2229-12 School and Workshop on Marke Microsrucure: Design, Efficiency and Saisical Regulariies 21-25 March 2011 Some mahemaical properies of order book models Frederic ABERGEL Ecole Cenrale Paris Grande Voie des Vignes 92290 Chaenay Malabry FRANCE
Some empirical and mahemaical properies of limi order books Frédéric Abergel Chair of Quaniaive Finance École Cenrale Paris hp://fiquan.mas.ecp.fr
Limi order books Join works (some in progress) wih I. Muni Toke, A. Jedidi References I. Muni Toke, Marke making behaviour and is impac on he Bid-Ask spread, in Econophysics of Order-driven Markes, Abergel, F.; Chakrabari, B.K.; Chakrabori, A.; Mira, M. (Eds.), Springer, 2011 F. Abergel, A. Jedidi, A mahemaical approach o order book modelling, hp://papers.ssrn.com/sol3/papers.cfm?absrac_id=1740889 F. Abergel, A. Chakrabori, I. Muni Toke, M. Pariarca, Econophysics I: empirical facs and Econophysics II: agen-based models, o appear in Quaniaive Finance
Limi order books Summary Empirical properies of he order book Saionary saisical properies Dynamical saisical properies Mahemaical models Mahemaical framework Price dynamics
Limi order book
Saisical properies I A hos of empirical sudies going back o ~20 years, addressing he wo following quesions: When will he nex even ake place? Where will he nex even ake place? Under independence assumpions: Zero-inelligence models
Saisical properies I Such uncondiional saisics do no fully reflec he dynamics of a limi order book. Many ineresing phenomena are no described his way Volailiy clusering Leverage Auocorrelaion of he order flow In real markes, agens observe he sae of he marke and adap o i An example (Muni-Toke): empirical evidence of marke making and marke aking
Saisical properies II Marke making Following a marke order New limi orders arrive more rapidly han uncondiional limi orders No significan correlaion beween he respecive signs of he marke and limi orders
Saisical properies II Marke making
Saisical properies II Marke aking Following a limi order New marke orders do no arrive more rapidly excep when he limi order fell wihin he spread
Saisical properies II Marke aking
Saisical properies II Several recen sudies accouning for dependencies (Large 2007, Muni Toke 2010, Eisler 2010) Condiional iner-even duraion Lead and lag relaionship Condiional price and volume disribuions lead o models involving Sae-dependen inensiies and placemen Muually excied processes
Mahemaical framework The limi order book: a vecor valued poin process Main quesions o be addressed Saionariy Price and spread dynamics Scaling and long ime asympoics
Mahemaical framework Back o he simples example: zero-inelligence model wih limi orders, marke orders and cancellaions (Farmer, Smih, Guillemo, Krishnamurhy, 2003) dl dm dc λ i ± i ± L λ ± ± M λ Δ P τ 1 τ a i, τ λ i ± i + i C C b i τ
Mahemaical framework Two ses of variables Coupled dynamics a,..., a ; b,..., b 1 N 1 i i ( ) ( ) a a iδp b b iδp A = a B = b i i i i k= 1 k= 1 N i i Two basic ypes of evens Jump: a change in he quaniies Shif : renumbering afer a change of one of he bes quoes
Saionariy In his simple model, here exiss a Lyapunov funcion (he oal available volume), hanks o he exponenial damping effec of cancellaions Therefore, here exiss a saionary disribuion wih exponenial convergence This resul can be generalized o sae-dependen inensiies
Exension o Hawkes processes Hawkes processes: a poin process wih sochasic inensiy The inensiy is excied by he previous jumps (auoregressive process) N j j ( ) P = 0 + jp s dns p = 1 λ λ ϕ Typical choice: exponenial kernels N j j β jp ( s) P = + 0 jpe dns p= 1 λ λ α Becomes a Markov process in 1D (or higher wih equal decay raes)
Exension o Hawkes processes Clusering of orders easily described Leverage modelled hanks o asymeric kernels Saionariy condiions relaed o he values of he Hawkes parameers α 1 ( ( j )) jp ( j E λ = Id λ ) 0 β jp Saionariy condiions are found saisfied in empirical sudies (Muni Toke, Hewle, Large )
Hawkes processes Spread disribuion A consequence of beer modelling: spread disribuion
Price dynamics Price dynamics depend on Evens affecing he bes limis The firs gap process A useful represenaion for he bes Ask and Bid prices: 1 1 ia + 1 dp P (( A ( τ ) A ( 0) )( dm dc )) ( ( 0) ) B i dl 1 ( 0) + A + i+ =Δ + i< B 1 1 ib 1 dp P (( B ( τ ) B ( 0) )( dm dc )) ( ( 0) ) A i dl 1 ( 0) + B i = Δ + + i< A
Price dynamics The expressions above provide a naural inerpreaion of he price changes: hey are due o New limi orders ha fall wihin he spread, for which one can safely assume some independence assumpions Evens ha modify he bes quoes (eiher cancellaions or marke orders), for which he price changes depends on he firs gaps 1 A τ A 1 0 and 1 B τ B 1 0 ( ( ) ( )) ( ) ( ) ( ) The price process has he following represenaion i X dn A Bachelier marke has a similar represenaion wih i.i.d. marks The marks may be assumed o be idenically disribued (under saionariy), bu no independen. dp = i i The long ime dynamics is sensiive o he dependence srucure of hese processes
Price dynamics A mahemaical resul The cenered price process in a zero-inelligence model wih proporional cancellaion rae scales o a brownian moion in he long ime limi No a surprise from he physicis s poin of view A firs general resul relaing order book models and classical price models The spurrious randomness of he volailiy due o he memory of he order book vanishes exponenially fas in his simple case Exensions o sae dependen inensiies, Hawkes processes
Price dynamics The case of local (endogenous) or sochasic (exogenous) inensiies allows one o mimick some classical local and sochasic volailiy models The leaner he order book, he closer he dynamics is o sandard diffusion models Long memory may appear in he case of slow cancellaion raes, slow decay kernel
Order book modelling Conclusion Empirical sudies of he order book A large body of empirical resuls Condiional quaniies conain a lo of relevan informaion The behaviour of marke paricipans a he bes limis ends o conrol he dynamics of price and spread Mahemaical modelling A general framework suiable for many exensions An approach bridging he gap beween order book dynamics and price process