0 P − a.s., from where it follows that { ν ∞ t } t ∈ U c . Likewise, for each { ν t } (cid:54) t (cid:54) T ∈ U c , we once more employ Lemma 4.4 and Fatou’s Lemma, now togetherwith relation (4.4), to infer the following J ν ∞ ∞ ( q, y ) (cid:62) lim sup A →∞ E ,q, y (cid:34) − (cid:90) T (cid:110) κ ( y t ) (cid:12)(cid:12) ν ∗ At (cid:12)(cid:12) φ + γσ φ ( y t ) (cid:12)(cid:12) Q ∗ At (cid:12)(cid:12) φ (cid:111) dt (cid:35) (cid:62) lim sup A →∞ J ν ∗ A A ( q, y ) (cid:62) J ν ∞ ( q, y ) . (4.5)Therefore, (4.5) implies { ν ∞ t } t ∈ argmax ν ∈U c J ν ∞ ( q, y ) . (4.6)In fact, { ν ∞ t } t is the unique solution of (4.6), since the functional ν ∈ U c (cid:55)→ J ν ∞ ( q, y ) ∈ R is strictlyconcave. 19 . Conclusions We investigated the problem of optimal portfolio execution under a framework suited to illiquid markets.The market friction we considered took the form of a temporary price impact, determined by the trader’sturnover rate according to a power law. Furthermore, we modeled the slope corresponding to this cost asa stochastic process. Likewise, we considered the volatility of the price of the asset to be uncertain. Thedynamic assumption we made over these two processes is that their driver is a multidimensional Markovdiffusion.To obtain our optimal trading strategy in the regularized setting, in which we did not require completeexecution of the initial inventory, we proposed performance criteria under the Implementation Shortfallparadigm, leading us to derive the HJB PDE that the value function should solve. Under an adequateansatz, we simplified this PDE. We were able to apply an iterative monotonicity technique to show, undermild model assumptions, that this equation admitted a unique continuous and bounded solution. Weproved that the optimal trading rate thus obtained did not lead the agent to engage in speculative trading.Furthermore, our method yielded an iterative algorithm for solving the PDE numerically. We presented anumber of numerical experiments.In the last part of the work, we considered the constrained problem, where we required complete executionof the initial portfolio. We were able to show that the functions determining the optimal strategies in theregularized framework satisfied a monotonicity relation, allowing us to define their pointwise limit. We couldretain a certain degree of integrability, when passing the corresponding strategies to the limit, thanks to theform of the performance criteria. By Then, we used a comparison argument between the optimal rates ofthe original framework and the admissible strategies of the constrained one to establish that the limitingstrategy is indeed a solution of the latter. The fact that it is the unique one followed from a concavityargument.
Acknowledgement
This study was financed in part by Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior -Brasil (CAPES) - Finance code 001. MOS was partially supported by CNPq grant
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Appendix A. On terminal time asymptotics of the sub- and supersolutions
Under the notations of Subsection 3.3, we observe that1 b (cid:90) ξ − A du | u | r (cid:54) − (cid:90) ξ − A dua − b | u | r (cid:54) (cid:18) b − a | ξ | r (cid:19) − (cid:90) ξ − A du | u | r , (A.1)for − A (cid:54) ξ (cid:54) − ( a/b ) /r . Since (cid:90) ξ − A du | u | r = | ξ | − r − A − r r − , we obtain from the first inequality in (A.1) that y ( t ) = F − ( T − t ) satisfies | y | (cid:62) (cid:2) A − r + b ( r − T − t ) (cid:3) − r . (A.2)From the second one, we estimate T − t (cid:54) (cid:18) b − a | y | r (cid:19) − (cid:90) y − A du | u | r = | y | r b | y | r − a (cid:18) | y | − r − A − r r − (cid:19) . (A.3)21arrying out some simple manipulations on (A.3), we deduce that | y | r − (cid:54) (cid:20) a ( T − t )( r − | y | + 1 (cid:21) (cid:2) b ( r − T − t ) + A − r (cid:3) − (cid:54) (cid:20) a ( T − t )( r − a/b ) /r + 1 (cid:21) (cid:2) b ( r − T − t ) + A − r (cid:3) − , from where we conclude | y | (cid:54) (cid:20) a ( r − T ( a/b ) /r + 1 (cid:21) r − (cid:2) b ( r − T − t ) + A − r (cid:3) − r . (A.4)From the definitions of z and z, see Subsection 3.3, the relations we stated in (3.12) follow promptly from(A.2) and (A.4). Appendix B. Continuity of z ∗ For each ( t, y ) ∈ [0 , T ] × R d and t (cid:54) u (cid:54) T, we write z t, y u := z ∗ ( u, y t, y u ) , where { y t, y u } u has dynamics(2.6) and initial condition y t, y t = y . Given t ∈ [0 , T ] and y , y ∈ R d , we can carry out simple estimates toshow that the quantity δz t t := sup t (cid:54) u (cid:54) T E (cid:12)(cid:12) z t , y u − z t , y u (cid:12)(cid:12) ( t (cid:54) t (cid:54) T )satisfies δz t t (cid:54) C E (cid:34)(cid:90) Tt (cid:12)(cid:12)(cid:12) e (cid:82) ut c ( τ, y t , y τ ) dτ σ φ (cid:0) y t , y u (cid:1) − e (cid:82) ut c ( τ, y t , y τ ) dτ σ φ (cid:0) y t , y u (cid:1)(cid:12)(cid:12)(cid:12) du (cid:35) + C E (cid:34)(cid:90) Tt (cid:12)(cid:12)(cid:12) e (cid:82) ut c ( τ, y t , y τ ) dτ κ − /φ (cid:0) y t , y u (cid:1) − e (cid:82) ut c ( τ, y t , y τ ) dτ κ − /φ (cid:0) y t , y u (cid:1)(cid:12)(cid:12)(cid:12) du (cid:35) + C E (cid:34)(cid:90) Tt (cid:12)(cid:12) c ( u, y t , y u ) − c ( u, y t , y u ) (cid:12)(cid:12) du + (cid:12)(cid:12)(cid:12) e (cid:82) Tt c ( u, y t , y u ) dτ − e (cid:82) Tt c ( u, y t , y u ) du (cid:12)(cid:12)(cid:12)(cid:35) + C (cid:90) Tt δz t u du, for t (cid:54) t (cid:54) T, whence Gronwall’s Lemma implies | z ∗ ( t , y ) − z ∗ ( t , y ) | (cid:54) δz t t (cid:54) Cω ( t , y , y ) , for a suitable continuous function ω : [0 , T ] × R d × R d → R such that ω ( t , y , y ) = 0 . We emphasize that wecan prove the continuity of ω through standard arguments using basic SDE estimates and the DominatedConvergence Theorem, cf. the proof of the continuity part of [32, Theorem 3.42]. Likewise, we show that t ∈ [0 , T ] (cid:55)→ z ∗ ( t, y ) ∈ R is continuous, for each y ∈ R d (in fact, locally uniformly in the spatial variable).In this way, we conclude that | z ∗ ( t , y ) − z ∗ ( t , y ) | (cid:54) | z ∗ ( t , y ) − z ∗ ( t , y ) | + | z ∗ ( t , y ) − z ∗ ( t , y ) | → , as ( t , y ) → ( t , y ) . Appendix C. Proof of the verification result
We notice that the strategy { ν ∗ t } t is clearly admissible, as we observe from its definition (3.18) that it isin fact uniformly bounded. Also, | Q ∗ t | (cid:54) | q | . Let { ( U t, y u , Z t, y u ) } t (cid:54) u (cid:54) T be the solution of the BSDE dU t, y u = − Φ( y t, y u , z (cid:0) u, y t, y u (cid:1) ) du + Z t, y u d W u , U t, y T = − A. y , z ) = γσ φ ( y ) − κH ( z/κ ) , and d y t, y u = α ( y t, y u ) du + β ( y t, y u ) d W u , y t, y t = y . Explicitly, U t, y u = E u, y t, y u (cid:34)(cid:90) Tu Φ (cid:0) y t, y r , z (cid:0) r, y t, y r (cid:1)(cid:1) dr − A (cid:35) . By the Feynman-Kac formula, see Theorem 3.5, the bounded function w ( t, y ) := U t, y t solves, in the viscositysense, a linear PDE that z also turns out to solve; hence, by Corollary 3.2, we infer that w ≡ z. By the Markovproperty, we have U t, y u = z ( u, y t, y u ) , for t (cid:54) u (cid:54) T. Next, we apply Itˆo’s formula to (cid:110) U t, y u | Q νu | φ (cid:111) u (cid:54) t (cid:54) T , for a given ν ∈ U t , and take expectations to derive z ( t, y ) | q | φ = U t, y t | q | φ = E (cid:34) − A | Q νT | φ − (cid:90) Tt (cid:8) κ (cid:0) y t, y u (cid:1) | ν u | φ + γσ φ (cid:0) y t, y u (cid:1) | Q νu | φ (cid:9) du (cid:35) + E (cid:34)(cid:90) Tt (cid:110) κ (cid:0) y t, y u (cid:1) H (cid:0) U t, y u /κ (cid:0) y t, y u (cid:1)(cid:1) | Q νu | φ − H (cid:0) Q νu , y t, y u , U t, y u , ν u (cid:1)(cid:111) du (cid:35) , (C.1)where H ( q, y , z, ν ) := (1 + φ ) z | q | φ sgn ( q ) ν − κ ( y ) | ν | φ . On the one hand, we notice that the maximum of ν (cid:55)→ H ( q, y , z, ν ) is attained at ν = − ( − z/κ ) /φ q, and furthermore H (cid:16) q, y , z, − ( − z/κ ) /φ q (cid:17) = κ ( y ) H ( z/κ ( y )) | q | φ , whence, from (C.1), we always have z ( t, y ) | q | φ (cid:62) E (cid:34) − A | Q νT | φ − (cid:90) Tt (cid:8) κ (cid:0) y t, y u (cid:1) | ν u | φ + γσ φ (cid:0) y t, y u (cid:1) | Q νu | φ (cid:9) du (cid:35) = J ν ( t, q, y ) , from where it follows that z ( t, y ) | q | φ (cid:62) sup ν ∈U t J ν ( t, q, y ) = J ( t, q, y ) . (C.2)On the other hand, by setting ν = ν ∗ in (C.1), we infer z ( t, y ) | q | φ = J ν ∗ ( t, q, yy