Chapter2
Stylized Facts
好词好句20个The name Stylized Facts refers to all non trivial statistical evidences which are obrved throughoutfinancial markets.Almost all price time ries offinancial stocks and indexes approximatively exhibit the same statistical properties(at least quali-tatively).In addition it has been shown that Stylized Facts are robust on different timescales and in different stock markets[1].
The systematic study of Stylized Facts has begun in very recent time(approxi-mately from’90)for two reasons:a technical and a cultural one.The former one is that the huge amount of empirical data produced byfinancial markets are now easily available in electronic format and can be massively studied thanks to the growth of computational power in the last two decades.In order to make a comparison with some traditionalfields of Physics,a similar quantity of information is obrved only in the output of a big particle accelerator.The latter instead is due to the fact that tradi-tional approaches to economic systems neglect empirical data as candidates respect to which a theory must be compared differently from Physics.From this point of view standard Economics is not an obrvational science.
Turning now our attention to the experimental evidences offinancial markets,the main Stylized Facts are
•the abnce of simple arbitrage,
•the power law decay of the tails of the return distribution,
•the volatility clustering.
In the following ctions we analyze them.
麻辣拌面2.1Abnce of Simple Arbitrage
The abnce of simple arbitrage infinancial markets means that,given the price time ries up to now,the sign of the next price variation is unpredictable on average.In other words it is impossible to make profit without dealing with a risky investment. This implies that the market can be en as an open system which continuously reacts to the interaction with the ading activity,flux of information,etc)and
M.Cristelli,Complexity in Financial Markets,19 Springer Thes,DOI:10.1007/978-3-319-00723-6_2,
©Springer International Publishing Switzerland2014
202Stylized Facts lf-organizes in order to quickly eliminate arbitrage opportunities.This property is also called arbitrage efficiency.
This condition is usually equivalent to the informational efficiency expresd in economic literature saying that the process described by the price p t is a martingale that is
E[p t|p s]=p s(2.1) where t>s.Here we are assuming that the price is a synthetic variable which reflects all the information available at time t.If this is not true the conditioning quantity is the available information I s at time s and not only the price p s.
However,the condition of martingale is uneasy from a practical point of view and the two-point autocorrelation function of returns is usually assumed as a good measure of the market efficiency
ρ(τ,t)=E[r t r t+τ]−E[r t]E[r t+τ]
E[r2t]−E[r t]2
.(2.2)
If the process{r t}is at least weakly stationary then Eq.2.2simply becomes ρ(τ)=(E[r t r t+τ]−μ2r)/σ2r whereμr=E[r t]andσr=E[r2t]−E[r t]2.If the autocorrelation function of returns is always zero we can conclude that the market is efficient.
In real markets the autocorrelation function is indeed always zero(e Fig.2.1) except for very short times(from few conds to some minutes)where the correlation is negative(e int of Fig.2.1).The origin of this small anti-correlation is well-known and due to the so-called bid ask bounce.This is a technical reason deriving from the double auction system which rules the order book dynamics(e[2]for further details).
扩充内存In the end we want to stress that the efficiency is a property that holds on average: locally some arbitrage opportunities can appear but,as they have been exploited,the efficiency is restored[3,4].
2.2Fat-Tailed Distribution of Returns
The distribution of price variations(called returns)is not a Gaussian and prices do not follow a simple random walk.In details very largefluctuations are much more likely in stock market with respect to a random walk and dramatic crashes are approximately obrved every5–10years on average.The large events cannot be explained by gaussian returns.Therefore to characterize the probability of the
events we introduce the complementary cumulative distribution function F(x)
F(x)=1−Prob(X<x)(2.3) which describes the tail behavior of the distribution P(x)of returns.
2.2Fat-Tailed Distribution of Returns
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A u t o c o r r e l a t i o n r Fig.2.1We report the autocorrelation function of returns for two time ries.
The ries of the main plot is the return ries of a stock of New York Stock Exchange (NYSE)from 1966to 1998while the ries of the int is the return ries of a day of trading of a stock of London Stock Exchange (LSE).As we can e the sign of prices are unpredictable that is the correlation of returns is zero everywhere.The time unit of the int is the tick,this means that we are studying the time ries in event time and not in physical time
The complementary cumulative distribution function F (x )of real returns is found to be approximately a power law F (x )∼x −αwith exponent in the range 2–4[5],i.e.the tails of the probability density function (pdf)decay with an exponent α+1.Since the decay is much slower than a gaussian this evidence is called Fat or Heavy Tails.Sometimes a distribution with power law tails is called a Pareto distribution.The right tail (positive returns)is usually characterized by a different exponent with respect to the left tail (negative returns).This implies that the distribution is asymmetric in respect of the mean that is the left tail is heavier than the right one (α+>α−).
电脑时间不准Moreover the return pdf is a function characterized by positive excess kurtosis,a Gaussian being characterized by zero excess kurtosis.In Fig.2.2we report the complementary cumulative distribution function F (x )of real returns compared with a pure power law decay with exponent α=4and with a gaussian with the same variance.
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When the tail behavior of the return distribution is studied varying the time lag at which returns are performed [1],a transition to a gaussian shape is obrved for yearly returns.However it is unclear if this transition is genuine or due to a lack of statistics or to the non stationary return time ries.
2.3Volatility Clustering
In the lower panel of Fig.2.3we report the return time ries of a NYSE stock (returns are here defined as log (p t +1/p t )).As we can e the behavior of returns appears to be intermittent in the n that periods of large fluctuations tend to be followed by
222Stylized Facts
Fig.2.2We report the complementary cumulative distribution function of the absolute value of returns(solid black line).The green dashed line(·−)is the complementary cumulative distribution function of a gaussian with the same variance of the real return distribution.The dashed black line is a pure power law decay with exponentα=4.The blue and red lines are instead the complementary cumulative distribution functions for positive and negative returns respectively.We can e that red curve has a slower decay with respect to the blue one.This asymmetry between positive and negative returns is the origin of the non zero skewness of the probability density function of returns largefluctuations regardless of the sign and the same behavior happens for small ones.
In Economics the magnitude of pricefluctuations is usually called volatility.It is worth noticing that a clustered volatility does not deny the fact that returns are arbitrage efficiency).Therefore the magnitude of the next price fluctuations is correlated with the prent one while the sign is still unpredictable. In other words stock prices define a stochastic process where the increments are uncorrelated but not independent.
西蓝花怎样做好吃Different proxies for the volatility can be adopted:widespread measures are the absolute value and the square of returns.As a conquence of the previous consider-ations about the clustering of volatility,the autocorrelation function of absolute(or square)returns is non zero.We alsofind that the a
utocorrelation is well-described by a power law decay with exponent ranging from−1to0as reported in Fig.2.4. The very slow decay means that volatility is correlated on very long time scales from minutes to veral months/years.The exponent of the autocorrelation function is not universal as the one of fat tails but it is typically around0.2–0.3.The volatility clustering was obrved thefirst time by Mandelbrot in1963[6].
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2.4Other Stylized Facts
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00,1l o g r e t u r n s Fig.2.3Return time ries of a stock of NYSE from 1966to 1998.The two figures reprent the same price pattern but returns are differently computed.In the top figure returns are calculated as simple t =p t −p t − t while in the bottom one returns are log returns that is r t =log p t −log p t − t .From the lower plot we can e that volatility appears to be clustered and therefore large fluctuations tend to be followed by large ones and vice versa.The visual impression that the return time ries appears to be stationary for log returns suggests the idea that real prices follow a multiplicative stochastic process rather than a linear process
2.4Other Stylized Facts
Beyond the Stylized Facts we can state other relevant effects which are widespread in financial markets such as
•the gain/loss obrves large drawdowns in stock prices and stock index values but not equally large upward movements.This is linked to the asymmetry of the return pdf.
•leverage effect:the volatility of an ast are negatively correlated with the returns of that ast.
•trading volume and volatility are correlated.
See also [1,3–5,7,8]for more details about Stylized Facts and their analysis.
2.5Stationarity and Time-Scales
Before turning our attention to the analysis of the models which try to interpret Stylized Facts,we want to discuss a final question:the stationarity and the time scales of the obrvation of financial markets.
