Market Micro Structure knowledge needed to control an intra-day trading process

Transcription

1 Market Micro Structure knowledge needed to control an intra-day trading process Gobal Head of Quantitative Research Crédit Agricole Chevreux Based on joint works with B. Bouchard and M. Dang (Impulse control); G. Pagès and S. Laruelle (stoch. algo); O. Guéant and J. Fernandez Tapia (HJB for MM); O. Guéant and J. Razafinimanana (MFG) Printed November 17, 2011

2 Optimal trading? Financial mathematics evolve to control (hedge / replicate) the first order risk of the major actors: Asset managment is about controlling the risk of holding a static portfolio, Derivative pricing is about replicating the risk of a non linear inventory, now it is the turn of the risk of modifying an inventory: liquidity risk highly related to market micro-structure This talk is a journey across the key elements of liquidity risk control using financial mathematics: optimal / quantitative trading.

3 Market micro-structure Outline Introduction Usual formal tools for optimal execution Market micro-structure Execution and trading Trade scheduling Practical and new elements Statistical invariants Algorithmic trading Stochastic algorithms Market making Back and stress-testing: our greeks Conclusion

4 Market micro-structure Market microstructure recent changes Automated trading takes place on electronic markets In statistics and modelling, the more knowledge you have on the phenomena, the less data you will need to achieve a given accuracy First understand (and model) the environment of your algorithms Micro-structure, order-books definition, auction mechanisms, market-impact...

5 Market micro-structure Previous organization of Equity market microstructure

6 Market micro-structure Actual market organisation of Equity markets US: Reg NMS (National Market System) / EU: MiFID (Market in Financial Instruments Directive)

7 Market micro-structure Electronic trading Before:

8 Market micro-structure Electronic trading Now:

9 Market micro-structure Continuous auctions For continuous auctions, once the bid-ask spread is created at the end of the open fixing, the match takes place continuously: The universe is discrete and event driven

10 Market micro-structure Continuous auctions Actors Liquidity provider Wait to be executed (not executed for sure) Execution price is better than the last quoted one (sometimes) pays less fees Liquidity consumer Executed for sure Pays around half a bid-ask spread more than the last quoted price Pays Price Impact for large quantities

11 Execution and trading Outline Introduction Usual formal tools for optimal execution Market micro-structure Execution and trading Trade scheduling Practical and new elements Statistical invariants Algorithmic trading Stochastic algorithms Market making Back and stress-testing: our greeks Conclusion

12 Execution and trading Market impact (volume driven) demands to trade slowly A good way to reduce your Market Impact is to trade on a larger time interval T. v cst V T with T trade as slowly as possible! } αψ + F ( σ, v V T ) with T

13 Execution and trading Market risk (volatility driven) demands to trade fast The more you wait and the more you tale market risk. For an arithmetic Brownian diffusion, the amplitude of the risk is: trade as fast as possible! a σ T

14 Execution and trading Those two effects are mixed

15 Trade scheduling Outline Introduction Usual formal tools for optimal execution Market micro-structure Execution and trading Trade scheduling Practical and new elements Statistical invariants Algorithmic trading Stochastic algorithms Market making Back and stress-testing: our greeks Conclusion

16 Trade scheduling The Almgren-Chriss framework (simplified) I have to buy V shares between time 0 and T ; I assume a regular temporal grid with a time step of δt (n goes from 1 to N = [T /δt]). My volume is splitted in N slices v n such that n v n = V ([Almgren and Chriss, 2000]) The price follows a Brownian motion: (1) S n+1 = S n + α δt + σ n+1 δt ξn+1 The additive, temporary only, market impact function is η n (v n ). Then the total cost is: N (2) W = v n (S n + η n (v n )) n=1 Of course, far more sophisticated models have been studied: [Almgren and Lorenz, 2007], [Almgren, 2009], [Bouchard et al., 2011], [Gatheral et al., 2010], Charles-Albert [Alfonsi Lehallet al., 2010]

17 Trade scheduling Classical solution When the market impact is linear with respect to the participation rate v n /V n and proportional to the volatility (change of variable x n = N k=n v n). W = V S 0 } {{ } immediate cost in a free world + N x n σ n ξ n + n=1 } {{ } market risk N n=1 ησ n v 2 n V n } {{ } market impact For a broker algo, we want to minimize a mean-variance criteria (no more than Markowitz through time): J λ = E(W V 1,..., V N, σ 1,..., σ N )+λv(w V 1,..., V N, σ 1,..., σ N ) We obtain a recurrence equation in x n : ( x n+1 = 1 + σ n 1 V n + λ σ n V n 1 η σ2 n ) x n σ n 1 σ n V n V n 1 x n 1

18 Statistical invariants Outline Introduction Usual formal tools for optimal execution Market micro-structure Execution and trading Trade scheduling Practical and new elements Statistical invariants Algorithmic trading Stochastic algorithms Market making Back and stress-testing: our greeks Conclusion

19 Statistical invariants We need to capture statistical invariants No risk-neutral measure available, so we will work under historical measure, any statistical knowledge is welcome!

20 Statistical invariants Intra day seasonality - what is stationary? Usual phases (in Europe): Open: uncertainty on prices and unwind of the overnight positions Macro economic news NY opens Volatility Volumes

21 Statistical invariants Measure or estimate: the case of Intra day volatility Proportion of volatility seems to be more stable by volatility itself How could we use proportion of volatility? But measuring volatility itself is difficult Our prices are discretized (rounded?) on a price grid A lot of interesting papers have been written on this subject: Jacod, Delattre, Aït-Sahalia, Zhang, Mykland, Shepard, Rosenbaum, Yoshida (covariance); [Aït-Sahalia and Jacod, 2007], [Zhang et al., 2005], [Hayashi and Yoshida, 2005], [Robert and Rosenbaum, 2010]

22 Statistical invariants A simple model: (3) X n+1 = X n + σ δt ξ n, S n = X n + ε (S n+1 S n ) 2 = 2nE(ε 2 ) + O( n) n Courtesy of Mathieu Rosenbaum

23 Statistical invariants Limit order book matching mechanism

24 Statistical invariants Dependencies: spread - volatility Spread effect on three stocks

25 Statistical invariants Dependencies: spread - volatility - traded volumes Charles-Albert Give Lehalle me two of the three curves, I will give you the missing one...

26 Algorithmic trading Outline Introduction Usual formal tools for optimal execution Market micro-structure Execution and trading Trade scheduling Practical and new elements Statistical invariants Algorithmic trading Stochastic algorithms Market making Back and stress-testing: our greeks Conclusion

27 Algorithmic trading Optimal liquidation under estimation assumptions: a practical point of view Volumes and volatility change from one day to another (here two years of volume and its log for a French large cap between 11:45 and 12:30): You have to consider that V n is a random variable/process, or that you have only access to ˆV n, an estimator of V n, with a variance.

28 Algorithmic trading A new set of equations When the V n and σ n are i.i.d. random variables, and when V n and ξ n are correlated: V(W ) = E(W ) = v S 0 + N xnσ 2 n 2 + n=1 + N η 2 V n=1 N η E n=1 N n=1 ( σn V n ( σn V n ) (x n x n+1 ) 2 ( ξn η x n (x n x n+1 )E(σ n ) E ) (x n x n+1 ) 4 It is easy to add a lot of other effects like: noise on expected market impact, auto-correlations, volume-volatility coupled-dynamic. No more closed-form formula. V n )

29 Algorithmic trading Results The obtained results are obviously different, but what does it means? When the noise is higher than the estimated market impact, the optimal trajectory focus on market risk (no use to try to optimise on a variable not accurate enough)

30 Algorithmic trading A real execution: organisation of an algo in two layers Now that you know your instantaneous optimal trading rate (if the market context is the expected one), how can you really capture this flow at the best possible price? A tactical layer has to follow the targeted trading rate.

31 Algorithmic trading Why tactics are so important? robustness of the whole algorithmic cinematics (a global optimisation could suffer from polluted HF data) an unique place dedicated to address the specificities of each trading pool a need to react quickly as possible to behavioural changes trade scheduling is about backward solving (stochastic control) with an horizon of several hours (very long) relying of statistical invariants through days, tactical trading is about forward solving (stochastic algorithms): filtering immediate past to optimise the present

32 Stochastic algorithms Outline Introduction Usual formal tools for optimal execution Market micro-structure Execution and trading Trade scheduling Practical and new elements Statistical invariants Algorithmic trading Stochastic algorithms Market making Back and stress-testing: our greeks Conclusion

33 Stochastic algorithms Stochastic algorithms The stationary solutions of the ODE: ẋ = h(x) contains the extremal values of F (x) = x 0 h(x) dx A discretized version of the ODE is (γ is a step): (4) x n+1 = x n + γ n+1 h(x n ) A stochastic version of this being (ξ n are i.i.d. realizations of a random variable, h(x ) = E(H(X, ξ 1 ))): (5) X n+1 = X n + γ n+1 H(X n, ξ n+1 ) the stochastic algorithms theory is a set of results describing the relationship between these 3 formula and the nature of γ, H, h and ξ [Hirsch and Smith, 2005], [Kushner and Yin, 2003], [Doukhan, 1994]

34 Stochastic algorithms Stochastic algorithms theory can be used when you only have a sequential access to a functional you need to minimize: to minimize a criteria E(F (X, ξ 1 )) of a state variable X if it is possible to compute: H(X n, ξ n+1 ) := F X (X n, ξ n+1 ) the results of the stochastic algorithms theory (like the Robbins-Monro theorem [Pagès et al., 1990]) can be used to study the properties of the long term solutions of the recurrence equation: X n+1 = X n + γ n+1 H(X n, ξ n+1 )

35 Stochastic algorithms Optimal flow capture in the dark: can be extended to any routing problem at high frequency, historical statistics are not so useful the limit price S and the quantity V are random variables, the executed quantity on dark pools has to be maximized (it is market impact free) and sometimes fees are different; this effect is modelled by a discount factor θ i (0, 1) (normalized with respect to a reference Lit pool) the quantity V is split into N parts (one for each DP): r i V is sent to the ith DP ( N i=1 r i = 1) (see [Pagès et al., 2009])

36 Stochastic algorithms Cost of the executed order The remaining quantity is to be executed on a reference Lit market, at price S. The cost C of the whole executed order is given by C = S = S ( N θ i min (r i V, D i ) + S V i=1 ( V ) N ρ i min (r i V, D i ) i=1 ) N min (r i V, D i ) i=1 where ρ i = 1 θ i (0, 1), i = 1,..., N.

37 Stochastic algorithms Mean Execution Cost Minimizing the mean execution cost, given the price S, amounts to: Maximization problem to solve { N } (6) max ρ i E (S min (r i V, D i )), r P N i=1 { where P N := r = (r i ) 1 i N R N + } N i=1 r i = 1. It is then convenient to include the price S into both random variables V and D i by considering Ṽ := V S and D i := D i S instead of V and D i. Assume that the distribution function of D/V is continuous on R +. Let ϕ(r) = ρ E (min (rv, D)) be the mean execution function of a single dark pool (Φ = i ϕ i(r i )), and assume that V > 0 P a.s. and P(D > 0) > 0

38 Stochastic algorithms The Lagrangian Approach We aim at solving the following maximization problem (7) max r P N Φ(r) The Lagrangian associated to the sole affine constraint is ( N ) (8) L(r, λ) = Φ(r) λ r i 1 So, L i I N, = ϕ r i(r i ) λ. i This suggests that any r arg max PN Φ iff ϕ i (r i ) is constant when i runs over I N or equivalently if (9) i I N, ϕ i(ri ) = 1 N ϕ N j(rj ). i=1 j=1

39 Stochastic algorithms Design of the stochastic algorithm Using the representation of the derivatives ϕ i yields that, if Assumption (C) is satisfied, then Characterization of the solution r arg max Φ i {1,..., N}, P N E V ρ i 1 {r i V <D i} 1 N ρ j 1 N {r j V <D j} = 0. Consequently, this leads to the following recursive zero search procedure (10) r n+1 i j=1 = r n i + γ n+1 H i (r n, Y n+1 ), r 0 P N, i I N,

40 Stochastic algorithms where for i I N, every r P N, every V > 0 and every D 1,..., D N 0, H i (r, Y ) = V ρ i 1 {ri V <D i } 1 N N ρ j 1 {rj V <D j} with (Y n ) n 1 a sequence of random vectors with non negative components such that, for every n 1, j=1 (V n, D n i, i = 1,..., N) d = (V, D i, i = 1,..., N). The underlying idea of the algorithm is to reward the dark pools which outperform the mean of the N dark pools by increasing the allocated volume sent at the next step (and conversely).

41 Market making Outline Introduction Usual formal tools for optimal execution Market micro-structure Execution and trading Trade scheduling Practical and new elements Statistical invariants Algorithmic trading Stochastic algorithms Market making Back and stress-testing: our greeks Conclusion

42 Market making Market making in equations Another style of trading is to be massively liquidity provider (see [Menkveld, 2010] for an in depth study of the behaviour of HF MM): maintain an inventory contributing to the bid S b and ask S a. Provided that the way your orders are consumed is a Poisson process which intensity being a function of your distance to the fair (i.e. mid) price: introduce a Bellman function u defined as: u(t, x, q, s) = sup S a,s b E [ exp ( γ(x T + q T S T )) x, s, q]

43 Market making The Hamilton-Jacobi-Bellman equation for u is (see [Avellaneda and Stoikov, 2008]): (HJB.1) 0 = t u(t, x, q, s) σ2 2 ssu(t, x, q, s) [ ] + sup λ b (s b, s) u(t, x s b, q + 1, s) u(t, x, q, s) s b + sup λ a (s a, s) [u(t, x + s a, q 1, s) u(t, x, q, s)] s a with the final condition: u(t, x, q, s) = exp ( γ(x + qs))

44 Market making With the change of variable u(t, x, q, s) = exp( γ(x + qs))v q (t) γ k it is enough (see [Guéant et al., 2011] ) to consider a family of strictly positive functions (v q ) q Z solution of the linear system of ODEs that follows: q Z, v q (t) = αq 2 v q (t) η (v q 1 (t) + v q+1 (t)) with v q (T ) = 1, where α = k 2 γσ2 and η = A(1 + γ k k ) (1+ γ ). Moreover, you have asymptotics: δ (q) b 1 ( γ ln 1 + γ ) + 2q + 1 σ 2 γ k 2 2kA δ (q) a 1 ( γ ln 1 + γ ) 2q 1 k 2 ( 1 + γ ) 1+ k γ k σ 2 γ ( 1 + γ ) 1+ k γ 2kA k

45 Market making Application Details for the quotes and trades when the strategy is used on AXA, 02/11/2010 with γ = Thin lines : the market ; bold lines : quotes of the market maker. Dotted lines : bid side ; plain lines: ask side. Black points : obtained trades.

46 Market making The associated inventory

47 Market making Extension: passive brokerage The Bellman function u is now defined as: u(t, x, q, s) = sup S a E [ exp ( γx T ) 1 qt =0 1 qt >0 x, s, q] The Hamilton-Jacobi-Bellman equation for u is: (HJB.2) 0 = t u(t, x, q, s)+µ s u(t, x, q, s)+ 1 2 σ2 ss u(t, x, q, s) + sup λ a (s a, s) [u(t, x + s a, q 1, s) u(t, x, q, s)] s a with the final condition: u(t, x, q, s) = exp ( γx) 1 q=0 1 q>0 and the boundary condition: u(t, x, 0, s) = exp ( γx)

48 Market making Again a change of variables: u(t, x, q, s) = A γq k exp( γ(x + qs))wq (t) γ k to obtain the linear system of ODEs (S) that follows: q Z, ẇ q (t) = (αq 2 βq)w q (t) ηw q 1 (t) with w q (T ) = 1 q=0 and w 0 = 1, where α = k 2 γσ2, β = kµ and η = (1 + γ k ) (1+ k γ ) and the optimal ask quote can be expressed as: ( ( ) 1 s a (t, q, s) = s + k ln wq (t) + 1k w q 1 (t) ln(a) + 1γ ( ln 1 + γ ) ) k

49 Market making Example of a passive sell order Trading example. The strategy is used with γ = 1 (euro 1 ) to get rid of a quantity of shares equal to 20 times the ATS within 2 hours. quotes of the trader (bold line), market best bid and ask quotes (thin lines).

50 Back and stress-testing: our greeks Outline Introduction Usual formal tools for optimal execution Market micro-structure Execution and trading Trade scheduling Practical and new elements Statistical invariants Algorithmic trading Stochastic algorithms Market making Back and stress-testing: our greeks Conclusion

51 Back and stress-testing: our greeks Formalisation of back and stress testing I A trading algorithm lives in a state space made of three kinds of variables: The instantaneous market data X t : it is usually the state of the Limit Order Book (LOB) on several trading venues, trades when they occur, news, etc. a trading algorithm has to filter the instantaneous market data X t to build its view on the state of the market: Y t. It takes the market data into account via an updating rule H θ : Y t = H θ (Y t δt, X t ) where θ is a set of static parameters.

52 Back and stress-testing: our greeks Formalisation of back and stress testing II And the internal state of the algorithm Z t containing for instance its inventory, its pending orders in the books, its risk constraints, etc. The trading algorithm has to take decisions based on available information: D t = F θ (Y t, Z t ) Trading between t = 0 and t = T, an algorithm tries to maximize: ( T ) V T (H, F, θ) = E g(z t ) dt + G(Z T ) t=0

53 Back and stress-testing: our greeks What will be the real outcome of an algo Its actions will change the behaviour of the market: you cannot be sure that an identified pattern in the order-books will be there once you identified it statistically So it is not usual backtesting... What can be done? Anyway, the use of a market replayer will garantee that even having no influence on the market, your behaviour will have a positive outcome You also need to know what to expect if the market behaviour changes

54 Back and stress-testing: our greeks How to control the impact of unexpected changes? you need a market simulator to inject assumptions on the influence of your algo on the market and measure what can appen (for instance you can base it on a Mean Filed Game-based model [Lehalle et al., 2010]) It is reasonnable to think that the real outcome will be close to: V T (H, F, θ) + V T α i dx i X i i (X i span the state space of the market, it is hidden inside V T via F θ H θ ) but you do not know the α i at least you can measure the V T / X i.

55 Back and stress-testing: our greeks Vega - sensitivity to intra day volatility: V := V T (H, F, θ) σ Psi - sensitivity to bid-ask spread: Ψ := V T (H, F, θ) (bid-ask spread) Phi - sensitivity to trading frequency: Φ := V T (H, F, θ) (trading frequency) Iota - sensitivity to imbalance between sell and buy orders: ι := V T (H, F, θ) (sell vs buy imbalance)

56 Back and stress-testing: our greeks Focus on back tests Naive backtesting

57 Back and stress-testing: our greeks Focus on back tests Really backtesting

58 Conclusion Two layers: a risk control one (trade scheduling) and a tactical one (interactions with order books, adding and removing liquidity) Formal models are important, but they must not hide that you work under historical measure (estimates) We cannot throw away the uncertainty / inaccuracy of parameters and conduct optimisation as if they were deterministic There is intra-day seasonality almost everywhere (use of statistics) We cannot afford to miss a temporary behaviour change of the market (use of on-line estimates / optimisation, i.e. stochastic algo) We need to confront our ideas to data flows reproducing market behaviour (use of order books dynamics modelling, for instance MFG-based)

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