Optimal Execution: IV. Heterogeneous Beliefs and Market Making

Transcription

1 Optimal Execution: IV. Heterogeneous Beliefs and Market Making René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Purdue June 21, 2012

2 The Agents Market Maker Nasdaq definition: agent that places competitive orders on both sides of the order book in exchange for privileges. Acts as a scaled-down version of the market. In this lecture: Liquidity provider, someone who posts an order book/transaction cost curve. Strategy: adapt pricing by reading client flows. Clients In this lecture: Liquidity takers, agents who trade with the Market maker. Are information driven.

3 Theoretical literature Early approaches: Hasbrouck(2007), Chakrborti - Toke - Patriarca - Abergel(2011) Inventory models: Garman(1976), Amihud - Mendelson(1980) Informed trader models: Kyle(1985), O Hara(1995) Zero-intelligence models: Gode - Sunder(1993), Maslov(2000), Cont(2008) Market impact models: Almgren - Chriss(2000), Bouchaud - Potters (2006), Schied(2007)

4 Objective Propose a stochastic, agent-based model in which existence and (tractable and realisitc) properties of the LOB appear as a result of the analysis (not as hypotheses) Client model Summarize sparsely the link between trade and price dynamics. Market maker model Tractable market making strategy based on previous result. R.C. - K. Webster (2012)

5 Setup: heterogeneous beliefs Let 1. (Ω, F, F = (F t ) t 0, P) with 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. 5. In L 2 and a.s. P t grows polynomially in t. NB Each agent has his /her own distinct filtration and probability measure. They are potentially mutually exclusive, but the price process is adapted to all of them

6 Anatomy of a trade Midprice P t announced by the market at time t Market maker proposes a transaction cost curve c t (l) around P t Market maker cannot differentiate clients pre-trade Client triggers a trade of volume l t Client obtains volume l t and pays cash flow P t l t + c t (l t ). Market maker tries to identify clients post-trade

7 Setup: transaction costs Agents behaviors Market maker controls transaction cost function l c t (l). Client i controls trading volumes/speeds l i t. Hypotheses 1. Marginal costs are defined: c C Clients may choose not to trade, c t (0) = 0 and the midprice is well defined, c t (0) = Marginal costs increase with volume: c t is convex. 4. c t has compact support.

8 Duality relationship Legendre transform γ t (α) := sup (αl c t (l)) l supp(c t ) Duality c t convex with compact support γ t is a positive finite measure. The distribution γ t represents the order book formed by the orders of the market maker.

9

10 Client model Client s Objective Summarize sparsely the link between trade and price dynamics in a general, theoretical framework. Not trying to build a optimal trading strategy. Assumptions The client only tries to predict, not cause price movements. The client s decision does not affect c t. Realistic if the client is small enough.

11 Client model Exogeneous state variables P t and c t are Itô processes. P t has polynomial growth and c t convex with compact support. Endogeneous state variables { dl i t = l i t dt dx i t = L i t dp t c t (l i t )dt L i i t is the total position of the client. Xt is his wealth, marked to the midprice. lt i, the rate at which he trades, is his control. Objective function [ ] J i = E P i U i (X i τ, p i τ i ) with τ i a stopping time.

12 Optimal trading strategy Theorem Under suitable integrability assumptions on U i and τ i, the optimal strategy is [ ] αt i := c t (lt i ) = E Q i p τ i P t Ft i with dqi = X U i (X i dp i τ i,p τ i ) [ E P i X U i (X i τ i,p τ i ) ].

13 Testing the client model Hypotheses Under Q i, τ i exp ( β i) independent of P t. σt i := c t (lt i ) }{{} (p τ i P t ) spread }{{} 2 Implied alpha Realized alpha This leads to a two parameter model linking trade to price dynamics: (β i, σ i ). Testing the hypotheses on data Assume all clients have one of two time scales. choose (β 1, β 2 ) that minimizes error between implied and realized alpha.

14 Source Nasdaq fullview data: all public quotes, all trades, nanosecond timestamps. Long parsing time: Data goes from 7:00-10:00am.

15 Two time scales implied realized L 1 regression used. Time scales: 9 ( 0.5 seconds) and 158 ticks. Mean error: Mean half-spread: Lower bound on error:

16 Market maker model Market Maker s Objectives Find a tractable market making strategy based on previous result. Build a theoretical model for the order book that replicates the empirical features described before. Strategy Exploit link between trade and price dynamics to dynamically adapt pricing.

17 Market maker model: endogenous variables With primal variables { dlt = 1 n i li t dt dx t = L t dp t + 1 n i c t(lt i)dt With dual variables { dlt = 1 n i γ t dx t = L t dp t + 1 n ( ) α i t [ dt ( ) ( )] α i t γ t α i t γt α i t dt i Assume the market maker is risk-neutral.

18 Model for the α i t Notation We will denote by µ t (α) the client belief distribution, that is, the empirically observed distribution of the ( α i t). Microscopic model(sde) dα i t = ρα i tdt + σdb i t + νdb t mean reversion corresponds to decay of information. Macroscopic 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

19 Approximate model for P t Intuition Do not want to make an explicit model for the price process. Instead, would like to infer the price from client trades. Implied alpha relationship α i t := c t (l i t ) = E Q i [ t ] e βi (t s) dp s Ft i Estimator with λ i = 1. dp λ t := n ( ) λ i β i αtdt i dαt i i=1

20 Estimation result Entropic feedback There exists λ s.t. E Pt p λ t 2 ɛ 2 1 n t E(Q i, P) ɛ 2 with E the entropy function and n ɛ = i (σi ) 2 1 n i i 0 log σ i ( ) γ s, µ s ds µ s

21 Approximate control problem State variables { 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 Objective function J λ = 0 under the constraint 0 e βt E [L t id, (βλ) t + L t βid + (id ᾱ t ) γ t γ t, µ t ] dt e βt log ( γ t µ t ), µ t dt C.

22 Pontryagin BSDE The solution to the Pontryagin BSDE gives rise to the market maker s shadow alpha : αt = id, λ t + (βλ) t βµ t β + ρ Hamiltonian H(γ, µ, α ) = (id α )γ γ + ɛ log γ, µ

23 Result Profitability of an order without feedback Define then we have: m(α) = (α α ) µ if α 0 }{{} α spread }{{} filling probability H(γ, µ, α ) = γ, m + ɛ log γ, µ Optimal strategy with feedback γ (α) µ(α) = ɛ C m(α) where C is a renormalization constant.

24 Simulated example Figure: Blue: Optimal order book γ. Green: Client alpha distribution µ.