Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis

Abstract

This paper deals with a stochastic order-driven market model with waiting costs, for orderbooks with heterogenous traders. Offer and demand of liquidity drives price formation and traders anticipate future evolutions of the orderbook. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and high frequency traders (HFT). We provide both analytical solutions and numerical experiments. Implications on classical quantities are explored: orderbook size, prices, and effective bid/ask spread. According to the model, in markets with Institutional Investors only we show the existence of inefficient liquidity imbalances in equilibrium, with two symmetrical situations corresponding to what we call liquidity calls for liquidity. During these situations the transaction price significantly moves away from the fair price. However this macro phenomenon disappears in markets with both Institutional Investors and HFT, although a more precise study shows that the benefits of the new situation go to HFT only, leaving Institutional Investors even with higher trading costs.

Appendix: Proofs

1.1 Proof of Proposition 5.2

Looking at equations $(A_{R^{++}})$ and $(A_{R^{-+}})$ we notice that at the boundary there is a jump causing a change of sign of the coefficient multiplying the derivatives (under the basic assumption $\lambda \ge \lambda ^-$). Therefore, at this point we must have

$$\begin{aligned} \frac{\lambda ^-}{x_0} \left( \frac{\delta }{x_0-q}-\tilde{u}\right) = \frac{c}{q}. \end{aligned}$$

(36)

On the other hand, as the seller’s routing decision $R^\oplus _s$ jumps from 1 to 0, we must have

$$\begin{aligned} \tilde{u} = \frac{- \delta }{y_0-q}. \end{aligned}$$

(37)

Combining (36) and (37) we get the equality:

$$\begin{aligned} \eta x_0 = \frac{1}{x_0-q}+\frac{1}{y_0-q}, \end{aligned}$$

where (same definition as in Proposition 5.2):

$$\begin{aligned} \eta \;{:=}\;\frac{c}{ \delta q\lambda ^-}. \end{aligned}$$

It follows that the diagonal point of the boundary $M_0$ is the point (Eq. 29 of Proposition 5.2)

$$\begin{aligned} (x_0^*,x_0^*) = (q+\sqrt{q^2+8/\eta })/2 \end{aligned}$$

and that the boundary is defined by the parametric equation of Proposition 5.2:

$$\begin{aligned} (x_0,y_0) = \Bigg (x_0,l(x_0){:=} q+ \Bigg (\eta x_0-\frac{1}{x_0-q}\Bigg )^{-1}\Bigg ), \; \forall x_0 \ge x_0^*. \end{aligned}$$

$\square$

1.2 Proof of Proposition 5.3

Unfortunately, looking at equations $(B_{R^{-+}})$ and $(B_{R^{--}})$ we conclude that we cannot adopt the same reasoning since the sign of the coefficients multiplying the derivative terms does not change.

We use another strategy. We solve $\tilde{v}$ analytically all along the characteristic line $y_1=x_1-k$, and then intersect the solution $\tilde{v}$ with $\frac{\delta }{x_1-q}$.

Along the characteristic $x=y+k$, we introduce the function

$$\begin{aligned} f(y) = \tilde{v}(y+k,y). \end{aligned}$$

Looking at Eq. (25), we get the generic form of the ordinary differential equation (ODE for short) satisfied by f:

$$\begin{aligned} f'+\frac{a}{y} f+\left( \frac{b}{y(y-q)}+d\right) =0, \end{aligned}$$

(38)

where

$$\begin{aligned} a =1+\lambda /\lambda ^-, \quad b = \delta (1+\lambda /\lambda ^-), \quad d= -\delta \eta , \; \text{ on } R^{-+}. \end{aligned}$$

We use the variation of constant method to solve Eq. (38).

The homogeneous solution is $f(y)=y^{-a}$ times a constant. Now let the constant varies as a function g(x). We have $f' = g'y^{-a}-agy^{-a-1}$. Substituting in (38) we obtain:

$$\begin{aligned} g' (y)= -b \frac{y^{a-1}}{y-q}-d y^a. \end{aligned}$$

This function is easy to integrate numerically. However in order to stay working with analytical formulas, we make the approximation $y-q \approx y$ for small q (recall that all this analytical part focus on the small q first order approximation). Now we are in the position to integrate the derivative $g'$.

We get

$$\begin{aligned} f(y) = g(y) y^{-a} = \left( \kappa \; y^{-a}-\frac{b}{a-1} y^{-1} - \frac{d}{a+1} y\right) . \end{aligned}$$

Now we have to compute the constant $\kappa$. Recall that we are working on the line $(y+k,y)$ and on the region $R^{-+}$ so that we are solving the ODE with an initial condition on $M_0$, which is known to be $(x_0,l(x_0))$.

Consequently we have to look at f as a family $(f_k)$ of functions indexed by $k \in \mathbb {R}^+$. On the characteristic line starting at $x_0-l(x_0)$, the function is given by

$$\begin{aligned} f_{x_0-l(x_0)}(y) = \left( \mathcal {C} (x_0) \; y^{-a}-\frac{b}{a-1} y^{-1} - \frac{d}{a+1} y\right) , \quad \forall y \ge l(x_0) . \end{aligned}$$

(39)

The core argument to compute the constant parameter $\mathcal {C} (x_0)$ for the solution on the characteristic $(y+k,y),$ with $k=x_0-l(x_0)$, is to remark that:

$$\begin{aligned} f_k(y) = \tilde{v}(y+k,y) = -\tilde{u}(y,y+k) =-f_k(y+k). \end{aligned}$$

Then, the initial condition equality

$$\begin{aligned} f_{x_0-l(x_0)}(l(x_0))=-f_{x_0-l(x_0)}(x_0), \end{aligned}$$

automatically gives the expression of $\mathcal {C}$:

$$\begin{aligned} \mathcal {C} (x_0) = \delta \frac{(1+\lambda ^-/\lambda ) [x_0^{-1}+l(x_0)^{-1}]-\frac{\eta }{1+\lambda /\lambda ^-} [x_0+l(x_0)]}{x_0^{-(1+\lambda /\lambda ^-)}+l(x_0)^{-(1+\lambda /\lambda ^-)} } , \end{aligned}$$

(40)

where the last equality holds since the equation of $\tilde{u}$ on $R^{+-}$ matches the equation of $\tilde{v}$ on $R^{+-}$.

Consequently, the analytical solution is given by (39)–(40).

Finally we are in the position to compute the parametric curve of the boundary between the two regions $R^{-+}$ and $R^{--}$.

To do so we look for the point $(x_1,y_1) = (y_1+k,y_1)$ such that $\tilde{v}(x_1,y_1) = \frac{\delta }{x_1-q}$.

More precisely, $M_1$ is defined by: $(y_1+x_0-l(x_0),y_1), \; \forall x_0 \ge x_0^*,$ where (Eq. 31 of Proposition 5.3)

$$\begin{aligned} y_1 \text{ verifies } f_{x_0-l(x_0)}(y_1) = \frac{\delta }{y_1+x_0-l(x_0)-q}. \end{aligned}$$

$\square$

1.3 Local equations of the four regions (second order equations)

Define $\varLambda = \lambda +\lambda ^-$. Let us now give the local equations on the same four regions.

$$\begin{aligned} (A_{R^{++}}) \quad 0= & {} \left[ \frac{\lambda ^-}{x} (p^b(x) - u) - c \right] + \left[ \lambda -\lambda ^-\right] \left( \partial _x u + \partial _y u\right) + q\left( \frac{\lambda ^-}{x } \partial _x u + \frac{\varLambda }{2} \Delta u\right) ,\nonumber \\ (B_{R^{++}}) \quad 0= & {} \left[ \frac{\lambda ^-}{y}( p^s(y) - v) + c \right] + \left[ \lambda -\lambda ^-\right] \left( \partial _x v+ \partial _y v\right) + q\left( \frac{\lambda ^-}{y } \partial _y v + \frac{\varLambda }{2} \Delta v\right) , \nonumber \\ (A_{R^{-+}}) \quad 0= & {} \left[ \frac{\lambda ^-}{x} (p^b(x) - u) - c \right] + \left[ -\lambda ^-\right] \left( \partial _x u + \partial _y u\right) \nonumber \\&+\, q\left( \frac{\lambda ^-}{x } \partial _x u + \frac{\lambda ^-}{2} \Delta u+ \lambda \partial _{yy} u \right) ,\nonumber \\ (B_{R^{-+}}) \quad 0 \!= & {} \! \left[ \frac{\varLambda }{y}( p^s(y) - v) \!+\! c \right] \!+\! \left[ -\lambda ^-\right] \left( \partial _x v+ \partial _y v\right) \!+\! q\left( \frac{\varLambda }{y } \partial _y u + \frac{\lambda ^-}{2} \Delta u+ \lambda \partial _{yy} v \right) , \nonumber \\ (A_{R^{+-}}) \quad 0 \!= & {} \! \left[ \frac{\varLambda }{x} (p^b(x) - u) - c \right] \!+\! \left[ -\lambda ^-\right] \left( \partial _x u \!+\! \partial _y u\right) \!+\! q\left( \frac{\varLambda }{x } \partial _x u \!+\! \frac{\lambda ^-}{2} \Delta u+ \lambda \partial _{xx} u \right) ,\nonumber \\ (B_{R^{+-}}) \quad 0 \!= & {} \! \left[ \frac{\lambda ^-}{y}( p^s(y) - v) \!+\! c \right] + \left[ -\lambda ^-\right] \left( \partial _x v\!+\! \partial _y v\right) \!+\! q\left( \frac{\lambda ^-}{y } \partial _y u \!+\! \frac{\lambda ^-}{2} \Delta u\!+\! \lambda \partial _{xx} v \right) , \nonumber \\ (A_{R^{--}}) \quad 0= & {} \left[ \frac{\varLambda }{x} (p^b(x) - u) - c \right] + \left[ -\varLambda \right] \left( \partial _x u + \partial _y u\right) +\,q\left( \frac{\varLambda }{x } \partial _x u + \frac{\varLambda }{2} \Delta u\right) ,\nonumber \\ (B_{R^{--}}) \quad 0= & {} \left[ \frac{\varLambda }{y}( p^s(y) - v) + c \right] + \left[ -\varLambda \right] \left( \partial _x v+ \partial _y v\right) +q\left( \frac{\varLambda }{y } \partial _y v + \frac{\varLambda }{2} \Delta v\right) . \end{aligned}$$

(41)

Where $\Delta$ stands for the Laplacian operator:

$$\begin{aligned} \Delta f=\partial _{xx} f + \partial _{yy} f. \end{aligned}$$

Remark that, compared to equations (28), both a diffusion term and a drift term appear.