Shape of the return probability density function and extreme value statistics

Shape of the return probability density function and extreme value statistics 13/09/03 Int. Workshop on Risk and Regulation, Budapest

Overview I aim to elucidate a relation between one field of research of econophysics and one aspect of risk management I discuss results obtained in the modeling of the shape of tails of asset return pdf and their relation with VaR and extreme value statistics Conclusion. 13/09/03 Int. Workshop on Risk and Regulation, Budapest 2

Asset return pdf Which is the shape of asset return probability density function? This is a longstanding open problem in finance!!! Physicists have recently contribute to the solving of this puzzle 13/09/03 Int. Workshop on Risk and Regulation, Budapest 3

Leptokurtosis Return pdfs are leptokurtic. There is today a large evidence that the second moment of the unconditional pdf is FINITE!! One-minute index returns from Mantegna & Stanley, Nature 376, 46-49 (1995) 13/09/03 Int. Workshop on Risk and Regulation, Budapest 4

From Mantegna & Stanley, Nature 383, 587-588 (1996) The degree of leptokurtosis varies at different time horizons. There is a progressive convergence towards the geometric Brownian motion at longer time horizons. 13/09/03 Int. Workshop on Risk and Regulation, Budapest 5

Tail behavior In 1996 T. Lux at that time at Bamberg University observed that stock return tails could be described by a power-law behavior 13/09/03 Int. Workshop on Risk and Regulation, Budapest 6

T. Lux observed that the stocks composing the DAX index were characterized by positive and negative tails of the cumulative return distribution of exponent α 3 13/09/03 Int. Workshop on Risk and Regulation, Budapest 7

A power-law tail A similar result was obtained in Boston by P. Gopikrishnan, M. Meyer, L.A.N. Amaral and H.E.Stanley by investigating high-frequency data of the NYSE 13/09/03 Int. Workshop on Risk and Regulation, Budapest 8

The data were extracted from the TAQ database. They analyzed transactions of the 1000 most capitalized stocks traded in the NYSE in 1993-1994 13/09/03 Int. Workshop on Risk and Regulation, Budapest 9

Tails could also be described by different stochastic models characterized by different tails. One example is a Truncated Lévy Flight with an exponential tail Mantegna & Stanley, PRL 73, 2946-2949 (1994) Bouchaud & Potters, Theory of financial risk, CUP 13/09/03 Int. Workshop on Risk and Regulation, Budapest 10

Progressive convergence to a normal attractor is consistent both for return pdfs with power-law tails and for Truncated Lévy Flight with an exponential tail Bouchaud-Potters, Theory of financial risk, CUP 13/09/03 Int. Workshop on Risk and Regulation, Budapest 11

Key question Are these details of the asset return pdf relevant for risk management? 13/09/03 Int. Workshop on Risk and Regulation, Budapest 12

Shape of return probability density and extreme value statistics Extreme value theory Extreme value theory answer the key question: What are the possible (non-degenerate) limit laws for the maxima M n when properly normalized end centered? Let {X n } be a sequence of i.i.d. rvs. If there exist constants c n > 0 and d n R such that c 1 n ( ) d M d H n n n 13/09/03 Int. Workshop on Risk and Regulation, Budapest 13 Then H belongs to the type of one of the following attractors Fréchet Weibull Gumbel An excellent text on this theory is Embrechts et al, Modeling Extremal Events, Springer

Attractors in the functional space The specific form of these distribution functions is Fréchet Weibull Gumbel Φ Ψ α α = exp exp = 0 x 0 { α x } { ( ) } α 1 { e x } x > 0 x x 0 x > 0 Λ = exp x R α > 0 α > 0 13/09/03 Int. Workshop on Risk and Regulation, Budapest 14

Maximum Domain of Attraction (MDA) A question to be considered is Given an extreme value distribution H, what conditions on the df F(x) imply that the normalized maxima M n converge weakly to H? Another question concerning norming constants c n and d n is How may we choose the norming constants c n > 0 and d n R such that: c 1 n ( ) d M d H n n 13/09/03 Int. Workshop on Risk and Regulation, Budapest 15

MDA of Fréchet distribution It consists of distribution function F(x) whose right tail is regularly varying with index -α A distribution tail F c (x) is regularly varying with index -α (α 0) if F ( xt) ( x) c α lim = t t > 0 x Fc an example of regularly varying function is a power-law function 1 F c ( y) = α y 13/09/03 Int. Workshop on Risk and Regulation, Budapest 16

Examples Examples of distribution belonging to the Fréchet MDA Pareto Cauchy Stable with index Pareto < 2 Log-gamma α ( x) kx F c k, α > 0 f 1 1 = π 1+ x ( x) 2 α f ( x) = Γ β ln x β 1 x ( ) ( ) α 1 β x > 1 α, β > 0 13/09/03 Int. Workshop on Risk and Regulation, Budapest 17

MDA of the Weibull distribution An important aspect of the dfs F(x) which are in the MDA of the Weibull distribution is that they have a finite right endpoint x F Hence the Weibull distribution is not relevant for the present problem and we will not consider it. 13/09/03 Int. Workshop on Risk and Regulation, Budapest 18

MDA of the Gumbel distribution Λ ( x) = exp{ exp{ x } A Taylor expansion argument shows that the tail behavior of the Gumbel distribution is 1 Λ x ( x) exp{ x} Hence the MDA will be characterized by rvs with an exponential-like tail In this domain of attraction both cases x F < and x F are possible 13/09/03 Int. Workshop on Risk and Regulation, Budapest 19

The characterization of the domain of attraction is somewhat less direct than in the previous cases but also in this case a theorem exists Theorem: The df F(x) with right endpoint x F belongs to the MDA of the Gumbel distribution if and only if there exists some z<x F such that F c (x) has representation x g() t F ( ) ( ) c x = c x exp dt z<x<x a() F t z c ( x) c > 0, g( x) 1, x xf Where a(x) is a positive, absolutely continuous function 13/09/03 Int. Workshop on Risk and Regulation, Budapest 20

Examples of rvs being in the Gumbel MDA Exponential-like Normal Lognormal f ( x) f F c ( x) k exp{ λx} 1 ( ) ( ) 2 x = exp x 1 exp 2 ( ln x µ ) = 2 2πσx 2π 2σ x > 0 µ R σ > 0 2 k, λ > 0 x R 13/09/03 Int. Workshop on Risk and Regulation, Budapest 21

Norming constants of the Gumbel distribution The general form of the normalizing centering constants is d n ( 1) = n d n is the quantile of the distribution F(x) Hence the normalizing centering constant coincides with the Value at Risk (VaR) 13/09/03 Int. Workshop on Risk and Regulation, Budapest 22

Value at Risk VaR is the maximum loss over a target horizon such that there is a low, pre-specified probability that the actual loss will be larger 13/09/03 Int. Workshop on Risk and Regulation, Budapest 23

Norming constants of the Gumbel distribution The general form of the normalizing centering constants is d n ( 1) = n d n is the quantile of the distribution F(x) Hence the normalizing centering constant coincides with the Value at Risk (VaR) 13/09/03 Int. Workshop on Risk and Regulation, Budapest 24

Gumbel pdf and VaR Pdf of maximal loss VaR is the most probable loss at the chosen confidence level d n =VaR 13/09/03 Int. Workshop on Risk and Regulation, Budapest 25

How much Gumbel and Fréchet pdfs differs? H ξ A direct comparison can be obtained by using the Generalized Extreme Value (GEV) distribution x = ξ where 1 + x > 0 { ( ) } 1/ ξ + ξx { { x } ξ = α 1 > Fréchet ξ = if if ξ ξ = Gumbel 13/09/03 Int. Workshop on Risk and Regulation, Budapest 26

How much Gumbel and Fréchet pdfs differs? 13/09/03 Int. Workshop on Risk and Regulation, Budapest 27

The important difference is on tails VaR 13/09/03 Int. Workshop on Risk and Regulation, Budapest 28

Shape of return probability density and extreme value statistics Can we detect empirically maximal loss pdfs? A case study, GE daily data from 1987 to 1998 13/09/03 Int. Workshop on Risk and Regulation, Budapest 29

Statistical robustness is this an outlier? Unfortunately, it is not enough to assess empirically the kind of the maximal loss pdf 13/09/03 Int. Workshop on Risk and Regulation, Budapest 30

Mission improbable? By considering the intrinsic difficulty in empirically determining the kind of maximal loss pdf Can we conclude something about maximal loss? Title of a section of the paragraph 6.5.2 of Embrechts et al, Modeling Etremal Events, Springer 13/09/03 Int. Workshop on Risk and Regulation, Budapest 31

Shape of return probability density and extreme value statistics We can point out an interval of the maximal loss distribution Focusing on the tail behavior 13/09/03 Int. Workshop on Risk and Regulation, Budapest 32

S&P 500 Jan 1984-Dec 1989 hourly recorded Strictly speaking, stock returns are not independent and are not identically distributed 13/09/03 Int. Workshop on Risk and Regulation, Budapest 33

Is the observation of a power-law tail due to the non stationary nature of the return pdf? 13/09/03 Int. Workshop on Risk and Regulation, Budapest 34

An attempt to estimate the behavior of the tail a index return pdf during a highly non stationary time period 13/09/03 Int. Workshop on Risk and Regulation, Budapest 35

The Omori law in finance The observation of the Omori law just after a big market crash Lillo and Mantegna, PRE 68, 016119 (2003) 13/09/03 Int. Workshop on Risk and Regulation, Budapest 36

Shape of return probability density and extreme value statistics 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, 6609-6619 (2001)) 13/09/03 Int. Workshop on Risk and Regulation, Budapest 37

Shape of return probability density and extreme value statistics 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, 228701 (2001). 13/09/03 Int. Workshop on Risk and Regulation, Budapest 38

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 13/09/03 Int. Workshop on Risk and Regulation, Budapest 39

We fit empirical data with the Omori functional form Specifically, we measure the cumulative number of S&P500 index returns exceeding a given threshold nσ Lillo and Mantegna, PRE 68, 016119 (2003) N(t)=K[(t+τ) 1-p - τ 1-p ]/(1-p) 13/09/03 Int. Workshop on Risk and Regulation, Budapest 40

Shape of return probability density and extreme value statistics 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 13/09/03 Int. Workshop on Risk and Regulation, Budapest 41

Shape of return probability density and extreme value statistics In addition to the 1987 crash we also investigate other two major financial crashes: (a) the 27.10.97 crash; (b) the 31.08.98 crash. Other crashes 13/09/03 Int. Workshop on Risk and Regulation, Budapest 42

Shape of return probability density and extreme value statistics 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 13/09/03 Int. Workshop on Risk and Regulation, Budapest 43

Shape of return probability density and extreme value statistics 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 13/09/03 Int. Workshop on Risk and Regulation, Budapest 44

Shape of return probability density and extreme value statistics Internal consistency By using the estimated values of α, β and p We verify the relation 13/09/03 Int. Workshop on Risk and Regulation, Budapest 45

Shape of return probability density and extreme value statistics 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=3.14 13/09/03 Int. Workshop on Risk and Regulation, Budapest 46

Dynamics of process scale α 3.2 in the high-frequency index return pdf in a period of very high volatility and by taking into accont a deterministic dynamics of the typical scale of the random process 13/09/03 Int. Workshop on Risk and Regulation, Budapest 47

Shape of return probability density and extreme value statistics 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 13/09/03 Int. Workshop on Risk and Regulation, Budapest 48

GARCH(1,1) 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. 13/09/03 Int. Workshop on Risk and Regulation, Budapest 49

Conclusion The knowledge of stylized facts of financial markets is instrumental to achieve a satisfactory modeling of this fascinating complex system Some physicists are contributing to their observation and interpretation Knowledge of stylized facts together with fascinating results of probability theory provide quantitative tools to key questions of risk management 13/09/03 Int. Workshop on Risk and Regulation, Budapest 50

Shape of return probability density and extreme value statistics The OCS website http://lagash.dft.unipa.it 13/09/03 Int. Workshop on Risk and Regulation, Budapest 51