Prices and Volumes on the Stock Market

Prices and Volumes on the Stock Market Krzysztof Karpio Piotr Łukasiewicz Arkadiusz Orłowski Warszawa, 25-27 listopada 2010 1

Data selection Warsaw Stock Exchange WIG index assets most important for investors different sectors content: every 3 months (volume, capitalization) 1995-01-02 to 2010-10-21 >= 27 assets 3961 trading days ~ 15 years. Warszawa, 25-27 listopada 2010 2

WIG: prices and volumes Warszawa, 25-27 listopada 2010 3

WIG: prices and volumes Warszawa, 25-27 listopada 2010 4

WIG: prices and volumes Prices 3-7 times larger Volumes 45 times larger! greater fluctuations than for prices some spikes Prices & volumes seem to go together. Warszawa, 25-27 listopada 2010 5

Questions How to compare prices/volumes to themselves? 15 years of history How to define a big/small? differences: few orders of magnitude. Warszawa, 25-27 listopada 2010 6

Current value vs history small price Not so small 45 days 300 days Warszawa, 25-27 listopada 2010 7

Current value vs history If you can not look forward. just look backward! How far? We don t know. Lets look k - days. Warszawa, 25-27 listopada 2010 8

Warszawa, 25-27 listopada 2010 9 Variables: current value vs average p t, v t - daily closure price, trading volume k - investor memory (k<=401). ) ( 1 2 1 k t t t t k t p p p k p P + + + = K ) ( 1 2 1 k t t t t k t v v v k v V + + + = K

P k distribution ln(p k ) Warszawa, 25-27 listopada 2010 10

ln(p 11t ) Warszawa, 25-27 listopada 2010 11

ln(p k ) Gaussian fit f ( x) = a exp( ( x μ) σ 2 2 ), μ = 0.037 (0.032 ; 0.0425) σ = 0.138 (0.131 ; 0.1455) R 2 = 0.93 Warszawa, 25-27 listopada 2010 17

ln(p k ) Gaussian fit parameters f ( x) = a exp( ( x μ) σ 2 2 ), Warszawa, 25-27 listopada 2010 18

Autocorrelations No autocorrelations -> R 2 = 0.99 b = 0.55±0.01 σ ( k) k σ ( k) = a k b small positive autocorrelations. Warszawa, 25-27 listopada 2010 19

V k distribution f ( x) = a x exp( (ln( x) μ) σ 2 2 ) μ = 0.053 (0.042 ;; 0.065) σ = 0.564 (0.550 ;; 0.579) R 2 =0.988 Warszawa, 25-27 listopada 2010 20

ln(v 11t ) Warszawa, 25-27 listopada 2010 21

ln(v k ) - Gaussian fit parameters f ( x) = a exp( ( x μ) σ 2 2 ), Warszawa, 25-27 listopada 2010 27

Autocorrelations No autocorrelations -> R 2 = 0.96 b = 0.101 ± 0.003 σ ( k) k big negative autocorrelations. Warszawa, 25-27 listopada 2010 28

Autocorrelations: prices vs volumes Prices Small positive Trend to keep changes Prices are unlimited Volumes Big negative Counteraction against changes Volume is limited Demand is restrained by the price growth Suppply is restrained by the price descrease. Warszawa, 25-27 listopada 2010 29

Correlations ln(p k ) vs ln(v k ) local localmaximum at at75 75 days days daily dailylogarithmic return return rates rates Warszawa, 25-27 listopada 2010 30

Correlation plot, k=101 Warszawa, 25-27 listopada 2010 31

Warszawa, 25-27 listopada 2010 32 Parametrization of correlation plots, 2 exp ), ( 2 2 2 2 2 1 2 1 2 1 2 1 2 2 2 1 ) ( ) )( ( ) ( ) 2(1 1 1 2 + = σ μ σ σ μ μ σ μ ρ ρ σ πσ ρ v v p p A v p f 2-dim Gauss Weighted χ 2 error = sqrt(bin content) χ 2 /ndf slightly bigger than 1

Correlations ln(p k ) vs ln(v k ) wide widelocal localmaximum around 100 100 days days local localminimum 210 210 days days Warszawa, 25-27 listopada 2010 33

ln(p k ) - ln(v k ), k=1 Warszawa, 25-27 listopada 2010 34

ln(p k ) - ln(v k ), k=401. Warszawa, 25-27 listopada 2010 46

Slope coefficient vs k stabilization Warszawa, 25-27 listopada 2010 47

Causality Correlations exist What s the direction? ln(p k ) -> ln(v k ) ln(v k ) -> ln(p k ) ln(v k ) <->ln(p k ) Granger test. Warszawa, 25-27 listopada 2010 48

Granger test What is better forecasted based on the past data? prices volumes ln( V t ), ln( P t ) = B l l 0 + αi ln( Vt i ) + βi ln( Pt i ) i= 1 i= 1 How far to look backward? What is the range l. Warszawa, 25-27 listopada 2010 49

Granger test: range Very clear direction prices -> volumes stabilization around 10 range 12 for future analysis. Warszawa, 25-27 listopada 2010 50

Granger test Warszawa, 25-27 listopada 2010 51

Granger Prices -> Volumes Result independent on k Two directional relation for some k? Other studies Latin America emerging: vol -> prices Mexico, US: prices -> vol Some evidence on Polish market maturity? Warszawa, 25-27 listopada 2010 52

Overall glance again Warszawa, 25-27 listopada 2010 53

Distribution of price/volume f ( x) = a x exp( (ln( x) μ) 2 ) σ 2 R 2 = 0.95 μ = -2.72 ± 0.05 σ = 1.04 ± 0.06 Warszawa, 25-27 listopada 2010 54

Companies (σ) U.S. (NYSE) Poland (GPW) Solvay: 0.943 Pekao: 1.195 DaimlerChysler: 0.900 Budimex: 3.076 Microsoft: 0.673 Rafako: 1.566 Warszawa, 25-27 listopada 2010 55

P k /V k vs k Warszawa, 25-27 listopada 2010 56

Final remarks Event prices & volumes signifitantly changed over time: P kt & V kt are stationary P kt & V kt are log-normal distributed P kt - small positive autocorrelations V kt - big negative autocorrelations Moderate correlations P kt -V kt local maximum (100) & minimum (210) 2dim plots tend to rotate: some stabilization at 250. Warszawa, 25-27 listopada 2010 57

Final remarks (cont.) Strong Granger causality P -> V similarity to mature markets opposite behavious than for emerging ones at least in Latin America Simply P/V values conform log-normal distribution Width of ln(p k /V k ) exchibites local maximum & minimum. Warszawa, 25-27 listopada 2010 58

Thank You! Warszawa, 25-27 listopada 2010 59