How To Calculate A Non Random Walk Down Wall Street

Transcription

1 Detrending Moving Average Algorithm: a Non-Random Walk down financial series, genomes, disordered materials... Anna Carbone Politecnico di Torino September 13, 2013

2 The Efficient Market Hypothesis... A Random Walk Down Wall Street First edition published in the 70 s by Burton Malkiel on the wave of the academic success of the influential survey on the Efficient Market Hypothesis (EMH) by Eugene Fama (1970). The Efficient Market Hypothesis implies a price series where all subsequent price changes represent random departures from previous prices i.e. a random walk. The logic of the random walk idea is that the flow of information is undelayed, thus information is immediately and genuinely reflected in stock prices. Thus resulting prices changes must be unpredictable.

3 A Moving Average Walk Down Wall Street...

4 Surveys on the use of Technical Analysis by Finance Professionals Horizons min % max % 1 day 50% 100% 1 week 50% 80% 1 month 40% 70% 3 months 30% 50% 6 months 20% 30% 1 year 2% 25% 1 Charts, Noise and Fundamentals in the London Foreign Exchange Market. H.L. Allen and M.P. Taylor, The Economic Journal, 100, (1990) 2 Technical trading rule profitability and foreign exchange intervention, B. LeBaron, Journal of International Economics 49, 125 (1999) 3 The Obstinate Passion of foreign Exchange Professional: Technical Analysis. Lukas Menkhoff and Mark P. Taylor, Journal of Economic Literature 45, 4 (2007)

5 Detrending Moving Average Algorithm (DMA) 20 ~ y n (i) σ 2 DMA = 1 N n N i=n [ ] 2 y(i) ỹ n (i) \L FOXVWHU y(i) ỹ n (i) = 1 n n y(i k) k= i c (j) L i c (j+1) τ j =i c (j+1)-i c (j) 2680 σ 2 DMA n2h

6 Detrending Moving Average Algorithm (DMA) σ 2 DMA n2h The log-log plot is a straight-line whose slope can be used to calculate the Hurst exponent H of the time series.

7 Detrending Moving Average Algorithm (DMA) Self-similarity of a time series x(t) σdma 2 is generalized variance for nonstationary signals. It can be derived from the auto-correlation function C xx (t, τ), which measures the self-similarity of a signal: σ xx (t, τ) [x(t) x(t)][x (t + τ) x (t + τ)] By taking τ = 0 and x = x(t), x n (i) = 1 n n k=0 x(i k) the autocovariance C xx (t, τ) reduces to the function σdma S. Arianos and A. Carbone Physica A 382, 9 (2007)

8 DAX (Deutscher AktienindeX) (a) Prices P(t) (b) Log Returns r(t) = lnp(t + t ) lnp(t) (c) and (d) Volatility σ T (t) 2 = 1 T T 1 t=1 [r(t) r(t) T ] 2 (c) with T = 300min (one half trading day); (d) with T = 660min (one trading day)

9 FIB30 futures: PRICES The FIB30 is a future contract on the MIB30 index, which considers the thirty firms with higher capitalization and trading (the top 30 blue-chip index) of the MIBTEL ( Anna Carbone (SincePolitecnico 2004 di ittorino is named S&P MIB Detrending index). Moving Average Algorithm: a Non-Random Walk d

10 FIB30 futures: VOLATILITIES 10-3 T=660min (1 trading day) 10-4 σ DMA Fract. Brown. Motion H= n

11 Cross-correlation of two stochastic series x(t) and y(t) Joint-similarity of two time series x(t) and y(t) The degree of coupling between two systems might be of interest. A measure of cross-correlation can be implemented 5 by defining: σ xy (t, τ) [x(t) x(t)][y (t + τ) ỹ (t + τ)] Generally τ 0. x = x(t) and y = y(t) x n (i) = 1 n n k=0 x(i k) and ỹ n (i) = 1 n n k=0 y(i k). 5 S. Arianos and A. Carbone, J. Stat. Mech.: Theory and Experiment P03037, (2009).

12 Cross-correlation of return and volatility If τ = 0 σ 2 DMA nh 1+H 2 The log-log plot is a straight-line. The slope is given by the sum of the Hurst exponents H 1 and H 2 of the two time series. 6 6 S. Arianos and A.Carbone, J. Stat. Mech: Theory and Experiment P03037, (2009).

13 Leverage effect: DAX Cross-correlation as a function of the lag τ for the return and volatility of the DAX series. 7 7 S. Arianos and A.Carbone, J. Stat. Mech: Theory and Experiment P03037, (2009).

14 Let s walk down complexity science!

15 Scaling properties of the moving average clusters 8 9 n=200 n=600 y(t) y(t) n=1000 area n=2000 t duration t Anna Carbone 8 A.Carbone, Politecnico di G.Castelli Torino and H.E. Stanley, Detrending Phys. Moving Rev. Average E, 69, Algorithm: a(2004) Non-Random Walk d

16 Scaling of the cluster lengths l 10 5 (a) (a) l 10 2 ψ l H τ H l τ ψ l ψ l = 1

17 Scaling of the cluster areas s (b) 1.9 (b) s ψ s H H τ H s τ ψs ψ s = 1 + H

18 Scaling of cluster lengths and durations H= β=2-h ) P(τ) 10-3 β x x10-5 (a) τ (a) H P(τ, n) τ β F (τ, n) with β = 2 H the fractal dimension of the time series and F (τ, n) defined as: F(τ, n) = e τ/τ.

19 Scaling law of the pdf of the cluster area γ=2/(1+h) P(s) (b) H=0.80 H=0.20 H= s γ (b) H with γ = 2/(1 + H) and: P(s, n) s γ F (s, n) F(s, n) = e s/s.

20 Self-organized criticality of the moving average clusters t addition of particles/energy y j y addition of particles/energy Brownian path fractional Brownian path j+1 Brownian path cluster lifetime j+2 smoothed fractional Brownian path cluster diameter l dissipation of particles/energies dissipation of particles/energy Moving Average Clusters SOC Clusters l τ ψ l ψ l = 1 ψ l = 1 s τ ψs ψ s = 1 + H ψ s = 3/2 P(l, n) l α α = 2 H α = 3/2 P(τ, n) τ β β = 2 H β = 3/2 P(s, n) s γ γ = 2/(1 + H) γ = 4/3

21 Shannon Entropy of long-range correlated time series 10 1 l n n=500 A n=1000 n=2000 n=3000 S(l,n) A' A'' D log l n= l S(l, n) µ(l,n) P(l, n) log P(l, n). S(l, n) = S 0 + D log l + l n. 10 PRE (2004), Physica A (2007), Sci. Reports (2013)

22 Information measure for long range correlated series: the case of 24 Human Chromosomes The sequence of the nucleotide bases ATGC is mapped to a numeric sequence. If the base is a purine (A,G) is mapped to +1, otherwise if the base is a pyrimidine (C,T) is mapped to 1. The sequence of +1 and 1 steps is summed and a random walk y(x) is obtained.

23 Information measure for long range correlated series Moving Average Algorithm: a Non-Random Walk d Figure Detrending :

24 Information measure for long range correlated series Moving Average Algorithm: a Non-Random Walk d Figure Detrending :

25 Information measure for long range correlated series Moving Average Algorithm: a Non-Random Walk d Figure Detrending :

26 Information measure for long range correlated series Moving Average Algorithm: a Non-Random Walk d Figure Detrending :

27 Detrending Moving Average Algorithm for high-dimensional fractals 11 Fractal surfaces 0.5 f(i 1,i 2 ) H= f(i 1,i 2 ) i i i i A. Carbone Phys. Rev. E 76, (2007)

28 Detrending Moving Average Algorithm for high-dimensional fractals 12 Moving Averages for the fractal surface with H = ~ f(i 1,i 2 ) n 1 x n 2 =15 x ~ f(i 1,i 2 ) n 1 x n i i i i A. Carbone Phys. Rev. E 76, (2007)

29 Detrending Moving Average Algorithm for high-dimensional fractals 13 DMA N 1 x N 2 =4096 x s The plot of σ 2 DMA vs. S = (n n2 2 ) is a straight line. It corresponds to the scaling relation: i.e.: σ 2 DMA [ (n n2 2 ) ] 2H σ 2 DMA S2H 13 A. Carbone Phys. Rev. E 76, (2007)

30 Three-dimensional fractals Phys. Rev. E, (2010,2011,2013)

31 Three-dimensional fractals Phys. Rev. E, (2010), (2011a), (2011b), (2013)

32 Three-dimensional fractals Phys. Rev. E, (2010), (2011a), (2011b), (2013)

33 Three-dimensional fractals Phys. Rev. E, (2010),(2011a),(2011b),(2013)

34 THANK YOU FOR YOUR ATTENTION!