Some recent evolutions in order book modelling

Frédéric Abergel

Chair of Quantitative FinanceÉcole Centrale Paris

http://fiquant.mas.ecp.fr

Order book modelling

Joint work with I. Muni Toke, A. Jedidi Some references

I. Muni Toke, Market making behaviour and its impact on the Bid-Ask spread, in Econophysics of Order-driven Markets, Abergel, F.; Chakrabarti, B.K.; Chakraborti, A.; Mitra, M. (Eds.), Springer, 2011

F. Abergel, A. Jedidi, A mathematical approach to order book modelling, in Econophysics of Order-driven Markets, Abergel, F.; Chakrabarti, B.K.; Chakraborti, A.; Mitra, M. (Eds.), Springer, 2011

F. Abergel, A. Jedidi, Stability and long time behaviour of a markovian order book model, to appear

F. Abergel, A. Chakraborti, I. Muni Toke, M. Patriarca, Econophysics I: empirical facts and Econophysics II: agent-based models, to appear in Quantitative Finance

Order book modelling

Summary

Empirical studies of the order book Stationary statistical properties Dynamical statistical properties

Mathematical modelling Mathematical framework Price dynamics

Statistical properties I

A host of empirical studies going back to ~20 years Limit order price and volume distribution Spread distribution Inter-event durations

Under independence assumptions:

Zero-intelligence models

Statistical properties II

More recent studies (Bouchaud, Farmer, Large 2007, Muni Toke 2010, Eisler 2010) Studies of conditional distributions… … or conditional inter-event duration Structure of the order flow

Empirical measurement of Volatility Clustering Market making… … and market taking

Leading to models involving

State-dependent intensities Mutually excited processes

Statistical properties IIMarket making

Following a market order

New limit orders arrive more rapidly than unconditional limit orders

No significant correlation between the respective signs of the market and limit orders

Statistical properties IIMarket taking

Following a limit order

New market orders do not arrive more rapidly…

… except when the limit order fell within the spread

Statistical properties IISpread distribution

An obvious “counter-example” to the zero-intelligence assumption: spread distribution

Mathematical framework

The order book is a pure jump vector-valued stochastic processes Arrival of events of various types Conditional on their occurrence, the events have a probabilistic description

Choices have to be made Some technical:

Fixed price grid (Cont, Stoikov, Talreja), fixed number of ticks, fixed number of limits with gaps…

Initial distribution, boundary conditions… Some fundamental

Exogenous and endogenous driving factors Dependence structure

Main questions to be addressed Stationarity and long time behaviour Price and spread dynamics Scaling limits: lot size, tick size…

Mathematical framework

The simplest example: zero-intelligence with limit orders, market orders and cancellations (Farmer, Smith, Guillemot, Krishnamurthy, 2003)

1

,

i it L

t M

i i ii it C C

PdL

dM

a bdC

Mathematical framework

Two sets of variables

Coupled dynamics

Two basic types of events Jump: a change in the quantities Shift : renumbering after a change of one of

the best quotes

1 1

1 1

,..., ; ,...,

N N

i i

i i

i i i ik k

a a b b

a a i P b b i P

A a B b

Mathematical framework

The infinitesimal generator of the associated Markov process

1 1 1

1,..., ; ,...,

1

(+similar terms involving the b 's)

i

i

iN N L i M i i

i i

Li M iiC i M b L b

i i

Ci iC b i

i

Pf a a b b f a f f a A f

a Pf a f f J a f f J a f

bf J a f

Mathematical framework

Example: shift on the Ask side following a sell market order

1

pour k>N

Mb i B

k

J a a

a a

1 0 / PB Inf P B

Stationarity

In this simple model, there exists a stationary distribution (Lyapunov function)…

… thanks to the “damping” effect of cancellations

This result is easily extended to state-dependent intensities (provided the Lyapunov function still exists)

Remark: a methodological difficulty Cancellations are not always identified in financial databases… … therefore, one of the key parameters for stationarity may not be

observable! Model calibration (Mike, Farmer, 2008) may be impaired

Price dynamics

Price dynamics depend on

Events affecting the best limits

The “first gap” process

A useful representation for the best Ask and Bid prices:

1

1

1 1 1

0

1 1 1

0

0 0

0 0

A

t

t

B

t

t

iA it t t t t t

i B

iB it t t t t t

i A

dP P A A dM dC B i dL

dP P B B dM dC A i dL

Price dynamics

The expressions above provide a natural interpretation of the price changes: they are due to

New limit orders that fall within the spread, for which one can safely assume some independence assumptions

Events that modify the best quotes (either cancellations or market orders), for which the price changes depends on the first gaps and

The price process has the following representation

A Bachelier market has a similar representation with i.i.d. marks The marks may be assumed to be identically distributed (under stationarity), but not independent.

The long time dynamics is extremely sensitive to the dependence structure of these processes

1 1 0t tA A 1 1 0t tB B

i

t it t

i

dP X dN

Price dynamics

A mathematical result (finally…)

The centered price process in a zero-intelligence model with proportional cancellation rate scales to a brownian motion in the

long time limit

A first general result relating order book models and classical price models

The “spurrious” randomness of the volatility due to the memory of the order book vanishes exponentially fast in this simple case

Extensions to state dependent intensities, Hawkes processes

Some interesting cases (ex. “heavy traffic” situations) are not covered

Order book modelling

Conclusion

Empirical studies of the order book A large body of empirical results Conditional quantities contain a lot of relevant information The behaviour of market participants at the best limits tends to “control” the

dynamics of price and spread Cancellation rate difficult to estimate (depending on data quality!)

Mathematical modelling A general framework suitable for many extensions A result bridging the gap between order book dynamics and price process The structure of cancellations is the clue for stationarity Plenty of room for improvement: phenomenological models for the intensities,

mean field games, heterogenous agents…