Random Walks, liquidity molasses and critical response in financial markets

Transcription

1 Random Walks, liquidity molasses and critical response in financial markets J.P Bouchaud, Y. Gefen, O. Guedj J. Kockelkoren, M. Potters, M. Wyart

2 Introduction Best known stylized fact in financial markets: price changes are weakly correlated approximate diffusion (Bachelier 1900) Efficient market theory: prices are fully rational and correspond to the best anticipation of future dividends price changes can only be due to unpredictable news; but: excess volatility, with long range, multiscale memory! More fundamentally: ambiguous information, psychological and cognitive biases, herding (cf. prediction of financial analysts!) Zero intelligence investors: Each trade has a totally random motivation, but has a non zero impact on the price each trade is considered by others as containing some information

3 Volatility clustering % 0 20 Dow Jones Index Random Walk Volatility clustering: comparison between the Dow Jones and a Brownian Random Walk

4 Volatility correlation <[σ 2 (t+τ) σ 2 (t)] 2 > Data (Averaged over 500 stocks) Fit, ν=0.22 Fit, exp( t/4) Fit, exp( t/40) τ (days)

5 Power-law response to volatility shocks - HF adjusted volatility (%) time (minutes) excess adjusted volatility (%) time (minutes)

6 Analysts herding behaviour 0.1 Σ, σ Σ S&P Σ US Σ EU σ S&P σ US σ EU θ Months With O. Guedj

7 Empirical facts on trades and quotes data Paris Bourse: fully electronic. Data form Example of a liquid stock: France Telecom 5000 trades/day 1.2 M-trades in 2002 Quotes: Bid price + Ask price midpoint m = (Bid+Ask)/2 Trades: At the Ask ε = +1, at the Bid ε = 1

8 The order book refresh island home disclaimer help QQQ GET STOCK QQQ go Symbol Search LAST MATCH Price Time 11:42: TODAY S ACTIVITY Orders 67,212 Volume 12,778,400 BUY ORDERS SHARES PRICE , , , , , , , , , , , (237 more) SELL ORDERS SHARES PRICE , , , , , , , , , (119 more)

9 Order flow and order book: New Stylized facts Many new quantities can be analyzed Statistics of the rain of incoming orders as a function of distance from current bid/ask Average size of the queue as a function of distance from current bid/ask Probability distribution of the size of the queue Collective modes of the order book Also: interaction between order book and price changes, between order flow and price changes ( Impact ) see below.

10 Statistics of the rain of orders As a function of the distance from the current bid/ask: Probability that a new order is placed is very broad up to 50% away from current price! Power law distribution P ( ) 1 µ with: µ 0.6 for (liquid) CAC40 stocks µ 1. for (liquid) NASDAQ stocks µ 1.5 for LSE stocks (Farmer & Zovko) Conditional average size of the order: Φ Φ 0 for, Φ ν for, with ν 1.5

11 Statistics of the rain of orders Total Vivendi F.T. Power-law, µ= Ρ( ) Note: same distribution for buy and sell orders

12 Shape of the order book As a function of the distance from the current bid/ask: The average size of the queue ρ( ) has a characteristic humped shape, with a maximum away from the bid (ask) Symmetric shape for buy and sell orders The shape is found to be stock independent for French stocks The shape can be different on NASDAQ stocks but not a centralized market!

13 The shape of the order book Vivendi Total France Telecom Numerical model

14 A simple analytical model I Orders at distance at time t are those which were placed there at a time t < t, and have survived until time t, that is: (i) have not been cancelled; (ii) have not been touched by the price at any intermediate time t between t and t. Therefore: ρ(, t) = t dt dup ( + u) P ( u C(t, t ) ) e Γ(t t ), where P ( u C(t, t ) ) is the conditional probability for the ask difference u = a(t) a(t ), such that + a(t) a(t ) 0, t [t, t].

15 A simple analytical model II Assuming that the price follows a Gaussian ranodm walk: ρ st ( ) = e α dup (u) sinh(αu)+sinh(α ) dup 0 (u)e αu, where α 1 = D/2Γ measures the typical variation of price during the lifetime of an order. When µ < 1, α can be rescaled away, and: ρ st ( ˆ ) = e ˆ ˆ du 0 u 1 µ sinh(u)+sinh( ˆ ) with ˆ = α. Note: for 0, ρ st ( ) 1 µ 0 hump! ˆ du u 1 µ e u Reproduces the numerical results satisfactorily

16 Comparison numerical model - analytical approx

17 Price dynamics: Diffusion Price fluctuations in trade time: (pn+l ) 2 D(l) = p n Dl Note : D(1) 0.01 Euros: True for all stocks. precisely the bid-ask spread.

18 Price Diffusion [D(l)/l] 1/2 (Euros) FT (2001 1st semester) FT (2001 2nd semester) FT (2002) Time (Trades)

19 Price Diffusion [D(l)/l] 1/2 (Arbitrary units) Total Pechiney Vivendi Barclays TF Time (Trades)

20 Price dynamics: Response function/market impact Average response function: R(l) = ( p n+l p n ) εn Weak growth as a function of l and then declines for l > l Response to a trade of volume V : R(l, V ) = ( p n+l p n ) εn Vn =V. Approximate factorisation: R(l, V ) ln V R(l) large volumes affect prices less than small volumes! (cf. Hasbrouck (1991), Gopikrishnan et al., Lillo et al.)

21 Average response R(l), [D(l)/l] 1/2 (Euros) FT (2002) FT (2001 1st semester) FT (2001 2nd semester) Time (Trades)

22 Average response R(l) l

23 Response: factorisation 0.04 log V=[2,3] log V=[3,4] log V=[4,5] log V=[5,6] log V=[6,7] log V=[7,8] log V=[8,9] log V=[9,10] Average R(l,V) Time (Trades)

24 Price dynamics: Fraction of informed trades Full distribution of u l = (p n+l p n ).ε n : R(l) = u l D(l) = u 2 l Only very small assymetry that disappears when u l is shifted by 0.01 Euros; skewness decays as l 1/2. Very few trades can be qualified as informed, i.e. correctly anticipating short term moves to at least cover minimal costs (cf. Do investors trade too much? Odean 1999)

25 Impact distribution l= unshifted u l shifted by 0.01 Euros P(u l ) u l < u l

26 Skewness Skewness (FT 2002, A=0) Skewness (FT2002, A=0.01) τ 1/2 0.4 Skewness Trades

27 Price dynamics: A fluctuation-response relation For Brownian random walks: Mobility = Diffusion/Temperature Similar relation in financial markets? Rosenow 2001 D(l) l = AR 2 (l) + B

28 Fluctuation-Response Relation 0.02 l= D FT 2002 FT e 05 5e 05 R 2

29 Market order flow: Long term memory Trade correlations: C(l) = ε n+l ε n C 0 l γ. with γ < 1 (γ 1/4 for FT, 1/2 for Vodafone see Lillo- Farmer) Paradox: The effective number of identical trades grows with l: N e l=1 C 0 (l) 1 + C 0 1 γ γ 50 R(l) should increase by a large factor and one should observe superdiffusion.

30 Trade correlations C(l) 10 1 C 2 (l) C 1 (l) C 0 (l) Time (trades)

31 A micro-model of price fluctuations Linear superposition of impacts: p n = n <n G 0 (n n )ε n ln V n + n <n η n, where G 0 (.) is the bare, non permanent response function (or propagator) of a single trade. Alternative model Lillo-Farmer: p n = n <n ε n V β n λ n + n <n η n : permanent, but fluctuating impact depending on instantaneous liquidity see discussion and comparison in condmat/

32 A simple case first Simple case: no correlation in signs D(l) R(l) G 0 (l) G 2 0 (n) + [G 0 (l + n) G 0 (n)] 2 0<n l n>0, For a permanent impact: Constant response and pure diffusion

33 Role of correlations More generally: R(l) = ln V G 0 (l)+ G 0 (l n)c 1 (n)+ [G 0 (l + n) G 0 (n)] C 1 (n) 0<n<l n>0 (and a more complicated equation for D(l). If G 0 were constant, then R(l) l 1 γ and D(l) l 2 γ the impact of single trades is itself non- Only way out: permanent G 0 (n) = R 0 (n 0 + n) β

34 Role of correlations Asymptotic behaviour: D(l) l 2 2β γ, R(l) l 1 β γ For diffusion to be normal: β = (1 γ)/2 3/8 but R(l) l 1 3/8 1/4 l 3/8 incompatible with data?? In fact: R(l) Γ(1 γ) Γ(β)Γ(2 β γ) [ π sin πβ π sin π(1 β γ) ] l 1 β γ

35 Theoretical and empirical response function FT 2002 G 0 =G 0 * β=0.38 β=0.40 β=0.42 β=0.44 R(l) Time (Trades)

36 Theoretical and empirical response function 10 2 CA EX G 0 (l) FP 10 3 ACA l

37 Interpretation: two antagonist categories of traders Liquidity takers: place market orders, as a result of true/putative information, or urge to buy/sell. Must limit their impact orders are cut in small pieces and create serial correlations due to their size. Liquidity providers: place limit orders, but no long term positions in markets. Must limit the fluctuations of the price slow mean reversal force: liquidity molasses. How: order barrier at the ask + anticorrelated quotes. Both populations compete such as to remove arbitrage opportunities (linear correlations), and impose β (1 γ)/2. Volatility may come from these trading rules alone, and only weakly from external news.

38 Proximity of the critical line Fit parameters β=(1 γ)/2 β γ

39 Conclusion: a critical dynamical equilibrium Price diffusion: result from a subtle competition (compensation) between persistent effects (liquidity takers, correlated orders) and antipersistent effects (liquidity providers, mean reverting forces). Both effects are characterized by scale-less, power-law functions of time Dynamical equilibrium between the two can be temporarily broken large, intermittent fluctuations and crashes. cf. Regulation of heart beats and anomalous statistics [H. E. Stanley et al.]; On-Off Intermittency in stick balancing task, [J. L. Cabrera and J. G. Milton, Phys. Rev. Lett. 89, (2002)]