New stylized facts in financial markets: The Omori law and price impact of a single transaction in financial markets

Transcription

1 Observatory of Complex Systems, Palermo, Italy Rosario N. Mantegna New stylized facts in financial markets: The Omori law and price impact of a single transaction in financial markets work done in collaboration with F. Lillo and J.D. Farmer 08/09/03 Frontier Science 2003, Pavia

2 Observatory of Complex Systems, Palermo, Italy Overview New stylized facts have been recently observed in financial markets; I discuss two examples observed in the equity markets: (i) the Omori law detected after a market crash; (ii) the price-impact function of a single transaction; Conclusion. 08/09/03 Frontier Science 2003, Pavia 2

3 The Omori law is governing the dynamics of the number of aftershocks occurring after a major earthquake. n( t) p t ; (i)the Omori law in geophysics N( t) = t 0 n( s) ds Cumulative number of aftershocks in the earthquake occurring in eastern Pyrenees on February 18, 1996 (from Moreno et al., J. of Geophys. Res., 106 B4, (2001)) 08/09/03 Frontier Science 2003, Pavia 3

4 Relaxation after a market crash The stochastic dynamics of the price of an asset traded in a financial markets is altered after a major financial crash. Examples of dynamical changes have been detected in: - the dynamics of implied volatility (1) ; - the dynamics of the variety of a portfolio (2) ; - the leverage effect (3). (1) Sornette, Johansen and Bouchaud, J. Phys. I 6, 167 (1996). (2) Lillo and Mantegna, Eur. Phys. J. B20, 503 (2001). (3) Bouchaud, Matacz and Potters, Phys. Rev. Lett. 87, (2001). 08/09/03 Frontier Science 2003, Pavia 4

5 Observatory of Complex Systems, Palermo, Italy The time series of index returns after a major financial crash shows a non-stationary time pattern S&P 500 Index after the Black Monday financial crash (19 Oct 1987). one-minute return 08/09/03 Frontier Science 2003, Pavia 5

6 We fit empirical data with the Omori functional form Observatory of Complex Systems, Palermo, Italy Specifically, we measure the cumulative number of S&P500 index returns exceeding a given threshold nσ Lillo and Mantegna, PRE 68, (2003) N(t)=K[(t+τ) 1-p - τ 1-p ]/(1-p) 08/09/03 Frontier Science 2003, Pavia 6

7 We assume: - a return pdf with power-law tails f s (r s ) r s (-1-α) - a scale γ(t) of the return pdf power-law decaying γ(t) t -β A simple model Under these assumptions we have: n(t) (γ(t)/l ) α Consistent with Omori law n(t) (1/t ) p If 08/09/03 Frontier Science 2003, Pavia 7

8 In addition to the 1987 crash we also investigate other two major financial crashes: (a) the crash; (b) the crash. Other crashes 08/09/03 Frontier Science 2003, Pavia 8

9 A proxy for stationary index returns By investigating the quantity We have a proxy of the stationary return time series r s (t) We estimate α for r p (t) by determining the Hill s estimator r p (t) r(t)/< r(t) > ma 08/09/03 Frontier Science 2003, Pavia 9

10 Dynamics of process scale We estimate the exponent β by fitting the 1-minute r(t) with the functional form f(t)=c 1 t - β + c 2 During the 60-day time period after the crash. We verify that in the investigated time periods c 1 t - β >> c 2 08/09/03 Frontier Science 2003, Pavia 10

11 Internal consistency By using the estimated values of α, β and p We verify the relation 08/09/03 Frontier Science 2003, Pavia 11

12 Alternative estimation of α We also investigate the cumulative number N(t) for the almost stationary random variable r p (t) The slope η of each curve is η l -α This investigation allows us an independent estimation of α, which is α= /09/03 Frontier Science 2003, Pavia 12

13 Failure of simple autoregressive models Empirical findings are NOT consistent with an exponential decay of return pdf time scale. This implies that ARCH(1) and GARCH(1,1) processes are not able to model empirical findings just after major financial crashes σ t2 =α 0 +α 1 x 2 t-1 σ t2 =α 0 +α 1 x 2 t-1 +β 1 σ2 t-1 08/09/03 Frontier Science 2003, Pavia 13

14 GARCH(1,1) Observatory of Complex Systems, Palermo, Italy Indeed, numerical simulations of a GARCH(1,1) process show that exponentially auto-correlated autoregressive processes cannot describe accurately the post-crash dynamics of index returns. 08/09/03 Frontier Science 2003, Pavia 14

15 (ii) the price-impact function of a single transaction Observatory of Complex Systems, Palermo, Italy A stylized fact of microstructure of financial markets: the price-impact function of a single transaction Lillo, Farmer and Mantegna, Nature 421, (2003) 08/09/03 Frontier Science 2003, Pavia 15

16 How prices react to volumes? Observatory of Complex Systems, Palermo, Italy How price changes when a given volume is traded in an individual transaction? This problem has been studied by several authors by aggregating the trades in time or in the number of transactions. A concave impact curve has been observed. Our aim is to characterize the impact of a SINGLE transaction 08/09/03 Frontier Science 2003, Pavia 16

17 Motivation Observatory of Complex Systems, Palermo, Italy Several authors suggest that the relation between price return r and volume ω is r = ω β where β is a characteristic exponent and λ is the liquidity of the asset Is this relation true? How the relation depends on the properties of the asset (e.g. the capitalization of the company)? λ 08/09/03 Frontier Science 2003, Pavia 17

18 Method For each transaction we determine the prevailing quote according to the Lee and Ready algorithm. We classify trades in buyer initiated and seller initiated trades. In our analysis the price is defined as midq=(bid+ask)/2 For each transaction we observe two possible behavior TQT Q P..TQ.QTT.. return=log(midq P /midq) TQT Q P..TT.QTT.. return=0 The raw data consist in a scatter plot with points of coordinates (return, volume) 08/09/03 Frontier Science 2003, Pavia 18

19 1996 Single stocks Price impact function is not the same for all the stocks 08/09/03 Frontier Science 2003, Pavia 19

20 Grouping events by mean capitalization Observatory of Complex Systems, Palermo, Italy 1996 T group has 4 stocks A group has 315 stocks 1000 stocks are grouped in 20 sets depending on their capitalization 08/09/03 Frontier Science 2003, Pavia 20

21 1995 Observatory of Complex Systems, Palermo, Italy Data investigated are 1000 most capitalized stocks traded at NYSE during the period Transactions and best quotes are recorded in the Trade and Quote database 1000 stocks are grouped in 20 sets depending on their capitalization 08/09/03 Frontier Science 2003, Pavia 21

22 Liquidity and capitalization Observatory of Complex Systems, Palermo, Italy We determine λ from the values of the leftmost points of the 20 impact curves as a function of capitalization The liquidity λ is approximately proportional to (capitalization) /09/03 Frontier Science 2003, Pavia 22

23 Scaling transformation In order to find a universal impact curve we make the scaling transformation x x ( Cap) δ ( Cap) γ 08/09/03 Frontier Science 2003, Pavia 23 y and we look for the optimal values of δ and γ= which minimize the quantity ε = x x 2 y + y y 2 BIN

24 gamma Contour plot of log(ε) as a function of δ and γ delta 08/09/03 Frontier Science 2003, Pavia gamma For both 1995 and 1998 the optimal values of δ and γ are δ = 0.3 γ = 0.3 delta

25 A universal price-impact function exists Observatory of Complex Systems, Palermo, Italy Scaling is observed!!! 08/09/03 Frontier Science 2003, Pavia 25

26 Conclusion Observatory of Complex Systems, Palermo, Italy New stylized facts are discovered in financial markets Their knowledge is instrumental to achieve a satisfactory modeling of this fascinating complex system 08/09/03 Frontier Science 2003, Pavia 26

27 The OCS website 08/09/03 Frontier Science 2003, Pavia 27