Heterogeneous Beliefs and HF Market Making

Transcription

1 Heterogeneous Belefs and HF Market Makng René Carmona Bendhem Center for Fnance Department of Operatons Research & Fnancal Engneerng Prnceton Unversty Prnceton, June 21, 2013

2 The Agents Market Maker Nasdaq defnton: agent that places compettve orders on both sdes of the order book n exchange for prvleges. In ths lecture: Lqudty provder, someone who posts an order book (equvalently, a transacton cost curve). Strategy: adapt prcng and volumes by readng clent flows. Clents In ths lecture: Lqudty takers, agents who trade wth the Market maker. Clents place market orders. Each clent has hs/her own nformaton and acts accordngly.

3 Theoretcal lterature Early approaches: Hasbrouck(2007), Chakrbort - Toke - Patrarca - Abergel(2011) Inventory models: Garman(1976), Amhud - Mendelson(1980) Informed trader models: Kyle(1985), O Hara(1995) Zero-ntellgence models: Gode - Sunder(1993), Maslov(2000), Cont(2008) Market mpact models: Almgren - Chrss(2000), Bouchaud - Potters (2006), Sched(2007)

4 Objectve: Endogenous Order Book Propose a stochastc, agent-based model n whch exstence and (tractable and realstc) propertes of the LOB appear as a result of the analyss (not as hypotheses) Clent model Should capture the dependence between trades and prce dynamcs. Market maker model Assumes the clents are ratonal, and optmzes hs/her order book choce R.C. - K. Webster (2012)

5 Setup: Heterogeneous Belefs Mathematcally 1. (Ω, F, F = (F t ) t 0, P) wth W a P-BM that generates F. 2. F k F generated by a P-BM W k. 3. P k s.t. P k F k t P F k t. 4. P t an Itô process adapted to all ( F k). k=0...n NB Each agent has hs /her own fltraton & probablty measure. The fltratons (nformaton structures) are potentally dfferent, The prce process s adapted to all of them (.e each clent sees the prce)

6 Anatomy of a Trade Mdprce P t announced by the market at tme t Market maker proposes an order book around P t Market maker cannot dfferentate clents pre-trade Clent trggers a trade of volume l t Clent obtans volume l t and pays cash flow P t l t + c t (l t ) (l c t (l) transacton cost functon at tme t) Market maker learns the dentty of the clent post-trade (assumpton depends upon market, true for FX)

7 Setup: Transacton Costs Agents behavors Market maker controls transacton cost functon l c t (l). Clent controls tradng volumes/speeds l t. Hypotheses 1. Margnal costs are defned: l c t (l) s dfferentable n l. 2. Clents may choose not to trade, c t (0) = 0 3. The mdprce s well defned, c t (0) = Margnal costs ncrease wth volume: c t s convex. 5. c t has compact doman ( outsde an nterval)

8 Dualty Relatonshp Legendre transform γ t (α) := sup (αl c t (l)) l supp(c t ) Dualty c t convex wth compact doman γ t s a postve fnte measure. The dstrbuton γ t represents the order book formed by the orders of the market maker. If γ t has a densty f (x), t s the shape functon we used earler.

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10 Clent Model Dsclamer: We are NOT tryng to mplement an optmal tradng strategy. Assumptons The clent only tres to predct, not cause prce movements. The clent s decson does not affect c t.

11 Clent Optmzaton Problem Exogeneous state varables Pt non-negatve Itô process ct (random adapted) convex functon n a fxed doman Endogeneous state varables { dl t = lt dt dx t = L t dp t c t (l t )dt l t rate at whch clent trades (control varable). L t volume or total poston of the clent X t wealth, marked to the md-prce. Objectve functon U τ utlty functon stoppng tme J = E P [ ] U (X τ, P τ )

12 Optmal Tradng Strategy Theorem Under sutable ntegrablty assumptons on U and τ, the optmal strategy s [ ] αt := c t (lt ) = E Q P τ P t Ft wth dq = X U (X dp τ,p τ ) [ E P X U (X τ,p τ ) ].

13 Testng the Clent Model Hypotheses Under Q, τ exp ( β ) ndependent of P t. σt := c t (lt ) }{{} (p τ P t ) spread }{{} 2 Impled alpha Realzed alpha Ths leads to a two parameter model lnkng trade to prce dynamcs: (β, σ ). Testng the hypotheses on data Assume all clents have one of two tme scales. choose (β 1, β 2 ) that mnmzes error between mpled and realzed alpha.

14 Source Nasdaq fullvew data: all publc quotes, all trades, nanosecond tmestamps. Long parsng tme: Data goes from 7:00-10:00am.

15 Two Tme Scales mpled realzed L 1 regresson used. Tme scales: 9 ( 0.5 seconds) and 158 tcks. Mean error: Mean half-spread: Lower bound on error:

16 Market Maker Optmzaton Problem Wth prmal varables { dlt = 1 n l t dt dx t = L t dp t + 1 n c t(lt )dt Recall α t = c t (l t ) so equvalently l t = [c t ] 1 (α t ) = γ t (α t ) Wth dual varables { dlt = 1 n γ t dx t = L t dp t + 1 n ( ) α t [ dt ( ) ( )] α t γ t α t γt α t dt We assume the market maker s rsk-neutral

17 Model for the α t Notaton We wll denote by µ t (α) the clent belef dstrbuton, that s, the emprcally observed dstrbuton of the ( α t). Mcroscopc model(sde) dα t = ρα tdt + σdb t + νdb t mean reverson corresponds to decay of nformaton. Macroscopc model(spde) [ 1 ( dµ t (α) = σ 2 + ν 2) ] µ t (α) + ρ (αµ t (α)) dt ν µ t (α)db t 2

18 What does that tell us about P t? Intuton Do not want to make an explct model for the prce process. Instead, would lke to nfer the prce from clent trades. Impled alpha relatonshp Prce Proxy α t := c t (l t ) = E Q dp λ t := [ t ] e β (t s) dp s Ft n ( ) λ β αtdt dαt =1 for any set of weghts λ s.t. λ = 1.

19 Estmaton Result Entropc feedback There exsts λ s.t. E Pt P λ t 2 ɛ 2 1 n t E(Q, P) ɛ 2 0 log wth E the relatve entropy (Kullback - Lebler) and n ɛ = (σ ) 2 1 σ n ( ) γ s, µ s ds µ s

20 Approxmate Control Problem State varables { dlt = γ t, µ t dt dµ t (α) = [ ( 1 2 σ 2 + ν 2) µ t (α) + ρ (αµ t (α)) ] dt ν µ t (α)db t Objectve functon J λ = 0 under the constrant 0 e βt E [L t d, (βλ) t + L t βd + (d ᾱ t ) γ t γ t, µ t ] dt e βt log ( γ t µ t ), µ t dt C.

21 (Pontryagn) Stochastc Maxmum Prncple BSDE The soluton to the Pontryagn BSDE gves rse to the market maker s shadow alpha : αt = d, λ t + (βλ) t βµ t β + ρ Hamltonan H(γ, µ, α ) = (d α )γ γ + ɛ log γ, µ

22 Result Proftablty of an order wthout feedback Defne then we have: m(α) = (α α ) µ f α 0 }{{} α spread }{{} fllng probablty H(γ, µ, α ) = γ, m + ɛ log γ, µ Optmal Strategy wth Feedback γ (α) µ(α) = ɛ C m(α) where C s a renormalzaton constant.

23 Smulaton Example Fgure : Blue: Optmal order book γ. Green: Clent alpha dstrbuton µ.