Deep learning algorithms for hedging with frictions

Abstract

This work studies the deep learning-based numerical algorithms for optimal hedging problems in markets with general convex transaction costs. Our main focus is on how these algorithms scale with the length of the trading time horizon. Based on the comparison results of the FBSDE solver by Han, Jentzen, and E (2018) and the Deep Hedging algorithm by Buehler, Gonon, Teichmann, and Wood (2019), we propose a Stable-Transfer Hedging (ST-Hedging) algorithm, to aggregate the convenience of the leading-order approximation formulas and the accuracy of the deep learning-based algorithms. Our ST-Hedging algorithm achieves the same state-of-the-art performance in short and moderately long time horizon as FBSDE solver and Deep Hedging, and generalize well to long time horizon when previous algorithms become suboptimal. With the transfer learning technique, ST-Hedging drastically reduce the training time, and shows great scalability to high-dimensional settings. This opens up new possibilities in model-based deep learning algorithms in economics, finance, and operational research, which takes advantage of the domain expert knowledge and the accuracy of the learning-based methods.

Appendices

Appendix A: General Asymptotics Results

As already emphasized above, a general existence proof for the FBSDE system (2.13)–(2.14) remains a challenging open problem. To obtain tractable results with the general transaction costs under Assumption 2.1 and obtain explicit approximation trading strategies as in Cayé et al. (2018); Guasoni and Weber (2020), we focus on the financial market with the following assumptions:

Assumption A.1

• (i) The processes $\Lambda$ and $\sigma ^2$ are bounded away from zero;

• (ii) The processes ${\bar{a}}$, ${\bar{b}}$, $\Lambda$, and $\sigma$ are essentially bounded Itô processes.

In this section, we establish the asymptotic optimal strategy for the frictional mean-variance preference (2.3). For better readability, we introduce the construction of optimal strategy under Assumption A.1 in Appedix A.1. The proof for the main approximation result is in Appendix A.

1.1 Asymptotically optimal trading strategies

We show that, under Assumption (), the smallness assumption on the transaction costs level $\lambda$ should also be a relative quantity with respect to the trading time horizon T. The following two results are the main ingredients for the asymptotic trading strategy.

A nonlinear ODE The first ingredient to cook up the leading-order approximation is the solution to a nonlinear ODE, which is also essential to the analysis of Gonon et al. (2021), Lemma 3.4 and the formal analysis of Shi (2020), Lemma 2.5.

Lemma A.2

Suppose the instantaneous transaction cost G satisfies Assumption 2.1. Then the ordinary differential equation

$$\begin{aligned} (G')^{-1}\left( \frac{g(x;\gamma ,\sigma ,{\bar{a}},\lambda )}{\lambda }\right) g'(x;\gamma ,\sigma ,{\bar{a}},\lambda ) + \frac{{\bar{a}}^2}{2} g''(x;\gamma ,\sigma ,{\bar{a}},\lambda ) = \gamma \sigma ^2 x. \end{aligned}$$

(3.5)

has a unique solution g on $\mathbb {R}$ such that $xg(x)\le 0$ for all $x\in \mathbb {R}$. Moreover, g is odd, non-increasing on $\mathbb {R}$ and g satisfies the growth conditions

$$\begin{aligned} \lim _{x\rightarrow -\infty } \frac{g(x;\gamma ,\sigma ,{\bar{a}},\lambda )}{\lambda (G^*)^{-1}(\frac{\gamma \sigma ^2}{2\lambda }x^2)} = 1, \qquad \lim _{x\rightarrow +\infty } \frac{g(x;\gamma ,\sigma ,{\bar{a}},\lambda )}{\lambda (G^*)^{-1}(\frac{\gamma \sigma ^2}{2\lambda }x^2)} = -1, \end{aligned}$$

(A.1)

where $G^*$ is the Legendre transform of G.

Remark A.3

If the time horizon T is large, then the solution to the optimal trading strategy should become stationary. Such a stationary solution should in turn solve the essential nonlinear ODE (3.5). Far from the terminal time T, it is natural to expect that the correct solution is still identified by the same growth condition in the space variable as (A.1).

For power functions $G(x) = |x|^q /q$, $q\in (1,2]$, the Legendre transform is

$$\begin{aligned} G^*(x)&= \sup _{y} \{xy - G(y)\} = x (G')^{-1} (x) - G\left( (G')^{-1} (x)\right) = |x|^p/p, \end{aligned}$$

where $p = q/(q-1)$ is the conjugate of q. In this case, with proper inner and outer rescaling coefficients,  (3.5) is exactly the same ODE which plays an important role in Lemma 19 and Lemma 21 in Guasoni and Weber (2020) and Lemma 3.1 in Cayé et al. (2019).

A fast mean-reverting SDE The second ingredient is the existence and uniqueness of a strong solution to a fast mean-reverting SDE.

Lemma A.4

Let g be the solution to the ODE (3.5) from Lemma A.2. There exists a unique strong solution of the SDE

$$\begin{aligned} \textrm{d}\Delta _t&= \left( (G')^{-1}\left( \frac{g(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t)}{\lambda \Lambda _t}\right) - {\bar{b}}_t\right) \textrm{d}t -{\bar{a}}_t \textrm{d}W_t, \quad \Delta _0 = \varphi _{0-} +\frac{\xi _0}{\sigma _0}-\frac{\mu _0}{\gamma \sigma ^ 2_0}. \end{aligned}$$

(A.2)

Moreover, this process is a recurrent diffusion.

Remark A.5

When $G(x) = x^2/2$ and ${\bar{b}}$, ${\bar{a}}$, $\sigma$ and $\Lambda =1$ are all constants, the solution to (3.5) is $g(x;\gamma ,\sigma ,{\bar{a}},\lambda ) = -\sqrt{\gamma \sigma ^2 \lambda }x$, and the dynamics  (A.2) becomes

$$\begin{aligned} d\Delta _t = -\left( \sqrt{\frac{\gamma \sigma ^2}{\lambda }}\Delta _t-{\bar{b}}\right) \textrm{d}t -{\bar{a}} \textrm{d}W_t, \qquad \Delta _0 = \varphi _{0-} +\frac{\xi _0}{\sigma }-\frac{\mu _0}{\gamma \sigma ^ 2}. \end{aligned}$$

(A.3)

This is an Ornstein-Uhlenbeck process, which is mean-reverting. In general, the requirement of $xg(x)\le 0$ ensures that the dynamic (A.2) is indeed mean-reverting and converges to an ergodic limit.

With these two ingredients on hand, we now present our first results in the following theorem:

Theorem A.6

Let g be the solution to (3.5) and $\left( \Delta _t\right) _{t\ge 0}$ the solution to (A.2). Then under Assumption 2.1 and Assumption A.1, for all competing admissible strategies ${\dot{\psi }}$, we have

$$\begin{aligned} J_T({\dot{\psi }}) \le J_T\left( (G')^{-1}\left( \frac{g(\Delta ;\gamma ,\sigma ,{\bar{a}},\lambda \Lambda )}{\lambda \Lambda }\right) \right) + O\left( \frac{\sqrt{\lambda }}{T}\right) +O\left( \sqrt{\lambda }\right) . \end{aligned}$$

Theorem A.6 shows that, under Assumption A.1 the smallness is not only an absolute quantity on $\sqrt{\lambda }$, but also on the relative quantity $\sqrt{\lambda }/T$, i.e. the smallness of $\lambda$ should also be relative quantity with respect to the trading time horizon. Notice that the order of the smallness is derived as a coarse upper bound for every transaction costs function G that satisfies Assumption 2.1. In fact, with the specific form of the transaction costs G, we can have finer estimations, as in the example of quadratic costs shown in Corollary B.2. For the general power costs case and proportional costs case, we refer the reader to the discussion of Theorem 3.3 in Cayé et al. (2019) and Theorem 4.2 in Gonon et al. (2021).

1.2 Proof of Sect. A.1

The proof of Lemma A.2 follows the same procedure as the proof of Lemma 3.4 in Gonon et al. (2021), and Lemma A.4 follows the same procedure as the proof of Lemma 3.5 in Gonon et al. (2021).

Here we provide some auxiliary results on the function g from (3.5), which are immediate results from the proof of Lemma 3.4 in Gonon et al. (2021).

Corollary A.7

Let g be the solution to (3.5) from Lemma A.2. Then the following holds:

• 1. There exists a constant $C_G>0$ that only depends on G such that for all $x\in \mathbb {R}$,

  $$\begin{aligned} |g(x;\gamma ,\sigma ,{\bar{a}},\lambda )| \le C_G \sqrt{\lambda }\left( \sqrt{\lambda } + \sqrt{\gamma \sigma ^2} |x|\right) . \end{aligned}$$

  (A.4)

• 2. There exists a constant $K_G>0$ that only depends on G such that for all $x\in \mathbb {R}$,

  $$\begin{aligned} |g'(x;\gamma ,\sigma ,{\bar{a}},\lambda )| \le \sqrt{\gamma \sigma ^2\lambda } K_G. \end{aligned}$$

  (A.5)

Corollary A.8

Let g be the solution to (3.5) from Lemma A.2 and let the process $\Delta$ be the strong solution to (A.2) from Lemma A.4.

• 1. We have the following uniform moment bounds

  $$\begin{aligned} \sup _{T\ge 0}\mathbb {E}\left[ |\Delta _T|^k\right] <\infty , \qquad \text{ for } \text{ all } \quad k\in \mathbb {N}. \end{aligned}$$

  (A.6)

• 2. There exists $M>0$, such that for an arbitrary process X with dynamic

  $$\begin{aligned} dX_t = \mu _t^X \textrm{d}t + \sigma _t^X \textrm{d}W_t, \end{aligned}$$

  the following inequality holds a.s.:

$$\begin{aligned}&\Big | \int _0^T \left( \gamma \sigma _t^2\Delta _tX_t+\mu _t^Xg(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t) + \sigma ^X_t{\bar{a}}_t g'(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t) \right) \textrm{d}t \nonumber \\&\qquad + \int _0^TX_t{\bar{a}}_t g'(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t)\textrm{d}W_t -g(\Delta _T;\gamma ,\sigma _T,{\bar{a}}_T,\lambda \Lambda _T) X_T \Big | \le \sqrt{\lambda }M\int _0^T |X_t| \textrm{d}t. \end{aligned}$$

(A.7)

Proof of Theorem A.6

With the strategy, we write

$$\begin{aligned} {\hat{\varphi _t}} = \varphi _{0-} + \int _0^t \left( G'\right) ^{-1}\left( \frac{g(\Delta _u; \gamma ,\sigma _u,{\bar{a}}_u,\lambda \Lambda _u)}{\lambda \Lambda _u}\right) \ \textrm{d}u = \varphi _{0-} + {\bar{\varphi }}_t +\Delta _t - \left( \frac{\mu _0}{\gamma \sigma ^ 2} - \frac{\xi _0}{\sigma } +\Delta _0\right) = {\bar{\varphi }}_t + \Delta _t, \end{aligned}$$

hence with (2.10), we have

$$\begin{aligned} \gamma \sigma ^2\Delta _t = \gamma \sigma ^2\left( {\hat{\varphi _t}} - {\bar{\varphi }}_t\right) = \gamma \sigma \left( \sigma {\hat{\varphi _t}} +\xi _t\right) - {\mu _t}. \end{aligned}$$

(A.8)

Consider a competing admissible strategy $\psi$ and, to ease notation, set

$$\begin{aligned} \dot{\theta }_t = \dot{\psi }_t -\left( G'\right) ^{-1}\left( \frac{g(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t)}{\lambda \Lambda _t}\right) , \end{aligned}$$

hence

$$\begin{aligned} \theta _t = \int _0^t \dot{\psi }_u - \left( G'\right) ^{-1}\left( \frac{g(\Delta _u;\gamma ,\sigma _u,{\bar{a}}_u,\lambda \Lambda _u)}{\lambda \Lambda _u}\right) \textrm{d}u = \psi _t - {\hat{\varphi _t}}. \end{aligned}$$

Equation (A.8) and the convexity of G yield

$$\begin{aligned}& J_T(\dot{\psi }) - J_T \left( (G')^{-1}\left( \frac{g(\Delta ;\gamma ,\sigma ,{\bar{a}},\lambda \Lambda )}{\lambda \Lambda }\right) \right) \nonumber \\ {}&= \frac{1}{T} \mathbb {E}\left[ \int _0^T\theta _t \mu _t - \frac{\gamma }{2}\theta _t (\psi _t \sigma _t+ {\hat{\varphi _t\sigma _t}} +2\xi _t)\sigma _t +\lambda \Lambda _t\left( G\left( \left( G'\right) ^{-1} \left( \frac{g(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t)}{\lambda \Lambda _t}\right) \right) -G(\dot{\psi }_t) \right) \; dt \right] \nonumber \\&\le \frac{1}{T}\mathbb {E}\left[ \int _0^T-\frac{1}{2} \gamma \left( \theta _t \sigma _t\right) ^2 + \theta _t \big (\mu _t - \gamma ({\hat{\varphi _t\sigma _t}} + \xi _t)\sigma _t\big ) +\lambda \Lambda _t G'\left( \left( G'\right) ^{-1} \left( \frac{g(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t)}{\lambda \Lambda _t}\right) \right) {\dot{\theta }}_t \; dt\right] \nonumber \\&= \frac{1}{T}\mathbb {E}\left[ \int _0^T-\frac{1}{2} \gamma \left( \theta _t \sigma _t \right) ^2 - \gamma \theta _t \sigma ^2_t\Delta _t - g(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t)\dot{\theta }_t \; dt\right] . \end{aligned}$$

(A.9)

We now analyze the terms on the right-hand side of (A.9). The inequality (A.7) from Lemma A.8 in turn yields

$$\begin{aligned}&\mathbb {E}\left[ \int _0^T \left( \gamma \theta _t \sigma ^2_t\Delta _t + g(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t)\dot{\theta }_t\right) \textrm{d}t\right] \nonumber \\ {}&\qquad \qquad \qquad \qquad \ge \mathbb {E}[g(\Delta _T;\gamma ,\sigma _T,{\bar{a}}_T,\lambda \Lambda _T)\theta _T] - \sqrt{\lambda } M \mathbb {E}\left[ \int _0^T |\theta _t|\textrm{d}t\right] \end{aligned}$$

(A.10)

Here, the local martingale part is a true martingale. Indeed, by Hölder’s inequality, the integrability condition $\varphi \sigma \in \mathbb {H}^2$ and the boundedness of $g'$ established in Corollary A.7,

$$\begin{aligned} \mathbb {E}\left[ \int _0^t |\theta _u g'(\Delta _u;\gamma ,\sigma _u,{\bar{a}}_u,\lambda \Lambda _u)|^2 \textrm{d}u \right]&\le \gamma \lambda K_G^2\mathbb {E}\left[ \int _0^t\sigma ^2_u \theta _u^{2}\textrm{d}u \right] <\infty . \end{aligned}$$

Also taking into account that

$$\begin{aligned} \left| g(\Delta _t;\gamma ,\sigma _t,{\bar{a}}_t,\lambda \Lambda _t) \right| \le \sqrt{\gamma \sigma ^2_t\lambda \Lambda _t} C_G |\Delta _t| + \lambda \Lambda _t C_G, \end{aligned}$$

we can therefore use (A.10) to replace the second and the third terms on the right-hand side of (A.9), obtaining

$$\begin{aligned}&J_T(\dot{\psi }) - J_T \left( (G')^{-1}\left( \frac{g(\Delta ;\gamma ,\sigma ,{\bar{a}},\lambda \Lambda )}{\lambda \Lambda }\right) \right) \\&\le - \frac{1}{T}\mathbb {E}[g(\Delta _T;\gamma ,\sigma _T,{\bar{a}}_T,\lambda \Lambda _T)\theta _T]\\&\quad - \mathbb {E}\left[ \int _0^T \frac{\gamma }{2} \left( \theta _t\sigma _t\right) ^2 \textrm{d}t \right] + \frac{\sqrt{\lambda }M}{T}\int _0^T \mathbb {E}[|\theta _t|]\textrm{d}t. \end{aligned}$$

The Cauchy-Schwartz inequality yields

$$\begin{aligned} \big |\mathbb {E}[g(\Delta _T;\gamma ,\sigma _T,{\bar{a}}_T,\lambda \Lambda _T)\theta _T]\big |&\le \left( \mathbb {E}[g(\Delta _T;\gamma ,\sigma _T,{\bar{a}}_T,\lambda \Lambda _T)^2]\mathbb {E}[\theta _T^2]\right) ^{1/2} \\ {}&\le \left( \mathbb {E}[2g(\Delta _T;\gamma ,\sigma _T,{\bar{a}}_T,\lambda \Lambda _T)^2](\mathbb {E}[({\hat{\varphi _T}})^2] + \mathbb {E}[(\psi _T)^2])\right) ^{1/2} \\ {}&\le 2C_G \sqrt{\lambda } \left( \mathbb {E}[({\hat{\varphi _T}})^2] + \mathbb {E}[(\psi _T)^2]\right) ^{1/2} \left( \gamma \sigma ^2 \mathbb {E}[|\Delta _t|^2]+ \lambda \right) ^{1/2}. \end{aligned}$$

Moreover, it follows thatϕ

$$\begin{aligned} \frac{1}{T}\mathbb {E}[|{\hat{\varphi _T}}|^2] =\frac{2}{T}\left( \mathbb {E}[|{\bar{\varphi }}_T|^2] + \mathbb {E}[|\Delta _T|^2]\right) \le 2\sup _{T>0} \frac{1}{T}\left( \mathbb {E}[|{\bar{\varphi }}_T|^2] + \mathbb {E}[|\Delta _T|^2]\right) <\infty . \end{aligned}$$

Together with the transversality condition (2.5), it follows that

$$\begin{aligned} \frac{1}{T} \left| \mathbb {E}[g(\Delta _T;\gamma ,\sigma _T,{\bar{a}}_T,\lambda \Lambda _T)\theta _T] \right| \le \frac{2C_G \sqrt{\lambda }}{T}\left( \mathbb {E}[({\hat{\varphi _T}})^2] \qquad \qquad+ \mathbb {E}[(\psi _T)^2]\right) ^{1/2} \left( \gamma \sigma ^2 \mathbb {E}[|\Delta _t|^2]+ \lambda \right) = O\left( \frac{\sqrt{\lambda }}{T}\right) , \end{aligned}$$

and again by Hölder’s inequality,

$$\begin{aligned} \frac{1}{T}\mathbb {E}\left[ \int _0^T |\theta _t| \textrm{d}t \right] \le \left( \frac{1}{T} \mathbb {E}\left[ \int _0^T\theta _t^2 \textrm{d}t\right] \right) ^{1/2} \le 2\left( \sup _{T>0} \frac{1}{T}\mathbb {E}\left[ \int _0^T {\hat{\varphi _t^2}} + \psi _t^2 \textrm{d}t\right] \right) ^{1/2}. \end{aligned}$$

Therefore, the trading rate $\dot{\varphi }$ is indeed asymptotically optimal as:

$$\begin{aligned}&J_T(\dot{\psi }) - J_T \left( (G')^{-1}\left( \frac{g(\Delta ;\gamma ,\sigma ,{\bar{a}},\lambda \Lambda )}{\lambda \Lambda }\right) \right) \\ {}&\qquad \qquad \qquad \le - \mathbb {E}\left[ \int _0^T \frac{\gamma }{2} \left( \theta _t\sigma _t\right) ^2 \textrm{d}t \right] - \frac{1}{T}\mathbb {E}[g(\Delta _T;\gamma ,\sigma _T,{\bar{a}}_T,\lambda \Lambda _T)\theta _T] + \frac{\sqrt{\lambda }M}{T}\int _0^T \mathbb {E}[|\theta _t|]\textrm{d}t\\&\qquad \qquad \qquad =- \mathbb {E}\left[ \int _0^T \frac{\gamma }{2} \left( \theta _t\sigma _t\right) ^2 \textrm{d}t \right] +O\left( \frac{\sqrt{\lambda }}{T}\right) +O\left( \sqrt{\lambda }\right) \\&\qquad \qquad \qquad \le O\left( \frac{\sqrt{\lambda }}{T}\right) +O\left( \sqrt{\lambda }\right) . \end{aligned}$$

$\square$

Appendix B: Asymptotic results for quadratic costs

When the transaction costs is considered to be quadratic, i.e. $G(x) = x^2/2$, we have a linear function as $\left( G'\right) ^{-1} (x) = x$. Hence the forward equation (2.13) becomes linear with respect to the backward component Y, and the existence and uniqueness can be established as in Delarue (2002), Kohlmann and Tang (2002), provided the coefficients satisfies certain regularities. However, for general function G satisfying Assumption (), the generator for the forward component is not globally Lipschitz hence no general theory is available for the FBSDE system (2.13)–(2.14).

1.1 A concrete example

As already emphasized above, a general existence proof for the FBSDE system (2.13)–(2.14) remains a challenging open problem. Let us just briefly sketch how the nonlinear FBSDE (2.13)–(2.14) reduces to a nonlinear PDE, with the following assumptions on the market:

Assumption B.1

• (i) the frictionless strategy satisfies that ${{\bar{b}}}_t=0$ and ${{\bar{a}}}_t = {{\bar{a}}}$;

• (ii) the volatility process of the stock price remain constant $\sigma >0$ in the models with and without transaction costs;

• (iii) the cost parameter is constant ($\Lambda =1$).

Under Assumption B.1, the forward-backward system (2.13)–(2.14) in turn becomes autonomous,

$$\begin{aligned}&d\Delta \varphi _t = (G')^{-1}\left( \frac{Y_t}{\lambda }\right) \textrm{d}t - {\bar{a}}\textrm{d}W_t,&\Delta \varphi _0&= \varphi _{0-} +\frac{\xi _0}{\sigma }-\frac{\mu _0}{\gamma \sigma ^ 2}, \end{aligned}$$

(B.1)

$$\begin{aligned}&\textrm{d}Y_t = \gamma \sigma ^2\Delta \varphi _t \textrm{d}t + Z_t \textrm{d}W_t,&Y_T&= 0. \end{aligned}$$

(B.2)

When the transaction costs is quadratic, i.e. $G(x) = x^2/2$, we can create the optimal strategy explicitly. Accordingly, the approximation can be made more precise, and we summarize the results as follows:

Corollary B.2

For $G(x) = x^2/2$, define the strategy

$$\begin{aligned} {\dot{\varphi }}_t = - \sqrt{\frac{\gamma \sigma ^2}{\lambda }}\tanh \left( \sqrt{\frac{\gamma \sigma ^2}{\lambda }}(T-t)\right) \Delta \varphi _t. \end{aligned}$$

(B.3)

Then under Assumption B.1, for all competing admissible strategies ${\dot{\psi }}$, we have

$$\begin{aligned} \sup _{{\dot{\psi }}} J_T({\dot{\psi }}) = J_T({\dot{\varphi }}) = J_T\left( -\sqrt{\frac{\gamma \sigma ^2}{\lambda }} \Delta \right) + O\left( \frac{\lambda }{T}\right) = J_T\left( -\sqrt{\frac{\gamma \sigma ^2}{\lambda }} \Delta \right) + o\left( \frac{\sqrt{\lambda }}{T}\right) , \end{aligned}$$

where $\Delta$ is the following Ornstein-Uhlenbeck process:

$$\begin{aligned} d\Delta _t = -\sqrt{\frac{\gamma \sigma ^2}{\lambda }}\Delta _t\textrm{d}t -{\bar{a}} \textrm{d}W_t, \qquad \Delta _0 = \varphi _{0-} +\frac{\xi _0}{\sigma }-\frac{\mu _0}{\gamma \sigma ^ 2}. \end{aligned}$$

Remark B.3

The optimality of ${\dot{\varphi }}$ by (B.3) is derived in Theorem 4.5 (Muhle-Karbe et al. 2020). The approximation order is a straight comparison between the candidate trading rate given by $-\sqrt{{\gamma \sigma ^2}/{\lambda }}\ \Delta$ and the optimal trading rate ${\dot{\varphi }}$.

When there is no liquidity risk, i.e. when $\Lambda =1$ throughout the trading horizon, the smallness assumption on the liquidity $\lambda$ is purely a relative quantity, comparing to the trading horizon T. Here with quadratic trading costs, the approximation is $\lambda /T$, which is finer than the overall approximation $\sqrt{\lambda }/T$.