Reducing Obizhaeva-Wang type trade execution problems to LQ stochastic control problems

In the literature on optimal trade execution in illiquid financial markets there arise stochastic control problems where the control is a process of finite variation (possibly with jumps) that acts as integrator both in the state dynamics and in the target functional. For brevity, we use the term finite variation stochastic control for such problems.¹¹1Notice that the class of finite variation stochastic control problems contains the class of singular stochastic control problems. In contrast, for control problems where the state is driven by a controlled stochastic differential equation (SDE) and the control acts as one of the arguments in that SDE and as one of the arguments in the integrand of the target functional, we use the term standard stochastic control problems.

In this article we present a general solution approach to finite variation stochastic control problems that arise in the literature on optimal trade execution. We set up a finite variation stochastic control problem of the type of the one in Obizhaeva and Wang [37] and its extensions like, e.g., Alfonsi and Acevedo [4], Bank and Fruth [13], Fruth et al. [24] and [25]. We then show how it can be transformed into a standard linear quadratic (LQ) stochastic control problem which can be solved with the help of state-of-the-art techniques from stochastic optimal control theory. In the introduction we first describe the finite variation stochastic control problem and showcase its usage in finance, before presenting our solution approach, summarizing our main contributions and embedding our paper into the literature.

Finite variation stochastic control problem: As a starting point we consider in this paper the following stochastic control problem. Let $T>0$ and let $(\Omega,\mathcal{F}_{T},(\mathcal{F}_{t})_{t\in[0,T]},P)$ be a filtered probability space satisfying the usual conditions. Let $\xi$ be an $\mathcal{F}_{T}$-measurable random variable and let $\zeta=(\zeta_{s})_{s\in[0,T]}$ be a progressively measurable process both satisfying suitable integrability assumptions (see (5) below). Further, let $\lambda=(\lambda_{s})_{s\in[0,T]}$ be a bounded progressively measurable process. Let $\gamma=(\gamma_{s})_{s\in[0,T]}$ be a positive Itô process driven by some Brownian motion and $R=(R_{s})_{s\in[0,T]}$ an Itô process driven by a (stochastically) correlated Brownian motion (see (3) and (4) below). Throughout the introduction we fix $t\in[0,T]$, $x,d\in\mathbb{R}$ and denote by $\mathcal{A}^{fv}_{t}(x,d)$ the set of all adapted, càdlàg, finite variation processes $X=(X_{s})_{s\in[t-,T]}$ satisfying $X_{t-}=x$, $X_{T}=\xi$, and appropriate integrability assumptions (see (A1)–(A3) below). To each $X\in\mathcal{A}^{fv}_{t}(x,d)$ we associate a process $D^{X}=(D^{X}_{s})_{s\in[t-,T]}$ satisfying

We consider the finite variation stochastic control problem of minimizing the cost functional

over $X\in\mathcal{A}^{fv}_{t}(x,d)$, where $E_{t}[\cdot]$ is a shorthand notation for $E[\cdot|\mathcal{F}_{t}]$.

Financial interpretation: Stochastic control problems with cost functional of the form (2) or a special case thereof play a central role in the scientific literature on optimal trade execution problems (see the literature discussion below). Consider an institutional investor who holds immediately prior to time $t\in[0,T]$ a position $x\in\mathbb{R}$ ($x>0$ meaning a long position of $x$ shares of a stock and $x<0$ a short position of $-x$ shares) of a certain financial asset. The investor trades the asset during the period $[t,T]$ in such a way that at each time $s\in[t-,T]$ the position is given by the value $X_{s}$ of the adapted, càdlàg, finite variation process $X=(X_{s})_{s\in[t-,T]}$ (satisfying $X_{t-}=x$). More precisely, $X_{s-}$ represents the position immediately prior to the trade at time $s$, while $X_{s}$ is the position immediately after that trade. The investor’s goal is to reach the target position

during the course of the trading period $[t,T]$. Note that we allow $\xi$ to be random to incorporate the possibility that the target position is not known at the beginning of trading but only revealed at terminal time $T$. Such situations may for example be faced by airline companies buying on forward markets the kerosene they need in $T$ months. Their precise demand for kerosene at that future time depends on several factors, such as ticket sales and flight schedules, that are not known today but only gradually learned.

We assume that the market the investor trades in is illiquid, implying that the investor’s trades impact the asset price. To model this effect, we assume (as is typically done in the literature on optimal trade execution) an additive impact on the price. This means that the realized price at which the investor trades at time $r\in[t,T]$ consists of an unaffected price $S^{0}_{r}$ plus a deviation $D^{X}_{r}$ that is caused by the investor’s trades during $[t,r]$. We assume that the unaffected price process $S^{0}=(S^{0}_{r})_{r\in[0,T]}$ is a càdlàg martingale satisfying appropriate integrability conditions. Then integration by parts and the martingale property of $S^{0}$ ensure that expected trading costs due to $S^{0}$ are given by

Thus, these costs do not depend on the investor’s trading strategy $X$ and are therefore neglected in the sequel (we refer to Remark 2.2 in [1] for a more detailed discussion in the case $\xi=0$). The deviation process $D^{X}$ associated to $X$ is given by (1). Informally speaking, we see from (1) that a trade of size $dX_{s}$ at time $s\in[t,T]$ impacts $D^{X}$ by $\gamma_{s}dX_{s}$. So, the factor $\gamma_{s}$ determines how strongly the price reacts to trades, and the process $\gamma$ is therefore called the price impact process. In particular, the fact that $\gamma$ is nonnegative entails that a buy trade $dX_{s}>0$ leads to higher prices whereas a sell trade $dX_{s}<0$ leads to smaller prices. The second component $-D^{X}_{s}dR_{s}$ in the dynamics (1) describes the behavior of $D^{X}$ when the investor is not trading. Typically, it is assumed that $R$ is an increasing process such that in the absence of trades $D^{X}$ is reverting to $0$ with relative rate $dR_{s}$. Therefore, $R$ is called the resilience process. We refer to [3] for a discussion of the effects of “negative” resilience, where $R$ might also be decreasing. We highlight that in the present paper we allow $R$ to have a diffusive part. In summary, we note that the deviation prior to a trade of the investor at time $s\in[t,T]$ is given by $D^{X}_{s-}$ whereas it is equal to $D^{X}_{s}=D^{X}_{s-}+\gamma_{s}\Delta X_{s}$ afterwards. We take the mean $D^{X}_{s-}+\frac{1}{2}\gamma_{s}\Delta X_{s}$ of these two values as the realized price per unit so that the investor’s overall trading costs due to $D^{X}$ amount to $\int_{[t,T]}\left(D^{X}_{s-}+\frac{1}{2}\gamma_{s}\Delta X_{s}\right)dX_{s}$. This describes the first integral on the right-hand side of (2). Under the assumption that $\lambda$ is nonnegative, the second integral $\int_{t}^{T}\lambda_{s}\gamma_{s}\left(X_{s}-\zeta_{s}\right)^{2}ds$ can be understood as a risk term that penalizes any deviation of the position $X$ from the moving target $\zeta$ in a quadratic way²²2The parametrization $\lambda_{s}\gamma_{s}$, $s\in[0,T]$, for the weight is chosen out of mathematical convenience since it makes some of the following assumptions and results shorter to state. Likewise, one can use $\tilde{\lambda}_{s}$, $s\in[0,T]$, as a weight and replace $\lambda$ by $\tilde{\lambda}/\gamma$ in the subsequent assumptions and results.. A possible and natural choice would be $\zeta_{s}=E_{s}[\xi]$, $s\in[0,T]$, so that the risk term ensures that any optimal strategy $X$ does not deviate too much from the (expected) target position $\xi$ in the course of the trading period.

Solution approach: The overarching goal of this paper is to show that the finite variation stochastic control problem (2) is equivalent to a standard LQ stochastic control problem (see Corollary˜2.3 and Corollary˜2.4 below). The derivation of this result is based on the following insights. The first observation is that, in general, the functional (2) does not admit a minimizer in $\mathcal{A}^{fv}_{t}(x,d)$ (see Section˜4.3 below for a specific example). In [1] the functional (2) was extended to a set of càdlàg semimartingales $X$ and it was shown that its minimum is attained in this set of semimartingales if and only if a certain process that is derived from the solution of an associated backward stochastic differential equation (BSDE) can be represented by a càdlàg semimartingale (see Theorem 2.4 in [1]). In this work we go even a step further and extend the functional (2) to the set $\mathcal{A}^{pm}_{t}(x,d)$ of progressively measurable processes $X=(X_{s})_{s\in[t-,T]}$ satisfying appropriate integrability conditions (see (A1) below) and the boundary conditions $X_{t-}=x$ and $X_{T}=\xi$. To do so, we first derive alternative representations of the first integral inside the expectation in (2) and the deviation in (1) that do not involve $X\in\mathcal{A}^{fv}_{t}(x,d)$ as an integrator (see ˜1.3). It follows that the resulting alternative representation of $J^{fv}$ (see ˜1.4) is not only well-defined on $\mathcal{A}^{fv}_{t}(x,d)$ but even on $\mathcal{A}^{pm}_{t}(x,d)$, and we denote this extended functional by $J^{pm}$ (see Section˜1.3). We next introduce a metric on $\mathcal{A}^{pm}_{t}(x,d)$ and prove that $J^{pm}$ is the unique continuous extension of $J^{fv}$ from $\mathcal{A}^{fv}_{t}(x,d)$ to $\mathcal{A}^{pm}_{t}(x,d)$ (see ˜1.7). In particular, it follows that the infimum of $J^{fv}$ over $\mathcal{A}^{fv}_{t}(x,d)$ and the infimum of $J^{pm}$ over $\mathcal{A}^{pm}_{t}(x,d)$ coincide.

Next, for a given $X\in\mathcal{A}^{pm}_{t}(x,d)$ we identify the process $\overline{H}^{X}_{s}=\gamma_{s}^{-\frac{1}{2}}D^{X}_{s}-\gamma_{s}^{\frac{1}{2}}X_{s}$, $s\in[t,T]$, as a useful tool in our analysis. Despite $X$ and $D^{X}$ having discontinuous paths in general, the process $\overline{H}^{X}$, which we call the scaled hidden deviation process, is always continuous. Moreover, we show that $\overline{H}^{X}$ can be expressed in feedback form as an Itô process with coefficients that are linear in $\gamma^{-\frac{1}{2}}D^{X}$ and $\overline{H}^{X}$ (see Lemma˜1.6). Subsequently, we reinterpret the process $\gamma^{-\frac{1}{2}}D^{X}$ as a control process $u$ and $\overline{H}^{X}$ as the associated state process. Since the cost functional $J^{pm}$ is quadratic in $\overline{H}^{X}$ and $u=\gamma^{-\frac{1}{2}}D^{X}$, we arrive at a standard LQ stochastic control problem (see (22) and (23)) whose minimal costs coincide with the infimum of $J^{pm}$ over $\mathcal{A}^{pm}_{t}(x,d)$ (see Corollary˜2.3). Importantly, there is a one-to-one correspondence between square integrable controls $u$ for this standard problem and strategies $X\in\mathcal{A}^{pm}_{t}(x,d)$, which allows to recover the minimizer $X^{*}\in\mathcal{A}^{pm}_{t}(x,d)$ of $J^{pm}$ from a minimizer $u^{*}$ of the standard problem and vice versa (see Corollary˜2.4).

We then solve the LQ stochastic control problem in (22) and (23) using techniques provided in the literature on stochastic optimal control theory. More precisely, we apply results from Kohlmann and Tang [34]³³3We moreover indicate in Remark 3.1 how we could alternatively use results from Sun et al. [40]. to provide conditions that guarantee that an optimal control $u^{*}$ exists (and is unique). This optimal control $u^{*}$ in the LQ problem is characterized by two BSDEs: one is a quadratic BSDE of Riccati type, the other one is linear, however, with unbounded coefficients (see ˜3.3). In Corollary˜3.4 we trace everything back and obtain a unique optimal execution strategy in the class of progressively measurable processes in a closed form (in terms of the solutions to the mentioned BSDEs).

Summary of our contributions: (a) The Obizhaeva-Wang type finite variation stochastic control problem (1)–(2) is continuously extended to the set $\mathcal{A}^{pm}_{t}(x,d)$ of appropriate progressively measurable processes $X$.

(b) Problem (1)–(2) is rather general. In particular, it includes the following features:

• •

  Presence of random terminal and moving targets $\xi$ and $(\zeta_{s})$;

• •

  Price impact is a positive Itô process $(\gamma_{s})$;

• •

  Resilience⁴⁴4To expand on this point, it is worth noting that in our current parametrization, only processes $(R_{s})$ with dynamics $dR_{s}=\rho_{s}\,ds$ without a diffusive component were considered by now in the literature on optimal trade execution in Obizhaeva-Wang type models. Moreover, in most papers $\rho$ is assumed to be positive, that is, only the case of an increasing $(R_{s})$ was extensively studied previously. is an Itô process $(R_{s})$ acting as an integrator in (1).

(c) Via introducing the mentioned scaled hidden deviation process $(\overline{H}^{X}_{s})$ and reinterpreting the process $(\gamma_{s}^{-\frac{1}{2}}D^{X}_{s})$ as a control in an (a priori, different) stochastic control problem, the extended to $\mathcal{A}^{pm}_{t}(x,d)$ problem is reduced to an explicitly solvable LQ stochastic control problem. Thus, a unique optimal execution strategy in $\mathcal{A}^{pm}_{t}(x,d)$ is obtained in a closed form (in terms of solutions to two BSDEs).

Literature discussion: Finite variation stochastic control problems arise in the group of literature on optimal trade execution in limit order books with finite resilience. The pioneering work⁵⁵5Posted 2005 on SSRN. Obizhaeva and Wang [37] models the price impact via a block-shaped limit order book, where the impact decays exponentially at a constant rate. This embeds into our model via the price impact process $\gamma$ that is a positive constant and the resilience process $(R_{s})$ given by $R_{s}=\rho s$ with some positive constant $\rho>0$. Alfonsi et al. [5] study constrained portfolio liquidation in the Obizhaeva-Wang model. Subsequent works within this group of literature either extend this framework in different directions or suggest alternative frameworks with similar features. There is a subgroup of models which include more general limit order book shapes, see Alfonsi et al. [6], Alfonsi and Schied [7], Predoiu et al. [38]. Models in another subgroup extend the exponential decay of the price impact to general decay kernels, see Alfonsi et al. [8], Gatheral et al. [27]. Models with multiplicative price impact are analyzed in Becherer et al. [17, 18]. We mention that in [18], the (multiplicative) deviation is of Ornstein-Uhlenbeck type and incorporates a diffusion term (but this is different from our diffusion term that results from a diffusive part in the resilience $R$). Superreplication and optimal investment in a block-shaped limit order book model with exponential resilience is discussed in Bank and Dolinsky [11, 12] and in Bank and Voß [16].

The present paper falls into the subgroup of the literature that studies time-dependent (possibly stochastic) price impact $(\gamma_{s})$ and resilience $(R_{s})$ in generalized Obizhaeva-Wang models. In this connection we mention the works Alfonsi and Acevedo [4], Bank and Fruth [13], Fruth et al. [24], where deterministically varying price impact and resilience are considered. Fruth et al. [25] allow for stochastically varying price impact (resilience is still deterministic) and study the arising optimization problem over monotone strategies. Optimal strategies in a discrete-time model with stochastically varying resilience and constant price impact are derived in Siu et al. [39]. In Ackermann et al. [1, 3, 2] both price impact and resilience are stochastic. We now describe the differences from our present paper in more detail. In [2] optimal execution is studied in discrete time via dynamic programming. In [1] the framework is the closest to the one in this paper. Essentially, our current framework is the framework from [1] extended by a risk term with some moving target $(\zeta_{s})$, a possibly non-zero (random) terminal target $\xi$, and a larger class of resilience processes (in [1], as in many previous papers, $(R_{s})$ is assumed to have the dynamics $dR_{s}=\rho_{s}\,ds$, and $(\rho_{s})$ is called resilience). In [3] the framework is similar to the one in [1], while the aim is to study qualitative effects of “negative” resilience (in the sense that $\rho_{s}\leq 0$ with $(\rho_{s})$ as in the previous sentence). Now, to compare the approach in the present paper with the one in [1], we first recall that in [1] the finite variation stochastic control problem of the type (1)–(2) is extended to allow for càdlàg semimartingale trading strategies $X$ and the resulting optimal execution problem over semimartingales is studied. The approach in [1] is based on (1)–(2) (extended with some additional terms), but this does not work beyond semimartingales, as $X$ acts as integrator there. In contrast, our continuous extension needs to employ essentially different ideas since we want to consider the set $\mathcal{A}^{pm}_{t}(x,d)$ of progressively measurable strategies (in particular, beyond semimartingales). This extension is indeed necessary to get an optimizer (see the discussion in the end of Section˜4.3).

Especially with regard to our extension result we now mention several papers where, in different models with finite resilience, trading strategies are not restricted to be of finite variation. The first instance known to us is Lorenz and Schied [35], who discuss dependence of optimal trade execution strategies on a drift in the unaffected price. In order to react to non-martingale trends they allow for càdlàg semimartingale trading strategies. Gârleanu and Pedersen [26, Section 1.3] allow for strategies of infinite variation in an infinite horizon portfolio optimization problem under market frictions. Becherer et al. [19] prove a continuous extension result for gains of a large investor in the Skorokhod $J_{1}$ and $M_{1}$ topologies in the class of predictable strategies with càdlàg paths. As discussed in the previous paragraph in more detail, in [1] the strategies are càdlàg semimartingales. In Horst and Kivman [29] càdlàg semimartingale strategies emerge in the limiting case of vanishing instantaneous impact parameter, where the initial modeling framework is inspired by Graewe and Horst [28] and Horst and Xia [31].

To complement the preceding discussion from another perspective, we mention Carmona and Webster [22], who examine high-frequency trading in limit order books in general (not necessarily related with optimal trade execution). It is very interesting that one of their conclusions is a strong empirical evidence for the infinite variation nature of trading strategies of high-frequency traders.

Finally, let us mention that, in the context of trade execution problems, risk terms with zero moving target have been included, e.g., in Ankirchner et al. [9], Ankirchner and Kruse [10], Graewe and Horst [28]. Inequality terminal constraints have been considered in Dolinsky et al. [23], and risk terms with general terminal and moving targets appear in the models of, e.g., Bank et al. [14], Bank and Voß [15], Horst and Naujokat [30], Naujokat and Westray [36]. In particular, [10], [15], and [23] consider random terminal targets $\xi$ within trade execution models where position paths are required to be absolutely continuous functions of time. This restriction of the set of position paths entails technical difficulties that make these problems challenging to analyze. In particular, existence of admissible paths that satisfy the terminal constraint is far from obvious and can in general only be assured under further conditions on $\xi$. Since in our model position paths are allowed to jump at terminal time we do not face these challenges in our framework.

The paper is structured as follows. Section˜1 is devoted to the continuous extension of our initial trade execution problem to the class of progressively measurable strategies. Section˜2 reduces the problem for the progressively measurable strategies to a standard LQ stochastic control problem. In Section˜3 we present the solution to the obtained LQ problem and trace it back up to the solution to the (extended to progressively measurable strategies) trade execution problem. In Section˜4 we illustrate our results with several examples. Finally, Section˜5 contains the proofs together with some auxiliary results necessary for them.

In this section we first set up the finite variation stochastic control problem (see Section˜1.1). In Section˜1.2 we then derive alternative representations of the cost functional and the deviation process which do not require the strategies to be of finite variation. We use these results in Section˜1.3 to extend the cost functional to progressively measurable strategies. In Section˜1.5 we show that this is the unique continuous extension. Section˜1.4 introduces the hidden deviation process as a key tool for the proofs of Section˜1.5. All proofs of this section are deferred to Section˜5.