统计物理学对经济科学的应用 (I)

陈志

4/11/2010

Why do we need statistical physics?

• Article on the 23 August 1997 issue of The Economist: “The Puzzling Failure of Economics”

the fundamental principles governing the complex system called economics are not completely uncovered.

• Wall Street has been the biggest single recruiter of physics PhDs.

Many physicists work as quants (quantitative analysts) --- very

common in large investment banks and other financial firms.

Physicists welcome the research related finance topics

Roughly 10 physics PhDs each year have addressed finance topics with their doctoral research.

Why?Jean-Phillipe Bouchaud (a pioneer in this area and the advisor of several such students) said:

“Somebody has to train all the physics graduates going into banking and finance, and we want it to be us, not people from other disciplines. To do this we need to establish a scientific presence in the field.”

Tools from statistical physics provides a fresh point of view to understanding finance topics.

Materials can be found on the web

Econophysics : http://polymer.bu.edu/~hes/econophysics/

Basic knowledge can be found at Wiki: en.Wikipedia.org

What we look at?

For example, we study a time series the S&P 500 index (an index of the New York Stock Exchange that consists of 500 companies representative of the US economy) Z(t), the time lag for recordings is Δt.

We usually look at the increments and the log return: G(t,Δt)=returns: Z(t+Δt)/Z(t)-1, volatility: standard deviation of time series over a suitable time interval.

Why? The fluctuations of Z(t) are usually proportional to Z(t). log return satisfies the addition law: G(t,Δt1+Δt2)=G(t,Δt1)+G(t+Δt1,Δt2)

S&P 500 index from 1984-1997.

An example

(I)Distributions: increments

Traditional view:

The increments are independent and identically distributed (IID) with a well defined second moment.From the central limit theorem: the distribution of G(t) is Gaussian.

What we found, for increments of both stock indices and foreign exchange rates:• distribution is symmetric;• distribution has fat tails, deviate from Gaussian distribution, however satisfying a Levy stable distribution;• increment time series is nonstationary.

See for example: R.N. Mantegna, H.E. Stanley, Nature 376 (1995) 46.

Increments of S&P 500 indices y(t), fat tails

Here Z(t)=y(t)-y(t-Δt).

Levy stable symmetrical distribution

where α is the index and γ is the scale factor.

Stable distribution (from en.Wikipedia.org): Let X1 and X2 be independent copies of a random variable X. Random variable X is said to be stable if for any constants a and b the random variable aX1 + bX2 has the same distribution as cX + d with some constants c and d. The distribution is said to be strictly stable if this holds with d = 0

Examples of stable distributions: Gaussian distribution, Cauchy distribution, Levy distribution, etc.

Δt= 1min:Solid line: Levy distribution of index α=1.40 and scaling factor γ=0.00375;Dotted line: Gaussian distribution.

Rescale by and

Verify Levy distribution: for different time intervals Δt

Is the distribution stable in time?

From the slope we can find the value of the index α.

Returns: distributions

Power law form?

More details

If in the Levy regime 0<α<2, distribution of Pn=∑xi will converge to a Levy stable stochastic process, see for example P. Gopikrishnan, V. Plerou, L. A. N. Amaral, M. Meyer, and H. E. Stanley, Phys. Rev. E 60, 5305-5316 (1999). .

Volatility

Why we are interested?

• Volatility is the key input of virtually all option-pricing models;

• There is long-time persistence in the volatility – much larger than the correlation time for price changes, see for example Y. Liu, P. Cizeau, M. Meyer, C.-K. Peng, H.E. Stanley, Physica A 245 (1997) 437.

Volatility time series is nonstationary.

We define volatility as: where .

See for example: Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, Phys. Rev. E 60, 1390-1400 (1999).

Distributions: Central part

Y axis: ,

X axis:

Gaussian or log-normal?

Δt= 30min, again deviate from the Gaussian distribution!

Tail of the distribution: power law

, where μ=3.10±0.08. Similar to that of returns! Out of the Levy regime.

(II) Correlations

We look at the normalized intraday return: ,

where , the sum is over all the trading days.

Here we study only the volatility.

See for example: Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, Phys. Rev. E 60, 1390-1400 (1999).

Simple autocorrelation function

Short time g(t): semi-log plot Long time |g(t)|:log-log plot

Different from well-studied examples of complex dynamical systems in physics such as turbulent flow where power law correlations on long time scales are commonly observed.

Other methods on |g(t)|

• Power spectrum

• Detrended fluctuation analysis (DFA)

Detail of DFA

We measure the fluctuation function F at different time window t:

|g(i)| is the original signal.

We look at the power law correlations in signals:

,~)( ttF

where α is the scaling exponent.

Relations with results of power spectrum and autocorrelation function

For a stationary signal with power law correlation,

• power spectrum

• autocorrelation function

,~)( ffS

.~)( ttC

We have: .

• When 0<α<0.5, power-law anticorrelations are present such that large values are more likely to be followed by small values and vice versa.• For white noise or the signals with short range correlations, α=0.5.• If α greater than 0.5 and less than or equal to 1.0 indicates persistent long-range power-law correlations, we have . The case of α=1 corresponds to 1/f noise.

Some special technique

• Data shuffing:

Since we find the volatility to be power-law distributed at the tail, to test that the power-law correlation is not a spurious artifact of the long-tailed probability distribution, we need to shuffle the time series (the distribution is unchanged).

• Remove outliers:

Will the results be different? We will discuss this in detail later.

Correlations of S&P 500 stock index: power spectrum

• There is a crossover (change of slope) at fx. • Autocorrelation function shows exponential decay with a characteristic time of the order of 4 min.

We set T=1 min and Δt=1 min.

Correlations of S&P 500 stock index: DFA

We find a similar crossover at tx~1/fx.

Volatility of individual companies: power spectrum

• There is a crossover at fx;• Autocorrelation function shows weak correlations up to 10 min.

We set Δt=5 min.

Volatility of individual companies: DFA

We find a similar crossover at tx~1/fx.