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Stylized Facts
ˇ ıd, Martin Sm´ you@martinsmid.eu, www.martinsmid.eu
´ ˇ UTIA AV CR

March 30., 2009

Outline
According to Cont (2001) (also avalable here).

What is Stylized Fact

Some stylized facts

Homework

References

Stylized facts
Stylized (empirical) facts are statistical properties of financial time series, common across a wide range of instruments, markets and timeperiods.

Stylized facts
Stylized (empirical) facts are statistical properties of financial time series, common across a wide range of instruments, markets and time periods. Caution: Assumptions required for the statistical estimation may fail!

Stylized facts
Stylized (empirical) facts are statistical properties of financial time series, common across a wide range of instruments, markets andtime periods. Caution: Assumptions required for the statistical estimation may fail! Namely, they are:)

Stylized facts
Stylized (empirical) facts are statistical properties of financial time series, common across a wide range of instruments, markets and time periods. Caution: Assumptions required for the statistical estimation may fail! Namely, they are:) Stationarity of the time series(violated e.g. by seasonal effects, which may however be corrected by a “time change”)

Stylized facts
Stylized (empirical) facts are statistical properties of financial time series, common across a wide range of instruments, markets and time periods. Caution: Assumptions required for the statistical estimation may fail! Namely, they are:) Stationarity of the time series (violated e.g. by seasonaleffects, which may however be corrected by a “time change”) Ergodicity (a weaker form of independence)

Stylized facts
Stylized (empirical) facts are statistical properties of financial time series, common across a wide range of instruments, markets and time periods. Caution: Assumptions required for the statistical estimation may fail! Namely, they are:) Stationarity of the time series (violatede.g. by seasonal effects, which may however be corrected by a “time change”) Ergodicity (a weaker form of independence) Requirements for Central limit theorems (hold for some dependent series but not for all)

Stylized facts
Stylized (empirical) facts are statistical properties of financial time series, common across a wide range of instruments, markets and time periods. Caution: Assumptionsrequired for the statistical estimation may fail! Namely, they are:) Stationarity of the time series (violated e.g. by seasonal effects, which may however be corrected by a “time change”) Ergodicity (a weaker form of independence) Requirements for Central limit theorems (hold for some dependent series but not for all) We are studying (asset) returns Rt = Pt /Pt−1 − 1.

No Autocorrelation
Returns oflonger periods do not show significant autocorreations For shorter periods (20 min) the microstructure effects take place (i.e. the bid-ask bounce causing a slight mean reversion)

Heavy tails
Random variable X is heavy-tailed with the tail index α if P(X > x) vanishes at infinity at order α.

Heavy tails
Random variable X is heavy-tailed with the tail index α if P(X > x) vanishes at infinityat order α. Only the moments of the order < α are finite

Heavy tails
Random variable X is heavy-tailed with the tail index α if P(X > x) vanishes at infinity at order α. Only the moments of the order < α are finite Warning: even though the high curtosis may indicate fat tails, it does not prove them

Heavy tails
Random variable X is heavy-tailed with the tail index α if P(X > x) vanishes atinfinity at order α. Only the moments of the order < α are finite Warning: even though the high curtosis may indicate fat tails, it does not prove them Normal distribution has α = ∞

Heavy tails
Random variable X is heavy-tailed with the tail index α if P(X > x) vanishes at infinity at order α. Only the moments of the order < α are finite Warning: even though the high curtosis may indicate fat...
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