1. 1 4. Empirical distributions & prediction of returns 4.1 Prices and returns Price (P) ACF decays slowly. Log price: p = log(P) Single-period (rate of) return: R(t) = [P(t) - P(t-1)]/P(t-1) = P(t) / P(t-1) – 1 Log return: r(t) = log[R(t) + 1] = p(t) – p(t-1) Log returns – stationary processes with (relatively) short memory. Log returns can become negative in simulations (negative P has no sense). Cumulative return: R(t, k) = P(t) / P(t-k) - 1 Cumulative log return: r(t, k) = r(t) + r(t-1) + … + r(t-k)

2. 2 4. Empirical distributions & prediction of returns 4.2 The efficient market hypothesis (EMH) EMH: markets instantly incorporate all new information in the asset prices. Hypothesis: explanation of empirical facts. Random walk hypothesis (RWH): prices follow the random walk. Bachelier (1900), Fama (1965), Malkiel (2003), Lo & MacKinlay (1999). EMH (RWH + rational investors): - “Weak” form: current price reflects all information on past prices. Technical analysis impossible; fundamental analysis may work. - “Semi-strong” form: prices reflect all publicly available information. Fundamental analysis doesn’t work either. - “Strong” form: even private information instantly is incorporated in price.

3. 3 4. Empirical distributions & prediction of returns 4.2 EMH (continued I) EMH criticisms: - Investors are not absolutely rational (greed & fear - “animal spirits”) Akelrof & Shiller (2009) - Grossman-Stiglitz paradox - Lo & MacKinlay (1999): prices do not always follow random walk Adaptive market hypothesis (Lo (2004)): Market is an “ecological system” in which investors compete for scarce resources. They have limited capabilities (“bounded rationality”). Prices reach their new efficient values not instantly but over some time during which investors adapt to new information by trial and error. Pragmatic view (Malkiel): OK, it’s not random walk – but can anyone consistently beat the market?

4. 4. Empirical distributions & prediction of returns 4.2 EMH (continued II) Interviews with Fama & Shiller: http://www.nytimes.com/2013/10/20/business/robert-shiller-a- skeptic-and-a-nobel-winner.htmlhttp://www.nytimes.com/2013/10/20/business/robert-shiller-a- skeptic-and-a-nobel-winner.html http://www.nytimes.com/2013/10/27/business/sharing-nobel-honors- and-agreeing-to-disagree.htmlhttp://www.nytimes.com/2013/10/27/business/sharing-nobel-honors- and-agreeing-to-disagree.html http://www.nytimes.com/2013/10/27/business/eugene-fama-king-of- predictable-markets.htmlhttp://www.nytimes.com/2013/10/27/business/eugene-fama-king-of- predictable-markets.html 4

5. 5 4. Empirical distributions & prediction of returns 4.3 Random walks (Not all random walks are equal...) p t = p t-1 + μ + ε t, E[ε t ] = 0; E[ε t 2 ] =  2 ; E[ε t ε s ] = 0, if t  s Differential form (Brownian motion): dp t = μ t dt + σ t dW t, W t = N(0, 1) E[p t | p 0 ] = p 0 + μt Var[p t | p 0 ] = σ 2 t Sample => ARMA => unit root? Also: augmented Dickey-Fuller (ADF) test 4.3.1 Random walk model RW1 RW1 when ε t is strict white noise: ε t = IID(0, σ 2 ). Gaussian (normal) white noise: ε t = N(0, σ 2 ). Any forecasting of RW1 is impossible.

6. 6 4. Empirical distributions & prediction of returns 4.3.2 Random walk model RW2 In RW2, ε t has independent but not identical innovations, i.e. innovations can be drawn from different distributions. RW2 is based on the martingale theory. A process X t is a martingale if E[X t | X t-1, X t-2,…] = X t-1 or E[X t - X t-1 | X t-1, X t-2,…] = 0 The latter difference represents the outcome of fair game. Gambling: St. Petersburg paradox, Gambler’s ruin, etc. Forecasting of expected value is impossible but higher moments (variance, etc.) may be predictable.

7. 7 4. Empirical distributions & prediction of returns 4.3.3 Random walk model RW3 In RW3, innovations remain uncorrelated, i.e. Cov(ε t, ε t-k ) = 0). Yet higher moments, e.g. variance, can be dependent, and hence forecastable: Cov(ε t 2, ε t-k 2 ) ≠ 0. RW3 is called white noise (but not the strict one). Hence RW3 exhibits conditional heteroskedasticity, which is well supported with empirical data on volatility. RW3 permits only non-linear forecasting of expectations (neural nets, genetic algorithms). Empirical returns show weak correlations. Hence, even RW3 can be technically rejected. As a result, stronger hypotheses on RW1 and RW2 can be rejected, too. Bottom line: financial markets are predictable to some degree.

8. 8

9. 9 4. Empirical distributions & prediction of returns 4.6 Empirical findings - Distributions of daily returns are non-Gaussian - Yet serial correlations are very small (hard to forecast) - Squared (or absolute) returns are well correlated - Volatility may be variable and clustered - Distributions have high kurtosis and may be skewed - Distributions have “fat” tails (power-law asymptotes) Some reports (e.g. Gabaix (2003)) estimate the power-law index of 3: equities in 1996 – 1998 on the grid from minutes to several days. Several theories offered (review in Schmidt (2004)). LeBaron (2001): power-law distributions can be generated by a mix of normal distributions with different time scales.

10. 10 4. Empirical distributions & prediction of returns Distribution of returns for S&P 500 in 1997 – 2009

11. 11 4. Empirical distributions & prediction of returns 4.7 Stable distributions (Levi flights) Sum of two copies from the same distribution has the same distribution. Fourier transform of probability distribution (characteristic function) F(q) = ∫ f(x) e iqx dx Levy distribution: iμq – γ|q| α [1 – iβδ tan(πα/2)], if α ≠ 1 ln F L (q) = { iμq – γ|q|[1 + 2iβδ ln(|q|)/π)], if α = 1 δ = q/|q|, 0 0. μ – mean; α – peakedness; β – skewness; γ – spread.

12. 12 4. Empirical distributions & prediction of returns 4.7 Stable distributions (continued) Normal distribution (α = 2): f SN (x) = exp[-z 2 /2] Cauchy distribution (α = 1, β = 0): f C (x) = Pareto distributions (α < 2) have power-law decay: f P (|x|) ~ |x| -(1 + α) The problem: moments are divergent... Solution (?): truncation. Mantegna & Stanley (2000); Bouchaud & Potters (2000); Schmidt (2004)

13. 13 4. Empirical distributions & prediction of returns 4.8 Fractals in finance Mandelbrot (1997); Peters (1996) Self-similarity (repeating patterns on smaller scale). Time series is not isotropic => self-affinity: X(ct) = c H X(t) H – Hurst exponent. Fractional Brownian motion: E[B H (t + T) - B H (t)] = 0; E[B H (t + T) - B H (t)] 2 =T 2H If H=0.5, regular Brownian motion. Correlation between E[B H (t) - B H (t - T)]/T and E[B H (t + T) - B H (t)]/T C = 2 2H-1 – 1

14. 14 4. Empirical distributions & prediction of returns 4.8 Fractals in finance (continued) Persistent process: C > 0 => 1/2 < H < 1 if grew in the past, probably will grow in future Anti-persistent process: C H < 1/2 if grew in the past, probably will fall in future Rescaled range (R/S) analysis: Consider a data set x i (i = 1,... N) with mean m N and variance σ N 2. S k =, 1 ≤ k ≤ N R/S = [max(S k ) - min(S k )]/σ N, 1 ≤ k ≤ N R/S = (aN) H Drawback: sensitivity to short-range memory