scenarii for endogenous liquidity crises - oecd€¦ · two scenarii of liquidity crises: one from...

Scenarii for endogenous liquidity crisesAntoine Fosset, Jean-Philippe Bouchaud & Michael BenzaquenEcole polytechnique & Capital Fund Management

NAEC-ECOLE POLYTECHNIQUE WORKSHOP

The Policy Implications of Econophysics

21 january 2020

www.EconophysiX.com CFM Chair of Econophysics & Complex Systems

https://www.econophysix.com/

An example of liquidity crisis: the flash crash 2/18

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9. 85%

Figure: Price of the SPMini future contract during the flash crash, 6th of May 2010



Endogeneous dynamic VS exogeneous dynamic 3/18

The study of Joulin et al.1 shows that anomalous price movements (> 4σ) on the scaleof one minute:I Only 5% are news related (i.e. exogenous).I 95% are not news related (i.e. endogenous).

Necessity to have a feedback in the price formation mechanism in order to take intoaccount the endogeneity!

1Armand Joulin, Augustin Lefevre, Daniel Grunberg, and Jean-Philippe Bouchaud. Stock price jumps:news and volume play a minor role.Wilmott Magazine, September/October, 1–7, 2008



Example of feedback in a physical system 4/18

Figure: Flock of birds: collective behaviour emerging from a zero-intelligence feedback.



Definition of the limit order book 5/18



Example of a limit order book 6/18

Figure: Example of a real order book.



Intuition 7/18Necessity to put some feedback of past price on the price formation process.Possible mechanism of feedback:

Feedback of past price changes on cancelations:

λct ∝ λc0︸︷︷︸base rate

+αK

(∫ t0

√2βe−β(t−s)dps

)2︸︷︷︸

feedback on past price changes



What does data tell us? 8/18

We would like to motivate this choice of feedback by looking at the data.

We aim to fit the following rate of events:

λt ∝ base rate + past trend contribution + square past trend contribution (1)

Take home messageThe past square trend dimishes the future liquity equaly at the bid and at the ask.This effect is mainly in the cancelations.The past trend has a slight effect on the bid-ask imbalance (i.e difference of liquiditybetween the bid and ask).



Transition from stable to unstable markets 9/18

Numerical simulationsI N size of the system i.e. the number of available price levels.I T time of simulation.

I αK feedback intensity: λct ∝ αc0 + αK(∫ t

0

√2βe−β(t−s)dps

)2.

I 1/β: time scale of the feedback.

Video of an order book

We measure the time τc of first liquidity crisis: first time when one side of the orderbook empties.



Snapshots of the order book 10/18

Figure: Snapshots of the order book. The left figure is taken at the beginning of the simulation and theright figure is taken during a period of high volatility. We can see that the liquidity almost dries out.



Stability map and phase transition 11/18

Figure: Stability map: Crisis probability P[τc ≤ T ] for T = 200, N = 280, λ`0 = 10, λc0 = 1 and λm0 = 20.The blue region corresponds to a stable order book, whereas the red region corresponds to liquidity crises

Numerical simulations show that we have an exact phase transition. For an infiniteorder book (N = T = +∞), if αK < α∗ there is no crisis and if αK > α∗ there arecrises. α∗ is the critical point.



Interpretation of crises via a phase transition 12/18

To make this scenario consistent, we have two possibilities:

I Real financial markets would have to sit below, but very close to the critical pointand his critical point is attractive (see Self-Organized Criticality).

I The feedback parameters αK is time dependent and occasionally visit theunstable phase.



Simple model of spread dynamics 13/18

Key ingredient: spread opening events trigger more spread opening events.

Model definition:I Number of spread opening events before t : S+tI Rate of spread closing event: λ−0I Rate of spread opening event: λ+t = λ

+0 + �X

2t = base rate + feedback

I Number of spread opening events on time scale 1/β: Xt :=∫ t

0 βe−β(t−s)dS+s .

Let’s call τc the time of liquidity crisis.I If � > 0 there is always a liquidity crises.I If � > 0 is small enough, before the liquidity crisis the spread seems stable: it is

metastable.I logE[τc] ≈ 1β� log

1�λ+0

for � > 0 small enough.



Properties of the metastability time 14/18

Figure: Set of parameters: λ+0 = 1, λ−0 = 0.5 and � = 0.2. (a) Survival function (sf) of the time of

metastability which is found to be exponential. (b) Evolution of the average metastability time with �. Thedotted red curve is the continuous time prediction. The plain red curve is obtained by multiplying the termin the exponential by a empirical factor 2.5. (c) Typical metastable trajectory.



Properties of the metastability time 15/18

Physical interpretation of metastability: the "particule" tries to minimize V . When it istrapped in Xeq , it needs a fluctuation large enough to cross the barrier V (X ∗)− V (Xeq)and go beyond X ∗.

0 1 Xeq 2 3 X * 4 5X/ +0

0.50

0.25

0.00

0.25

0.50

0.75

1.00V(

X)

= 0= 0.2



Conclusions and future work 16/18

ConclusionsOn empirical data, the past square trend dimishes the future liquity.Two scenarii of liquidity crises: one from a phase transition, the other from ametastable description.

OutlineWe could also design a empirical test that could help discriminating between thesecond order phase transition and activation scenarii.We also would design a complete protocol to estimate the price feedback and use itto predict liquidity crises.



References 17/18

Armand Joulin, Augustin Lefevre, Daniel Grunberg, and Jean-Philippe Bouchaud.Stock price jumps: news and volume play a minor role.Wilmott Magazine, September/October, 1–7, 2008.

Antoine Fosset, Jen-Philippe Bouchaud, and Michael Benzaquen.Endogenous liquidity crises.Arxiv 1912.00359, 2019.



Thank you !



Introduction and motivationThe limit order bookData analysisFirst scenario: liquidity crises and phase transitionSecond scenario: liquidity crises and metastability

scenarii for endogenous liquidity crises - oecd€¦ · two scenarii of liquidity crises: one from...

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