Optimal investment with transient price impact

The classical Merton problem [23], [22] of maximizing expected utility from terminal wealth by dynamically trading a risky asset in a financial market has by now been intensively studied and well understood in models with market frictions like transaction costs. We refer to the recent survey by Muhle-Karbe et al. [25] for an overview. In contrast, less is known about utility maximization problems in illiquid market models where the friction is induced by price impact: The investor trades at bid- and ask-prices which are adversely affected by the volume or speed of her current and past trades. Within these models, the vast majority of the existing literature is primarily concerned with the problem of optimally executing exogenously given orders; cf., e.g., the surveys by Gökay et al. [14] and Gatheral and Schied [13]. However, regarding more complex optimization problems such as optimal portfolio choice, explicit characterizations of optimal strategies seem to have been elusive so far. This is notably the case for optimal investment problems on a finite time horizon in the presence of a bid-ask spread and price impact that, rather than being purely temporary or fully permanent, is transient in the sense that the impact of the investors current and past trades on execution prices does not vanish instantaneously but persists and decays over time at some finite resilience rate.

Most of the currently available work on optimal portfolio choice problems in illiquid financial markets focuses on models with purely temporary price impact, i.e., infinite resilience, zero bid-ask spread, and restricts to long-term investors as, e.g., in Guasoni and Weber [18], [16], [17] with constant relative risk aversion, in Forde et al. [10] with constant absolut risk aversion or in Gârleanu and Pedersen [11], [12] with mean-variance preferences. In the latter papers, the authors also take into account finite resilience. For investors having a finite planning horizon but still solely facing temporary price impact, asymptotic results have been obtained by Moreau et al. [24] and in a more general setup in Cayé et al. [5]; cf. also Chandra and Papanicolaou [7] for a pertubation analysis. The results from [24] are also used as a building block to describe asymptotically optimal trading strategies under highly resilient price impact in Kallsen and Muhle-Karbe [20], or in Ekren and Muhle-Karbe [9] in the setting of [11]. In all the above cited papers, trading strategies are confined to be absolutely continuous.

In the present paper, we propose a price impact model which goes beyond the block-shaped limit order book model of Obizhaeva and Wang [26] by allowing for both selling and buying stock. Specifically, our model determines bid- and ask-prices via a coupled system of controlled diffusions, giving us the possibility to specify market depth, tightness and resilience: the three dimensions of liquidity identified in the seminal work by Kyle [21]. The coupled bid- and ask-price dynamics induce convex liquidity costs on the trading strategies which are allowed to be singular and comprise non-infinitesimal block trades as in [26]. In fact, our model is closely related to the one proposed in Roch and Soner [28] which is an extension of the illiquid market model approach introduced by Çetin et al. [6] in the sense that it additionally takes into account finite resilience and a bid-ask spread. In contrast, our model captures recovery of the bid- and ask-prices by a reversion to each other rather than towards some auxiliary reference price process. Moreover, our illiquidity parameters, i.e., market depth and resilience, are constant in order to preserve tractability.

We provide existence of an optimal solution to the corresponding classical problem of maximizing expected utility from terminal liquidation wealth at some finite planning horizon. In its simplest version, our price impact model is an illiquid variant of a Bachelier model with convex liquidity costs which are levied on the agent’s trading activity. For an investor who exhibits constant absolute risk aversion, it turns out that the resulting singular optimal stochastic control problem readily reduces to a deterministic optimal tracking problem of the optimal frictionless buy-and-hold Merton portfolio in the presence of convex costs. Instead of the more common dynamic programming methods which lead to the challenge of solving a three-dimensional free boundary problem induced by a Hamilton-Jacobi-Bellman partial differential equation, we exploit a convex analytic approach. Deriving first order conditions in terms of the (infinite dimensional) subgradients of the convex cost functional allows us to construct explicitly the solution to the singular control problem by calculus of variations. As a consequence, we are able to describe analytically the free boundaries of the buying-, selling and a no-trading region in the underlying three-dimensional state space for the optimally controlled dynamics of the spread and the risky asset holdings with respect to the remaining time to maturity.

Our explicit results make transparent how the optimal strategy has to comprise several aspects. As already expected by the work in Guasoni and Weber [18], [16], [17], Forde et al. [10], and Gârleanu and Pedersen [11], [12], it is indeed optimal to trade towards the optimal frictionless portfolio while taking into account the initial bid-ask spread as well as the available time horizon. Specifically, since liquidation is costly in the present setup, the optimizer also has to take care of optimally unwinding his accrued position when approaching terminal time. It turns out that already in this elementary illiquid Bachelier model the interaction of market tightness, finite resilience, desired position targeting and optimal liquidation at a finite time horizon permits a surprisingly rich phenomenology of possible trajectories for the optimal share holdings. In this regard, our optimization problem is substantially different from the infinite horizon and zero spread frameworks considered in the papers cited above. Our findings also complement and extend the explicit results on the optimal order execution problem as studied in Obizhaeva and Wang [26] in a similar Bachelier-type setting.

The paper most closely related to ours is Soner and Vukelja [30]. Therein, the authors adopt the model from Roch and Soner [28] without bid-ask spread in a Black-Scholes framework with constant resilience and stochastic market depth proportional to the risky asset price. Using the dynamic programming principle and the notion of viscosity solutions, the problem of maximizing expected utility from terminal liquidation wealth for CRRA investors with finite planning horizon is studied. Compared to our results, their more general framework comes at the cost that a characterization of the optimal strategy is only possible numerically via a discrete-time approximation scheme.

The rest of the paper is organized as follows. In Section 2 we introduce a price impact model. Section 3 outlines the problem of maximizing expected utility from terminal liquidation wealth in our model and provides existence of an optimal solution in a general setup. In the specific case when market uncertainty is generated by an arithmetic Brownian motion with drift and the investor exhibits constant absolute risk aversion, we show that the optimal singular stochastic control problem has a deterministic solution which we construct explicitly. This is presented in Section 4. Technical proofs are deferred to Section 5.

We fix a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ satisfying the usual conditions of right continuity and completeness and consider an investor whose trades in a risky asset affect its market prices in an adverse manner. For our specification of her price impact, we propose a variant of the block-shaped limit order book model introduced by Obizhaeva and Wang [26]. Specifically, the investor’s trading strategy is described by a pair $X=(X^{\uparrow},X^{\downarrow})$ of predictable, nondecreasing, right-continuous processes where $X^{\uparrow}=(X^{\uparrow}_{t})_{t\geq 0}$, $X^{\downarrow}=(X^{\downarrow}_{t})_{t\geq 0}$ denote , respectively, the cumulative purchases and sales of the risky asset until time $t\geq 0$. We set $X^{\uparrow}_{0-}\triangleq X^{\downarrow}_{0-}\triangleq 0$. Trading takes place via market orders in an idealized block-shaped limit order book at the best bid- and ask-prices $B^{X}$ and $A^{X}$. Their dynamics are specified as the solution to the following coupled system of controlled diffusions

with given parameters $\eta>0$, $\kappa>0$, $A^{X}_{0-}\triangleq A_{0}>0$ and $B^{X}_{0-}\triangleq B_{0}>0$. The interpretation of the bid- and ask-price dynamics in (1) is the following: Both processes $A^{X}$ and $B^{X}$ are driven by some common exogenous fundamental random shock $dP_{t}$ modeled by a continuous semimartingale $(P_{t})_{t\geq 0}$ with initial value $P_{0-}\triangleq(A_{0}+B_{0})/2$. The process $(P_{t})_{t\geq 0}$ can also be regarded as the unaffected price process. Due to finite market depth $1/\eta\in(0,\infty)$ which can be interpreted as the height of a block-shaped limit order book, a buy order $dX^{\uparrow}_{t}$ incurs an impact and increases the best ask-price $A^{X}$ by the amount $\eta dX^{\uparrow}_{t}$ whereas the best bid-price $B^{X}$ is not directly affected. After completion of each buy trade, ask- and bid-prices revert to each other at some resilience rate $\kappa>0$. The effects of sell orders $dX^{\downarrow}_{t}$ on the best bid-price $B^{X}$ in (1) are analogous. Note that price impact is transient and does not vanish instantaneously but persists and decays over time at a finite exponential rate $\kappa$. We will assume for simplicity that both illiquidity parameters, i.e., the instantaneous price impact factor $\eta$ as well as the resilience rate $\kappa$, are constant. According to the bid- and ask-price dynamics in (1), the controlled evolution of the bid-ask spread $\zeta^{X}_{t}\triangleq A_{t}^{X}-B_{t}^{X}$ is described by

with initial value $\zeta^{X}_{0-}\triangleq\zeta_{0}\geq 0$ and right-continuous solution

Let us now derive the investor’s wealth dynamics corresponding to a trading strategy $X=(X^{\uparrow},X^{\downarrow})$. First, we associate to $X$ the self-financing portfolio process $(\xi_{t}^{X},\varphi_{t}^{X})_{t\geq 0}$ with some given initial values $(\xi^{X}_{0-},\varphi^{X}_{0-})\in\mathbb{R}^{2}$ where $\xi_{t}^{X}$ denotes the amount of cash and $\varphi^{X}_{t}\triangleq\varphi^{X}_{0-}+X^{\uparrow}_{t}-X^{\downarrow}_{t}$ the number of shares of the risky asset held at time $t\geq 0$. Assuming zero interest rates, the self-financing condition dictates that the cash balance $\xi^{X}$ changes only due to trading activity $X$, i.e., we postulate that

with $\Delta X^{\uparrow,\downarrow}_{t}\triangleq X^{\uparrow,\downarrow}_{t}-X^{\uparrow,\downarrow}_{t-}$, respectively. Observe that the effective execution price to, e.g., buy a not necessarily infinitesimal quantity of $dX^{\uparrow}_{t}$ shares at time $t$ is given by $A^{X}_{t-}+\eta\Delta X^{\uparrow}_{t}/2$ where $\eta\Delta X^{\uparrow}_{t}/2$ accounts for the impact a non-infinitesimal order incurs; cf., e.g., also Alfonsi et al. [1] or Predoiu et al. [27]. Analogous considerations apply for sell orders. The investor’s total wealth at any time is now expressed in terms of the liquidation value of her current portfolio. That is, we define the investor’s liquidation wealth process $(V_{t}(X))_{t\geq 0}$ associated to her portfolio process $(\xi^{X},\varphi^{X})$ with trading strategy $X=(X^{\uparrow},X^{\downarrow})$ and initial endowment $(\xi^{X}_{0-},\varphi^{X}_{0-})\in\mathbb{R}^{2}$ as

We set the initial value to $V_{0-}(X)\triangleq\xi^{X}_{0-}+\varphi^{X}_{0-}(A_{0}+B_{0})/2-(\zeta_{0}|\varphi^{X}_{0-}|+\eta(\varphi^{X}_{0-})^{2})/2$. Note that the liquidation value $V_{t}(X)$ in (4) decomposes into two parts: The first part represents the portfolio’s book value $\xi^{X}_{t}+\varphi^{X}_{t}(A^{X}_{t}+B^{X}_{t})/2$, where the value of the position $\varphi^{X}_{t}$ in the risky asset is measured in terms of the mid-quote price $(A^{X}_{t}+B^{X}_{t})/2$. The second part $\zeta^{X}_{t}|\varphi^{X}_{t}|/2+\eta(\varphi^{X}_{t})^{2}/2$ accounts for the corresponding liquidation costs which are incurred by the bid-ask spread $\zeta^{X}_{t}$ as well as the instantaneous price impact $\eta$ when unwinding in one single block trade the $\varphi^{X}_{t}$ shares. Following lemma shows that the dynamics of the liquidation wealth process $(V_{t}(X))_{t\geq 0}$ in (4) conveniently separate into the common frictionless wealth and a nonnegative, convex cost functional.

Observe that the quantity $V_{0-}(X)+L_{0-}(X)=\xi^{X}_{0-}+\varphi^{X}_{0-}P_{0-}$ in (5) represents by definition the initial wealth’s book value or initial frictionless wealth of strategy $X$ with initial endowment $(\xi^{X}_{0-},\varphi^{X}_{0-})$.

We consider an investor who aims to trade optimally in the price impact model introduced in Section 2. The investor’s preferences are described by a utility function $u:\mathbb{R}\rightarrow\mathbb{R}$ in $C^{1}(\mathbb{R})$ which is strictly concave, increasing and bounded from above. She wants to maximize expected utility from her terminal liquidation wealth $V_{T}(X)$ at some finite planning horizon $T>0$ as defined in (4) by following a trading strategy $X=(X^{\uparrow},X^{\downarrow})$ with given initial endowment $\xi^{X}_{0-}\triangleq\xi_{0}\in\mathbb{R}$ in cash and $\varphi^{X}_{0-}\triangleq\varphi_{0}\in\mathbb{R}$ shares of the risky asset. Her corresponding initial wealth and the associated liquidation costs are denoted by $V_{0}\triangleq V_{0-}(X)$ and $L_{0}\triangleq L_{0-}(X)$ for some given initial bid-ask spread $\zeta^{X}_{0-}=\zeta_{0}\geq 0$. In other words, in view of Lemma 2.1, the agent’s aim is to solve the optimization problem

over all admissible trading policies

The main tool which allows us to provide existence of an optimal strategy to the maximization problem in (8) is given by the following convex compactness result for processes of finite variation.

Another important ingredient is provided by the continuity of the liquidation wealth $V_{T}(X)$ in $X\in\mathcal{X}$ given in (5).

As a consequence, due to convexity of the liquidity cost functional $L_{T}(X)$ in $X\in\mathcal{X}$ by virtue of Lemma 2.1, we obtain the following existence and uniqueness result for the optimization problem in (8).

Let us investigate the utility maximization problem from terminal liquidation wealth as formulated in (8) in the specific case when market uncertainty $dP_{t}$ in our price impact model (1) is generated by a Brownian motion with drift $\mu>0$ and volatility $\sigma>0$. That is, we assume that the unaffected price process $(P_{t})_{t\geq 0}$ is given by

where $(W_{t})_{t\geq 0}$ denotes a standard Brownian motion on the given filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$. In addition, we assume that the inverstor’s preferences are prescribed by an exponential utility function

with constant absolute risk aversion parameter $\alpha>0$. In this setup, the optimization problem in (8) becomes

Note that for exponential utility, the optimal strategy in (11) does not depend on the investor’s initial frictionless wealth $V_{0}+L_{0}$. By virtue of Theorem 3.3, there exists a unique optimal solution to the maximization problem in (11) for any time horizon $T>0$, initial position $\varphi_{0}\in\mathbb{R}$ in the risky asset and any initial bid-ask spread $\zeta_{0}\geq 0$

When taking into account illiquidity frictions as in our setup, that is, price impact induced by finite market depth as well as market tightness imposed by the bid-ask spread, it is intuitively sensible to expect the following: Instead of directly implementing the desired frictionless Merton position in (12), the optimal frictional portfolio for problem (11) will gradually trade towards the latter. In fact, in the presence of price impact $\eta>0$, it turns out that problem (11) readily translates into a deterministic optimal tracking problem of the frictionless optimal portfolio position $\varphi^{0}$.