How should you trade your favourite alpha signal? This simple-sounding question is of pivotal importance for quantitative asset managers, as fees, bid/ask spreads and market impact chip away at their gains. In this paper, Adam Rej, Raphael Benichou, Joachim de Lataillade, Gilles Zérah and Jean-Philippe Bouchaud find the optimal trading strategy for a rich family of predictive signals in the presence of these costs, extending previous papers in which they were only considered in isolation

Determining the optimal trading strategy in the presence of a predictive signal and transaction costs is of the utmost importance for quantitative asset managers, since too much trading (both in volume and frequency) can quickly cause a strategy’s performance to deteriorate, or even make it a money-losing machine. The detailed structure of these costs is actually quite complex. Some are called ‘linear’ because they grow as $\varGamma Q$, where $Q$ is the traded volume and $\varGamma$ is the linear cost parameter. These are due to various fees (market fees, brokerage fees, etc) or the bid/ask spread, and they usually represent a small fraction of the amount traded (typically $\smash{10^{-4}}$ on liquid markets, but sometimes much more in over-the-counter/illiquid markets). More subtle are impact-induced costs, which come from the fact that a large order must be split into a sequence of small trades that are executed gradually. However, since each executed trade, on average, affects the price in the direction of the trade, the average execution price is higher (if one buys) than the decision price, which leads to what is called ‘execution shortfall’. This cost clearly increases more quickly than $Q$, since the price impact itself increases with the size of the trade. There now seems to be a wide-ranging consensus that impact-induced costs are of the order $\smash{\sigma Q^{3/2}/V^{1/2}}$, where $\sigma$ is the daily volatility and $V$ is the daily turnover (see Tóth et al (2011), Donier et al (2015) and Donier & Bonart (2014) for recent accounts).

From a theoretical point of view, however, the $\smash{Q^{3/2}}$ dependence of the costs makes analysis difficult. As a simplifying assumption, one often replaces the empirical $\smash{Q^{3/2}}$ behaviour with a ‘quadratic cost’ formula $\eta Q^{2}$, so the price impact is proportional to $Q$ (see, for example, Almgren & Chriss 2000; Obizhaeva & Wang 2013). In the absence of linear costs ($\varGamma=0$), the optimal strategy may be found as a result of a simple quadratic optimisation problem (see, for example, Almgren & Chriss 2000; Gârleanu & Pedersen 2013). The optimal policy is to rebalance at finite speed towards the target portfolio. This results in a position that is an exponential moving average of the trading signal. The pure linear cost problem (ie, $\eta=0$) was independently solved, in slightly different contexts, in Martin & Schöneborn (2011) and de Lataillade et al (2012). It requires instantaneous rebalancing towards a finite band around the ideal position as well as no action inside the band, also called the no-trade (NT) region. The case where both linear and quadratic costs are present is, of course, highly interesting, and no exact solution is known at this stage. An approximate solution was proposed in Passerini & Vázquez (2015). A method for constructing the exact solution in the small cost limit, where both $\varGamma$ and $\eta$ tend to zero, can be found in Liu, Muhle-Karbe & Weber (2014). The aim of our paper is to show that one can relax the assumption that $\varGamma$ is small and expand around the general solution for linear cost, with the expansion parameter being $\eta\to 0$. The solution defines four different regions (see figure 1), as follows.

• An NT region: this is still present inside a band around the ideal position, but the band shrinks by an amount $\sim\eta^{1/3}$.

• A small ‘boundary layer’: this has a width of $\eta^{1/3}$ and surrounds the band. The trading speed is of order $\eta^{-1/3}$ and takes a scaling form.

• Rebalancing region (i): this is further away from the band, but still within its zone of influence. The trading speed is of order $\eta^{-1/2}$ and behaves as a square root of the distance to the band.

• Rebalancing region (ii): this is even further away from the band. The trading speed is a linear function of the distance to the ideal position, and one recovers the exact $\varGamma=0$ solution as expected.

Our method readily generalises to other non-linear cost structures, particularly to the $Q^{3/2}$ law alluded to above. We briefly discuss how our results extend to this case in the final section of this paper.

## 1 Setup of the problem and the $\varGamma=0$ solution

Following Martin & Schöneborn (2011), we assume the value $\smash{X_{t}}$ of the traded instrument has a dynamics governed by the following drift-diffusion equation:11The diffusion constant $\smash{\sigma^{2}}$ can also depend on $\smash{X_{t}}$, as in Martin & Schöneborn (2011), without materially affecting the following results. For simplicity, we keep $\sigma$ constant.

 $\mathrm{d}X_{t}=\mu(X_{t})\,\mathrm{d}t+\sigma\,\mathrm{d}W_{t}$ (1)

where $\mu(X)$ is the ‘signal’, eg, $\mu(X)=-\kappa X$ for classical mean-reverting statistical arbitrage (StatArb) strategies. We will call $\smash{\mathcal{L}_{x}}$ the associated Itô differential operator:

 $\mathcal{L}_{x}[f]=\mu(x)\frac{\partial f}{\partial x}+\frac{\sigma^{2}}{2}% \frac{\partial^{2}f}{\partial x^{2}}$ (2)

The position (number of shares/lots, etc) of the manager at time $t$ is denoted by $\theta_{t}$. For a given rebalancing policy, the expected risk-adjusted profit and loss (P&L) per unit time, conditional on $X_{t}=x$ and $\theta_{t}=\theta$, is given by:

 $W(\theta,x\mid t)=\lim_{T\to\infty}\frac{1}{T}\int_{t}^{\infty}\mathrm{d}t^{% \prime}\,\mathrm{e}^{-(t^{\prime}-t)/T}\times[\mu(X_{t^{\prime}})\theta_{t^{% \prime}}-\lambda\theta^{2}_{t^{\prime}}-\varGamma|\dot{\theta}_{t^{\prime}}|-% \eta\dot{\theta}^{2}_{t^{\prime}}]$ (3)

where $\lambda$ is the cost of risk (which includes a factor $\sigma^{2}$). The first term is the average gain of the position; the last two terms are rebalancing costs. We now introduce the value function $V(\theta,x)$, defined as $W(\theta,x\mid t)$ for the optimal future rebalancing policy. Note that, because we assume a stationary process for $X_{t}$, the value function is independent of $t$.22Technically, it can be useful to keep $T$ large but finite, which amounts to regularising the differential operator $\mathcal{L}_{x}[f]$ with a term $-f/T$ and taking the limit $T\to\infty$. As is well known, $V(\theta,x)$ then obeys a Hamilton-Jacobi-Bellman (HJB) equation, which in the present case reads as follows:

 $0=\mu(x)\theta-\lambda\theta^{2}+\max_{\dot{\theta}}\bigg\{{-}\varGamma|\dot{% \theta}|-\eta\dot{\theta}^{2}+\frac{\partial V}{\partial\theta}\dot{\theta}+% \mathcal{L}_{x}[V]\bigg\}$ (4)

The maximisation with respect to $\dot{\theta}$ is very simple and leads to:

 $v=\dot{\theta}^{*}=\frac{1}{2\eta}\bigg[\frac{\partial V}{\partial\theta}-% \varGamma\,\text{sign}(\dot{\theta}^{*})\bigg]\quad\text{or}\quad v=0$ (5)

where the NT region ($v=0$) is defined by $|\partial V/\partial\theta|\leq\varGamma$. In this region, the HJB equation simplifies to:

 $\mathcal{L}_{x}[V_{\mathrm{NT}}]=-\mu(x)\theta+\lambda\theta^{2}$ (6)

In the rebalancing (RB) region, on the other hand, the HJB equation becomes a non-linear partial differential equation (PDE):

 $\bigg(\frac{\partial V^{\pm}}{\partial\theta}\pm\varGamma\bigg)^{\!2}=4\eta^{2% }v^{2}=4\eta[\lambda\theta^{2}-\mu(x)\theta-\mathcal{L}_{x}[V^{\pm}]]$ (7)

where the $\pm$ sign corresponds to either large enough positive $\theta$, such that the optimal policy is to sell ($v<0$), or large enough negative $\theta$, such that the optimal policy is to buy ($v>0$).

Figure 1: Trading speed as a function of the distance to the boundary of the NT region. The outer solution behaves as a square root close to the band and linearly further out. The square root singularity is regularised in a boundary layer of width η 1/3 , so the velocity vanishes linearly at the (shifted) boundary of the band Θ η

## 2 The $\eta=0$ solution

For $\eta=0$, the solution of the corresponding HJB equation has been worked out by Martin & Schöneborn (2011), and it will be denoted by $V_{0}(\theta,x)$. The NT region is parameterised by two functions, $\varTheta^{-}_{0}(x),\varTheta^{+}_{0}(x)$, such that, for a given $x$, the speed of trading $v$ vanishes inside the interval $[-\varTheta^{-}_{0}(x),\varTheta^{+}_{0}(x)]$, hereafter referred to as a band. Outside the band, the $\eta=0$ solution to (7) is given by:

 $V_{0}^{\pm}(\theta,x)=V_{\mathrm{NT},0}(\pm\varTheta^{\pm}_{0}(x),x)-\varGamma% |\theta\mp\varTheta^{\pm}_{0}(x)|$ (8)

Inside the band, the general solution to the linear equation (6) can be constructed using a Green’s function $\mathcal{G}_{x}$ of the operator $\mathcal{L}_{x}$ and the two independent solutions $\psi_{1,2}(x)$ of the homogeneous equation $\mathcal{L}_{x}f=0$ (see Martin & Schöneborn (2011) for details). Schematically:

 $V_{\mathrm{NT},0}(\theta,x)=\mathcal{G}_{x}[-\mu(x)\theta+\lambda\theta^{2}]+% \alpha_{1}(\theta)\psi_{1}(x)+\alpha_{2}(\theta)\psi_{2}(x)$ (9)

where $\alpha_{1,2}(\theta)$ are two yet-to-be-determined functions. Martin & Schöneborn (2011) propose fixing these functions in two steps. First, one imposes that, at the (still unknown) boundaries of the NT zone, the derivatives of $V_{0}(\theta,x)$ are continuous, ie:

 $\frac{\partial V_{\mathrm{NT},0}(\theta,x)}{\partial\theta}\bigg|_{\theta=\pm% \varTheta^{\pm}_{0}(x)}=\mp\varGamma$ (10)

This allows one to solve $\alpha_{1,2}(\theta)$ as functionals of the boundary positions $\varTheta^{\pm}_{0}(x)$. Second, one determines these boundaries by invoking the variational argument, ie, that these boundaries should maximise the value function $V_{0,\text{NT}}(\theta,x)$ everywhere in the NT region. This second condition allows us to fully determine $\varTheta^{\pm}_{0}(x)$. While we fully agree with the final expressions obtained by Martin & Schöneborn (2011; hereafter M&S), we argue their second condition does not generalise to the case $\eta\neq 0$. The general condition should, rather, be that the second derivative of the value function with respect to $\theta$ is continuous everywhere, including the boundaries between NT and RB regions. This is a direct consequence of the continuity of the trading speed across the boundary and the fact that the HJB equation contains second derivatives of the value function. In fact, we show in appendix B of Rej et al (2015) that the M&S solution obeys:

 $\frac{\partial^{2}V_{\mathrm{NT},0}(\theta,x)}{\partial\theta^{2}}\bigg|_{% \theta=\pm\varTheta^{\pm}_{0}(x)}\equiv 0$ (11)

a property that apparently went unnoticed in Martin & Schöneborn (2011), which is actually much simpler than the variational condition. In the next section, we will attempt to construct a consistent solution to the HJB for arbitrary $\varGamma$ but small quadratic costs $\eta\to 0$. We will make use of the continuity of the first and second derivatives of the value function to determine the new location of the boundaries.

## 3 The small $\eta$-matched asymptotic expansions

### 3.1 The outer region

Let us assume that, far enough from the new band positions $[-\varTheta^{-}_{\eta}(x),\varTheta^{+}_{\eta}(x)]$ (the ‘outer region’), the trading solution for small $\eta$ reads:

 $V_{\eta}^{\pm}(\theta,x)=V_{0}^{\pm}(\theta,x)+\sqrt{\eta}V_{1}^{\pm}(\theta,x% )+\cdots\quad(\eta\to 0)$ (12)

Below, we will confine ourselves to the ‘$+$’ sector, where the trading speed is negative; we will also drop the superscripts on the position of the band $\varTheta_{0}$ as well as on $V_{0},V_{1}$. Plugging our ansatz into (7) and retaining leading terms in $\eta$ gives us:

 $\frac{\partial V_{1}}{\partial\theta}=-2\sqrt{\lambda\theta^{2}-\mu(x)\theta-% \mathcal{L}_{x}[V_{0}]}$ (13)

where we have used $\partial V_{0}/\partial\theta=-\varGamma$ so the zero-order term on the left-hand side vanishes. Using the solution outside the band (8), we write:

 $\mathcal{L}_{x}[V_{0}]=\mathcal{L}_{x}[V_{\mathrm{NT},0}(\varTheta_{0}(x),x)]+% \varGamma\mathcal{L}_{x}[\varTheta_{0}(x)]$ (14)

Note:

 $\mathcal{L}_{x}[V_{\mathrm{NT},0}(\theta,x)]=-\mu(x)\theta+\lambda\theta^{2}$ (15)

from which we deduce (the dependence of $\mu$ and $\varTheta_{0}$ on $x$ is henceforth suppressed):

 $\displaystyle\mathcal{L}_{x}[V_{\mathrm{NT},0}(\varTheta_{0},x)]$ $\displaystyle=-\mu\varTheta_{0}+\lambda\varTheta_{0}^{2}+\mu\varTheta_{0}^{% \prime}(V_{\mathrm{NT},0})_{\theta}$ $\displaystyle\qquad+\frac{\sigma^{2}}{2}[\varTheta_{0}^{\prime 2}(V_{\mathrm{% NT},0})_{\theta\theta}+\varTheta_{0}^{\prime\prime}(V_{\mathrm{NT},0})_{\theta% }+2\varTheta_{0}^{\prime}(V_{\mathrm{NT},0})_{x\theta}]$ (16)

where the prime stands for the derivative of $\varTheta_{0}$ with respect to $x$, and the subscripts indicate variables ($x$ or $\theta$) with regard to which derivatives are taken. The boundary conditions at $\varTheta_{0}$ are:

 $(V_{\mathrm{NT},0})_{\theta}(\varTheta_{0}(x),x)\equiv-\varGamma,\quad(V_{% \mathrm{NT},0})_{\theta\theta}(\varTheta_{0}(x),x)\equiv 0\quad\text{for~{}all% ~{}}x$ (17)

implying:

 $(V_{\mathrm{NT},0})_{x\theta}=-\varTheta_{0}^{\prime}(V_{\mathrm{NT},0})_{% \theta\theta}\equiv 0$ (18)

so $\mathcal{L}_{x}[V_{0}]$ finally simplifies to:

 $\mathcal{L}_{x}[V_{0}]\equiv-\mu\varTheta_{0}+\lambda\varTheta_{0}^{2}$ (19)

Therefore, the equation for $V_{1}$ becomes:

 $\frac{\partial V_{1}}{\partial\theta}=-2\sqrt{\lambda(\theta^{2}-\varTheta_{0}% ^{2})-\mu(\theta-\varTheta_{0})}$ (20)

Now, the velocity in the trading zone is simply:

 $v=\frac{1}{2\sqrt{\eta}}\frac{\partial V_{1}}{\partial\theta}$ (21)

We thus find:

• $v$ diverges as $\eta^{-1/2}$ when $\eta\to 0$, recovering the instantaneous rebalancing in this limit;

• $v$ behaves linearly for large $|\theta|$;

• $v$ behaves as a square root close to the unperturbed band $\varTheta_{0}$ (see discussion below):

 $v\approx-\sqrt{\frac{\lambda(\varTheta_{0}+\theta)-\mu}{\eta}}\sqrt{\theta-% \varTheta_{0}}$ (22)

This square root singularity is interesting, because it means there must be a region very close to the band where this naive perturbative solution breaks down. Indeed, the second derivative of $\sqrt{\eta}V_{1}$ with respect to $\theta$ diverges and, thus, may not be neglected. One has to analyse this region by zooming in on the immediate proximity of the band, conventionally dubbed the ‘inner region’ or the boundary layer (see Hinch 1991).

### 3.2 The inner region

To make a start on the analysis, we take the derivative of (7) with respect to $\theta$ and introduce:

 $f:=\frac{\partial V^{+}}{\partial\theta}+\varGamma\equiv 2\eta v$

This leads to the exact equation:

 $ff_{\theta}=-2\eta(\mu-2\lambda\theta+\mathcal{L}_{x}[f])$ (23)

We postulate that, close to the (new) band $\varTheta_{\eta}(x)$, the function $f$ exhibits the following scaling:

 $f=-\eta^{\alpha}F(y)\quad\text{where~{}}y:=\frac{\theta-\varTheta_{\eta}(x)}{% \eta^{\beta}}>0,~{}\alpha,\beta>0$ (24)

The parameters $\alpha$ and $\beta$ are two exponents that need to be determined, while $F(y)$ is a positive function (note that $f\propto v<0$ in the ‘$+$’ sector we are considering here). A first condition comes from the fact that when $y\to\infty$, this scaling form must reproduce the above square root solution, (22). Writing $F(y)\approx A\sqrt{y}$ for large $y$, one finds:

 $v=-\frac{1}{2\eta}\,\eta^{\alpha}F(y)\approx_{y\gg 1}-\frac{\eta^{\alpha-\beta% /2}}{2\eta}A\sqrt{\theta-\varTheta_{\eta}}$ (25)

Up to leading order in $\eta$, $\varTheta_{\eta}=\varTheta_{0}$, so the identification with (22) directly yields:

 $\alpha-\beta/2=1/2,\qquad A=2\sqrt{2\lambda\varTheta_{0}-\mu}$ (26)

Injecting the scaling form into (23), and noting that derivatives with respect to $x$ supply factors of $\eta^{-\beta}$, we find the leading terms are:

 $\eta^{2\alpha-\beta}FF_{y}=2\eta\bigg({-}\mu+2\lambda\varTheta_{0}+\frac{% \sigma^{2}}{2}\eta^{\alpha-2\beta}\varTheta_{0}^{\prime 2}F_{yy}\bigg)$ (27)

Matching the $\eta$ powers of the two sides of the equation leads to:

 $2\alpha-\beta=1\quad\text{or}\quad 2\alpha-\beta=1+\alpha-2\beta$ (28)

The first equality can only hold if $\alpha\geq 2\beta$, in which case the last term on the right-hand side is negligible. This would, however, lead to:

 $(F^{2})_{y}=4(2\lambda\varTheta_{0}-\mu)\equiv A^{2}$ (29)

which still has a square root singularity (where $F$ goes to zero); so, close enough to the singularity, the last term on the right-hand side diverges and cannot be neglected. This contradicts our original assumption.

The only other possibility is:

 $2\alpha-\beta=1+\alpha-2\beta$

which, together with:

 $\alpha-\beta/2=1/2$

leads to $\alpha=2/3$ and $\beta=1/3$. The leading ordinary differential equation (ODE) for $F$ takes the following form:

 $(F^{2})_{y}-BF_{yy}=A^{2},\qquad B\equiv 2\sigma^{2}\varTheta_{0}^{\prime 2}$ (30)

Upon integration:

 $F^{2}-BF_{y}=A^{2}(y-y_{0})$ (31)

The solution to the above may be expressed in terms of Airy functions. Writing $F=-B\varPsi_{y}/\varPsi$, the above equation reads:

 $\varPsi_{yy}=\frac{A^{2}}{B^{2}}(y-y_{0})\varPsi$ (32)

We want a solution to this equation such that $\varPsi_{y}(y=0)=0$ (ie, $F(y=0)=0$), such that $F(y>0)>0$. The general solution is:

 $\varPsi(y)=c_{1}\operatorname{Ai}(z(y-y_{0}))+c_{2}\operatorname{Bi}(z(y-y_{0}% )),\qquad z=\bigg(\frac{A}{B}\bigg)^{\!2/3}$ (33)

but the asymptotic behaviour of the Airy functions imposes $c_{2}=0$. The condition $\text{Ai}^{\prime}(-zy_{0})=0$ selects the first maximum of Ai that occurs for $-zy_{0}\approx-1.018\dots$, thereby fixing $y_{0}$. Finally, the solution is:

 $F(y)=-Bz\frac{\text{Ai}^{\prime}(z(y-y_{0}))}{\text{Ai}(z(y-y_{0}))}$ (34)

For large $y$, one uses the asymptotic behaviour $\ln\operatorname{Ai}(u)\approx-2u^{3/2}/3$ to obtain:

 $F(y)\approx Bz^{3/2}\sqrt{y}\equiv A\sqrt{y}$ (35)

as desired. We have found a solution that goes smoothly to zero when $y\to 0$, ie, in a region of width $\smash{\eta^{1/3}}$ immediately outside the boundary of the band $-\varTheta_{\eta}$ (see figure 1). Note that $F_{y}(0)=1.018\dots A^{4/3}B^{-1/3}$.

In the small $\varGamma$ limit, the NT region has a width of order $\smash{\varGamma^{1/3}}$ around the ideal (Markowitz) position, therefore leading to $\smash{2\lambda\varTheta_{0}}-\smash{\mu\sim\varGamma^{1/3}}$, or $A\sim\smash{\varGamma^{1/6}}$ and $z\sim\smash{\varGamma^{1/9}}$. Plugging this into (34) leads to a width of the inner region scaling as $\smash{(\eta^{3}/\varGamma)^{1/9}}$. This allows us to make the connection between our results and those of Liu, Muhle-Karbe & Weber (2014), who consider the double limit $\varGamma,\eta\to 0$ with $\smash{\eta/\varGamma^{4/3}}$ fixed. In that regime, $\smash{(\eta^{3}/\varGamma)^{1/9}}\sim\smash{\varGamma^{1/3}}$, which recovers the predictions of Liu, Muhle-Karbe & Weber (2014). Our results are, we believe, quite interesting, as they are valid for arbitrary values of $\varGamma$, with a universal shape of the scaling function $F(y)$ in the inner region, independent of the precise problem at hand.

### 3.3 Boundaries shift inwards

Next, we need to find the shifted band position $\varTheta_{\eta}$. We will use the fact that the second derivative of the value function should be continuous at the boundary. Integrating $f$ leads to the following equality for the value function in the trading zone:

 $V^{+}(\theta,x)=\int_{\varTheta_{\eta}}^{\theta}\mathrm{d}\theta^{\prime}\,f(% \theta^{\prime},x)-\varGamma(\theta-\varTheta_{\eta})+V_{\mathrm{NT},\eta}(% \varTheta_{\eta},x)$ (36)

that by construction coincides with $V_{\mathrm{NT},\eta}$ at the boundaries of the band. Observe $\smash{V_{\mathrm{NT},\eta}}$ differs from the original M&S solution by the change of boundary conditions. Because of the optimality of the $\eta=0$ boundaries, one immediately infers $|\varTheta_{\eta}-\varTheta_{0}|=\operatorname{O}(\eta^{a})$ translates to $|V_{\mathrm{NT},\eta}(\theta)-V_{\mathrm{NT},0}(\theta)|=\operatorname{O}(\eta% ^{2a})$.

We will now show that $a=1/3$, ie, the shift of the band is of the same order as the width of the boundary layer. The continuity of the second derivative at $\theta=\varTheta_{\eta}$ imposes, up to leading order in $\eta$:

 $-\eta^{1/3}F_{y}(0)=(V_{\mathrm{NT},0})_{\theta\theta}(\varTheta_{\eta})% \approx(V_{\mathrm{NT},0})_{\theta\theta\theta}(\varTheta_{0})(\varTheta_{\eta% }-\varTheta_{0})$ (37)

where we use the condition $(V_{\mathrm{NT},0})_{\theta\theta}(\varTheta_{0})=0$, derived in appendix B of Rej et al (2015). This immediately yields:

 $\varTheta_{\eta}=\varTheta_{0}-\frac{F_{y}(0)}{(V_{\mathrm{NT},0})_{\theta% \theta\theta}(\varTheta_{0})}\eta^{1/3}$ (38)

ie, the inward shift of the band is of order $\eta^{1/3}$, exactly like the width of the boundary layer. This is reasonable, as it means both effects of the quadratic term on the immediate proximity of the band are of the same order. As we show in appendix B of Rej et al (2015), the third derivative of $\smash{V_{\mathrm{NT},0}}$ turns out to be positive at the unperturbed band; hence, the above expression implies the boundary shifts inwards, ie, the NT region is reduced by the presence of quadratic costs.

The alert reader might wonder how the solution given by (36) still has a continuous first derivative at $\varTheta_{\eta}$. This is discussed in full detail in Rej et al (2015), which establishes that our boundary layer solution (36) is indeed $\smash{C^{2}}$ across the NT-RB boundary.

## 4 Discussion and extensions

All the results above are compatible with the numerical results of Liu, Muhle-Karbe & Weber (2014), who exhibit a square root-like trading speed close to the band and a band that shrinks when $\eta$ increases (see figure 1 therein; the parameters $\lambda$, $\varepsilon$ are identical to our $\eta$, $\varGamma$, respectively). We have ourselves solved the HJB equation numerically and found a band position that is compatible with our prediction above: $\smash{\varTheta_{0}-\varTheta_{\eta}\propto\eta^{1/3}}$ (see Liu, Muhle-Karbe & Weber 2014, figure 1).

In the above, we did not elaborate on the domain of validity of the small $\eta$ expansion. For perturbation theory to make sense, the shift in the position of the band must remain small compared with the width of the band itself, ie:

 $\bigg|\frac{\varTheta_{0}^{+}-\varTheta_{\eta}^{+}}{\varTheta_{0}^{+}+% \varTheta_{0}^{-}}\bigg|\ll 1$ (39)

Assuming an Ornstein-Uhlenbeck process for the prediction ($\kappa^{-1}$ is the mean-reversion time), $\mu(x)=-\kappa x$, we obtain in the small $\varGamma$ limit:

 $(V_{\mathrm{NT},0})_{\theta\theta\theta}(\varTheta_{0})\approx\frac{8\lambda^{% 2}}{\sigma^{2}\kappa}\bigg(\frac{3\varGamma\sigma^{2}}{2\kappa}\bigg)^{\!1/3}$ (40)

This, together with the results of Martin & Schöneborn (2011):

 $\varTheta_{0}^{+}+\varTheta_{0}^{-}\sim\frac{\sigma^{2}}{\lambda}\bigg(\frac{% \varGamma\kappa^{2}}{\sigma^{4}}\bigg)^{\!1/3},\qquad\varTheta_{0}^{\prime}% \sim\frac{\kappa}{\lambda}$ (41)

where we have dropped all $\operatorname{O}(1)$ numerical constants, allows us to recast the above as:33In the following discussion, we assume $\lambda\varTheta_{0}\ll\mu$, ie, the price of risk for a position at the edge of the band is small compared with the expected gains.

 $\frac{\eta}{\varGamma^{4/3}}\ll\frac{\lambda}{(\sigma\kappa)^{4/3}}$ (42)

so the relevant combination is indeed ${\eta}{\varGamma^{-4/3}}$, as anticipated in Liu, Muhle-Karbe & Weber (2014) in the $\varGamma,\eta\to 0$ limit. In order to make sense of the above inequality, it is useful to substitute the price of risk $\lambda$ with a risk target $\mathcal{R}$, corresponding to the volatility of the ideal position. In the absence of costs, this ideal position reads as:

 $\theta^{*}=\frac{\mu}{2\lambda}$ (43)

leading to a typical risk $\smash{\mathcal{R}\sim\sqrt{\kappa}\sigma^{2}/2\lambda}$. The above inequality can thus be rewritten as:

 $\frac{\eta}{\varGamma^{4/3}}\ll\frac{\sigma^{2/3}}{\mathcal{R}\kappa^{5/6}}$ (44)

Suppose the target risk $\mathcal{R}$ is a fraction $\varphi$ of the daily volume $V$, ie, $\mathcal{R}=\varphi V\sigma$. The quadratic cost parameter may be expressed using dimensional quantities $\eta=C\sigma T^{3/2}/V$, where $C$ is a number and $T=1$ day. The final dimensionless condition is, interestingly enough, independent of the volume $V$:

 $C\varphi\ll\bigg(\frac{\varGamma}{\sigma\sqrt{T}}\bigg)^{\!4/3}(\kappa T)^{-5/6}$ (45)

Taking, for example, $\varGamma=2$ basis points, $\sigma\sqrt{T}=2\%$ and $\kappa T=0.1$, we find $C\varphi\ll 10^{-2}$. In conclusion, our small $\eta$ expansion makes sense, for a given quadratic cost coefficient $C$, if the portfolio’s typical positions represent a small fraction of the daily traded volume.

Finally, we note that the minimal P&L induced by linear and quadratic costs can be written as $\varGamma^{2/3}G(\eta\varGamma^{-4/3})$, where $G(u)$ is a function that tends to a constant for $u\to 0$ and behaves as $\sqrt{u}$ for large $u$. This enables one to estimate the impact of both parameters on the final costs. It is also interesting to note the use of the simplified ‘outer’ solution for the trading speed (22) leads to an increased cost of order $\eta$ compared with the optimal solution.

Let us conclude by pointing out that the above matched asymptotic expansion around the M&S solution can be undertaken for arbitrary non-linear costs. If, instead of the quadratic cost term $\eta\dot{\theta}^{2}$ considered above, one worked with the more realistic $\zeta\dot{\theta}^{3/2}$ term, one would find the following HJB equation in the trading region:

 $\bigg(\frac{\partial V^{\pm}}{\partial\theta}\pm\varGamma\bigg)^{\!3}\propto% \zeta^{2}[\lambda\theta^{2}-\mu(x)\theta-\mathcal{L}_{x}[V^{\pm}]]$ (46)

requiring a perturbative expansion of the form:

 $V_{\zeta}^{\pm}(\theta,x)=V_{0}^{\pm}(\theta,x)+\zeta^{2/3}V_{1}^{\pm}(\theta,% x)+\cdots$ (47)

This leads to trading speed behaving as $\smash{(\theta-\varTheta_{\zeta})^{1/3}}$ in the outer region, close enough to the band, but crossing over to the following boundary layer solution:

 $f(\theta)=\zeta^{4/5}\hat{F}\bigg(\frac{\theta-\varTheta_{\zeta}(x)}{\zeta^{2/% 5}}\bigg)$ (48)

where $\hat{F}(y)$ now obeys the following Abel equation (see, for example, Panayotounakos & Zarmpoutis 2011):

 $\hat{F}^{3}-B^{\prime}\hat{F}_{y}=A^{\prime 2}(y-y_{0})$ (49)

One should impose $\smash{\hat{F}(y=0)=0}$ and $\smash{\hat{F}(y)\sim y^{1/3}}$ for large $y$. Since our asymptotic expansion in the quadratic case relies chiefly on the properties of the M&S solution in the NT region, all the results obtained readily transpose to the present case as well. The general shape of the optimal trading solution is, mutatis mutandis, similar to that drawn in figure 1.

Adam Rej is a research manager, Raphael Benichou is a directional portfolio manager, Joachim de Lataillade is head of execution, Gilles Zérah is head of research management and Jean-Philippe Bouchaud is chairman and head of research at Capital Fund Management in Paris. The authors thank J Donier, CA Lehalle, J Muhle-Karbe and M Potters for enlightening discussions on the subject matter of this paper.

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