Incorporating order-flow into optimal execution

Abstract

We provide an explicit closed-form strategy for an investor who executes a large order when market order-flow from all agents, including the investor’s own trades, has a permanent price impact. The strategy is found in closed-form when the permanent and temporary price impacts are linear in the market’s and investor’s rates of trading. We do this under very general assumptions about the stochastic process followed by the order-flow of the market. The optimal strategy consists of an Almgren–Chriss execution strategy adjusted by a weighted-average of the future expected net order-flow (given by the difference of the market’s rate of buy and sell market orders) over the execution trading horizon and proportional to the ratio of permanent to temporary linear impacts. We use historical data to calibrate the model to Nasdaq traded stocks and use simulations to show how the strategy performs.

Appendix: Proof of Proposition 1

To solve (9) we make the ansatz

$$\begin{aligned} H(t,x,S,{\varvec{\mu }},q) = x + q\,\left( S- \tfrac{1}{2}\Delta \right) + h(t,{\varvec{\mu }},q)\,, \end{aligned}$$

and upon substitution of the ansatz in the DPE above we see that $h(t,{\varvec{\mu }},q)$ satisfies

$$\begin{aligned} \partial _t h + \mathcal {L}^{\varvec{\mu }}h + b\, \mu \, q - \phi \,q^2 + \sup _\nu \left\{ -k\,\nu ^2 -(b\,q+ \partial _q h)\,\nu \right\} =0\,, \end{aligned}$$

subject to the terminal condition $h(T,{\varvec{\mu }},q)=-\alpha \,q^2$, and the optimal liquidation speed in feedback form is

$$\begin{aligned} \nu ^* = -\frac{1}{2\,k}\left( b\,q+\partial _q h \right) \,. \end{aligned}$$

(20)

Upon substitution back into the DPE we find that h satisfies the non-linear partial-integral differential equation (PIDE)

$$\begin{aligned} \left( \partial _t + \mathcal {L}^{\varvec{\mu }}\right) h + b\, \mu \, q - \phi \,q^2 + \frac{1}{4\,k}\left( b\,q+\partial _q h \right) ^2 =0\,. \end{aligned}$$

(21)

Due to the existence of linear and quadratic terms in q in (21), and its terminal conditions, we expect $h(t,{\varvec{\mu }},q)$ to be a quadratic form in q, and we assume the ansatz

$$\begin{aligned} h(t,{\varvec{\mu }},q) = h_0(t,{\varvec{\mu }}) + q\,h_1(t,{\varvec{\mu }})+q^2\,h_2(t,{\varvec{\mu }})\,. \end{aligned}$$

Inserting this into (21) and collecting like terms in q leads to the following coupled system of PIDEs

$$\begin{aligned}&\left( \partial _t +\mathcal {L}^{\varvec{\mu }}\right) h_0 + \frac{1}{4k}\,h_1^2 = 0\,, \end{aligned}$$

(22a)

$$\begin{aligned}&\left( \partial _t +\mathcal {L}^{\varvec{\mu }}\right) h_1 + b\,\mu + \frac{1}{2k}\,h_1\,\left( b+2h_2\right) =0\,, \end{aligned}$$

(22b)

$$\begin{aligned}&\left( \partial _t +\mathcal {L}^{\varvec{\mu }}\right) h_2 -\phi + \frac{1}{4k}\,\left( b+2h_2\right) ^2 =0\,, \end{aligned}$$

(22c)

subject to the terminal conditions

$$\begin{aligned} h_0(T,{\varvec{\mu }}) = 0,\quad h_1(T,{\varvec{\mu }})=0,\quad h_2(T,{\varvec{\mu }})=-\alpha \,. \end{aligned}$$

To solve for $h_2$ we note that since Eq. (22c) for $h_2$ contains no source terms in $\mu$ and its terminal condition is independent of $\mu$, the solution must be independent of $\mu$, i.e. $h_2$ is a function only of time. In this case, (22c) is an ODE of Riccati type and can be solved explicitly:

$$\begin{aligned} h_2(t,{\varvec{\mu }}) = \chi (t)-\tfrac{1}{2}\,b, \quad \text {where} \quad \chi (t)= \sqrt{k\,\phi }\,\frac{1+\zeta \,e^{2\gamma (T-t)} }{1-\zeta \,e^{2\gamma (T-t)}}\,, \end{aligned}$$

with the constants $\gamma$ and $\zeta$:

$$\begin{aligned} \gamma =\sqrt{\frac{\phi }{k}},\quad \text {and} \quad \zeta = \frac{\alpha -\frac{1}{2}b + \sqrt{k\,\phi }}{\alpha -\frac{1}{2}b - \sqrt{k\,\phi }}\,. \end{aligned}$$

Now we turn to solving (22b) which is a linear PIDE for $h_1$ where $h_2+\tfrac{1}{2}b$ acts as an effective discount rate and $b\,\mu$ is a source term. The general solution of such an equation can be represented using the Feynman-Kac theorem. Thus we write

$$\begin{aligned} h_1(t,{\varvec{\mu }}) = b\,{\mathbb {E}}_{t,{\varvec{\mu }}}\left[ \int ^T_t\exp \left\{ \frac{1}{k}\int ^u_t \,\left( h_2(s)+\tfrac{1}{2}b\right) \,ds \right\} \,\mu _u\,du \right] \end{aligned}$$

which can be simplified to

$$\begin{aligned} h_1(t,{\varvec{\mu }}) = b\, \int ^T_t \left( \frac{\zeta e^{\gamma (T-u)} - e^{-\gamma (T-u)}}{\zeta e^{\gamma (T-t)} - e^{-\gamma (T-t)} } \right) \,{\mathbb {E}}_{t,{\varvec{\mu }}} \left[ \, \mu _u\, \right] \,du\,. \end{aligned}$$

(23)

Finally, we can solve for $h_0(t,{\varvec{\mu }})$ by again noticing it is a linear PDE with non-linear source term and a straight forward application of Feynman-Kac, and interchanging integration and expectation, we obtain (11c). $\square$

Table 7 Permanent and temporary price impact parameters for Nasdaq stocks, average volume of MOs, average midprice, $\sigma$ volatility of returns (open-to-close and 5-min windows), mean arrival per hour of MOs $\lambda ^\pm$, and average volume of MOs $\mathbb {E}[\eta ^\pm ]$

Table 8 Permanent and temporary price impact parameters for Nasdaq stocks, average volume of MOs, average midprice, $\sigma$ volatility of returns (open-to-close and 5-minute windows), mean arrival per hour of MOs $\lambda ^\pm$, and average volume of MOs $\mathbb {E}[\eta ^\pm ]$

The remainder of the appendix contains tables with estimated parameters and simulation results. See Tables 7, 8, 9, 10, 11, 12.

Table 9 Relative performance of the strategy (in basis points)

Table 10 Relative performance of the strategy (in basis points)

Table 11 Relative performance of the strategy (in basis points)

Table 12 Relative performance of the strategy (in basis points)