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.

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Notes

  1. 1.

    It can be noted here that another cost function \({\mathcal {J}}\) could be defined here as:

    $$\begin{aligned} d{\mathcal {J}}(x_t)=\left[ \omega (q,x_t) P(x_t) +(1 -\omega (q,x_t) ) {\mathcal {J}}(x_t-q) \right] dN^{\mu (x)}_t - cq\, dt, \end{aligned}$$

    where \(\omega (q,x_t)\) is a random variable taking value 1 with a probability q / x and 0 otherwise. In such a case, instead of a prorata rule, we will have a trading rule for which an order is fully executed with a probability q / x, or not at all. This case covers the trading model of [41], in which the orderbook matching rule is FIFO (First In, First Out), but any agent can modify and reinsert his order at any time. In such a case the probability for one specific agent to be first in the queue (and thus be fully filled), is q / x.

    Since \({\mathbb {E}}dJ = {\mathbb {E}}d{\mathcal {J}}\), the emerging dynamics are the same.

References

  1. 1.

    Adlakha, S., Johari, R., Weintraub, G.Y.: Equilibria of dynamic games with many players: existence, approximation, and market structure. J. Econ. Theory 156, 269–316 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Alfonsi, A., Fruth, A., Schied, A.: Optimal execution strategies in limit order books with general shape functions. Quant. Financ. 10(2), 143–157 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Almgren, R., Thum, C., Hauptmann, E., Li, H.: Direct estimation of equity market impact. Risk 18, 57–62 (2005)

    Google Scholar 

  4. 4.

    Almgren, R.F., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3(2), 5–39 (2000)

    Google Scholar 

  5. 5.

    Bacry, E., Delattre, S., Hoffmann, M., Muzy, J.F.: Modeling microstructure noise with mutually exciting point processes. Quant. Financ. 13, 65–77 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Bouchard, B., Dang, N.-M., Lehalle, C.-A.: Optimal control of trading algorithms: a general impulse control approach. SIAM J. Financ. Math. 2(1), 404–438 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Brogaard, J., Baron, M., Kirilenko, A.: The Trading Profits of High Frequency Traders. Confronting Many Viewpoints, In Market Microstructure (2012)

    Google Scholar 

  8. 8.

    Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A.: Long time average of mean field games. Netw. Heterog. Media 7(2), 279–301 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Carmona, R., Delarue, F., Lachapelle, A.: Control of McKean–Vlasov dynamics versus mean field games. Math. Financ. Econ. 7, 131–136 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Carmona, R., Lacker, D.: A probabilistic weak formulation of mean field games and applications. Math. Financ. Econ. 7, 131–136 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Euronext. EuroNext rules: book I. Technical report, EuroNext (2006)

  13. 13.

    Field, J., Large, J.: Pro-rata matching and one-tick futures markets. CFS working paper 2008,40, Frankfurt, Main (2008)

  14. 14.

    Foucault, T., Kadan, O., Kandel, E.: Limit order book as a market for liquidity. Rev. Financ. Stud. 18(4), 1171–1217 (2005)

    Article  Google Scholar 

  15. 15.

    Foucault, T., Menkveld, A.J.: Competition for order flow and smart order routing systems. J. Financ. 63(1), 119–158 (2008)

    Article  Google Scholar 

  16. 16.

    Foucault, T., Pagano, M., Röell, A.: Market Liquidity: Theory, Evidence, and Policy. Oxford University Press, Oxford (2013)

    Google Scholar 

  17. 17.

    Ganchev, K., Nevmyvaka, Y., Kearns, M., Vaughan, J.W.: Censored exploration and the dark pool problem. Commun. ACM 53(5), 99–107 (2010)

    Article  Google Scholar 

  18. 18.

    Gareche, A., Disdier, G., Kockelkoren, J., Bouchaud, J.P.: A Fokker–Planck description for the queue dynamics of large tick stocks. Phys. Rev. E 88, 032809 (2013)

    Article  Google Scholar 

  19. 19.

    Gatheral, J.: No-dynamic-arbitrage and market impact. Quant. Financ. 10(7), 749–759 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Guéant, O., Lasry, J.-M., Lions, P.-L.: Paris-Princeton Lectures on Mathematical Finance 2010, chapter Mean Field Games and Applications. Springer, New York (2010)

  21. 21.

    Guéant, O., Lehalle, C.-A., Fernandez-Tapia, J.: Optimal execution with limit orders. SIAM J. Financ. Math. 13(1), 740–764 (2012)

    Article  MATH  Google Scholar 

  22. 22.

    Hall, A., Kofman, P., Mcculloch, J.: True Spreads Censored by Tick Size. Technical report (2005)

  23. 23.

    Ho, T., Stoll, H.R.: The dynamics of dealer markets under competition. J. Financ. 38(4), 1053–1074 (1983)

    Article  Google Scholar 

  24. 24.

    Huang, M.Y., Caines, P.E., Malhame, R.P.: Large population cost-coupled LQG problems with non-uniform individuals. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kirilenko, A. A., Kyle, A. P., Samadi, M., Tuzun, T.: The Flash Crash: The Impact of High Frequency Trading on an Electronic Market. Social Science Research Network Working Paper Series(2010)

  26. 26.

    Krusell, P., Smith, A.A.: Income and wealth heterogeneity in the macroeconomy. J. Polit. Econ. 106(5), 867–896 (1998)

    Article  Google Scholar 

  27. 27.

    Kyle, A.P.: Continuous auctions and insider trading. Econometrica 53(6), 1315–1335 (1985)

    Article  MATH  Google Scholar 

  28. 28.

    Lachapelle, A., Wolfram, M.-T.: On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transp. Res. Part B 15, 10 (2011)

    Google Scholar 

  29. 29.

    Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Lehalle, C.-A., Guéant, O., Razafinimanana, J.: High Frequency Simulations of an Order Book: a Two-Scales Approach. In: Abergel, F., Chakrabarti, B.K., Chakraborti, A., Mitra, M. (eds.) Econophysics of Order-Driven Markets. Springer, New Economic Windows, New York (2010)

    Google Scholar 

  31. 31.

    Lehalle, C.-A., Laruelle, S., Burgot, R., Pelin, S., Lasnier, M.: Market Microstructure in Practice. World Scientific publishing, Singapore (2013)

    Google Scholar 

  32. 32.

    Lions, P.-L. (2007–2012). Jeux à champ moyen et applications. Technical report, Collège de France

  33. 33.

    Lucas, R., Moll, B.: Knowledge growth and the allocation of time. J. Polit. Econ. 75, 352 (2013)

    Google Scholar 

  34. 34.

    Madhavan, A.: Exchange-Traded Funds, Market Structure and the Flash Crash. Social Science Research Network Working Paper Series (2011)

  35. 35.

    Mendelson, H., Amihud, Y.: How (Not) to Integrate the European Capital Markets. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  36. 36.

    Menkveld, A. J.: High Frequency Trading and The New-Market Makers. Social Science Research Network Working Paper Series (2010)

  37. 37.

    Pagès, G., Laruelle, S., Lehalle, C.-A.: Optimal split of orders across liquidity pools: a stochatic algorithm approach. SIAM J. Financ. Math. 2, 1042–1076 (2011)

    Article  MATH  Google Scholar 

  38. 38.

    Pequito, S., Aguiar, A. P., Sinopoli, B., Gomes, D. A.: Nonlinear Estimation Using Mean Field Games. In: 2011 5th International Conference on Network Games, Control and Optimization (NetGCooP) (2011a)

  39. 39.

    Pequito, S., Aguiar, A. P., Sinopoli, B., Gomes, D. A.: Unsupervised Learning of Finite Mixture Using Mean Field Games. In: 9th Allerton Conference on Communication, Control and Computing(2011b)

  40. 40.

    Robert, C.Y., Rosenbaum, M.: A new approach for the dynamics of ultra-high-frequency data: the model with uncertainty zones. J. Financ. Econom. 9(2), 344–366 (2011)

    Article  Google Scholar 

  41. 41.

    Roşu, I.: A dynamic model of the limit order book. Rev. Financ. Stud. 22(11), 4601–4641 (2009)

    Article  Google Scholar 

Download references

Acknowledgments

This work has been partially granted by the Crédit Agricole Cheuvreux Research Initiative in partnership with the Louis Bachelier Institute, the Collège de France and the Europlace Institute of Finance. Authors thank Ioanid Roşu for fruitful discussions about the orderbook model.

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Correspondence to Charles-Albert Lehalle.

Appendix: Proofs

Appendix: Proofs

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:

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

\(\square \)

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)

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

\(\square \)

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.

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Lachapelle, A., Lasry, JM., Lehalle, CA. et al. Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis. Math Finan Econ 10, 223–262 (2016). https://doi.org/10.1007/s11579-015-0157-1

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Keywords

  • Orderbook modeling
  • Mean field games
  • Order-driven market
  • Waiting cost
  • Liquidity equilibrium
  • High frequency trading
  • Market microstructure
  • Price formation process

JEL Classification

  • C730 (Stochastic and Dynamic Games)
  • G140 (Information and Market Efficiency)