Fin285a:Computer Simulations and Risk Assessment Section 3.2 Stylized facts of financial data Danielson,

Transcription

1 Fin285a:Computer Simulations and Risk Assessment Section 3.2 Stylized facts of financial data Danielson, Blake LeBaron Fall

2 Overview Autocorrelations and predictability Fat tails Volatility persistence Extreme dependence Summary 2

3 Autocorrelations and predictability 3

4 S&P price series Daily S&P return series January 4, July 29, 2016 Sample length: T = 16, 751 File: spret.mat 4

5 Corr(R t, R t-j ) Daily return autocorrelations Lag j 5

6 Fat tails 6

7 Frequency Daily return density and Normal Daily return 7

8 QQ plot Diagnostic test of distribution Plot quantiles for data versus test distribution matlab: qqplot(x) 8

9 Quantiles of Input Sample QQ return plot and Normal 0.15 QQ Plot of Sample Data versus Standard Normal Standard Normal Quantiles 9

10 Normal tail probabilties This table compares extreme negative returns for a normal distribution with the same mean and standard deviation as the Dow returns to the Dow itself. Probability of Returns Below Normal Dow -1% % % % %

11 Testing for Normality Jarque-Bera(JB) test T 6 (Skew)2 + T 24 (Kurtosis 3)2 χ 2 (2) For daily returns 3.08x10 5, p-value Probability data from null (Normal) = p-value matlab: [h, p, jbstat] = jbtest(ret) 11

12 Some tail quotes as you well know, the biggest problems we now have with the whole evolution of risk is the fat-tail problem, which is really creating very large conceptual difficulties. Because as we all know, the assumption of normality enables us to drop off the huge amount of complexity in our equations.. Alan Greenspan (1997) 12

13 Some tail quotes For reasons that are still unclear, shares began to move in ways that were the opposite of those predicted by computer models. These moves triggered selling by the funds as they attempted to cover their losses and meet margin calls from the banks. This in turn exacerbated the share price movements. FT (2007) 13

14 Student-t and returns Can a student-t distribution do better? 14

15 Frequency Daily return density and Student-t Daily returns 15

16 Quantiles of Input Sample QQ plot of returns and Student-t(4) 15 QQ Plot of Sample Data versus Distribution Quantiles of t Location-Scale Distribution 16

17 Quantiles of Input Sample QQ plot of returns and Student-t(3) 25 QQ Plot of Sample Data versus Distribution Quantiles of t Location-Scale Distribution 17

18 Volatility persistence 18

19 Daily return S&P daily returns Year 19

20 Stylized fact Persistent periods of high and low volatility in returns Volatility clusters Long range autocorrelations (years) Stochastic volatility R t = σ t e t (1) σ t = f (σ t 1,..., σ t j ) + η t (2) 20

21 Corr(R 2 t,r2 t-j ) Squared return autocorrelations Lag j (days) 21

22 Extreme dependence 22

23 Exeedance dependence Danielson: Nonlinear dependence Cross sections of returns more correlated in volatile markets Cross sections of returns more correlated in down markets Global market stress and connections Problems for diversification 23

24 Exeedance dependence Correlation of returns R 1,t, R 2,t with R 1,t and R 2,t < R Measure correlations in left tail When prices falling, does this impact correlations? Use quantiles on X-axis ρ(p) = corr[x, Y X Q X (p) and Y Q Y (p)] (3) 24

25 Exceedance correlation: daily Coke and Exxon Exceedance correlation Probability (returns < ) 25

26 Summary 26

27 Summary Fat tails: Large losses more probable than normal distribution Volatility persistence: long periods of unstable valuations/uncertainty Extreme dependence: diversification is weakest when you need it most! 27