Modelling ﬁnancial returns - Univr

Financial Econometrics, Modelling financial returns

Financial EconometricsLecture 1

Modelling financial returns

Roberto RenòUniversità di SienaFebruary 22, 2012

1.1

1 Stylized factsA theoretical benchmark: The Black and Scholes and Merton model

• Risk free money market account with interest rate r• Risky asset following a Geometric Brownian Motion:

dSt = µStdt +σStdWt

which implies a lognormal distribution for the price.• Option payoff (European) at maturity T : φ(ST )• Denote the option value by F(t,S)• Replication arguments: ∂F

∂ t+ rS

∂F∂S

+12

σ2S2 ∂ 2F

∂S2 = rF

F(T,S) = φ(S)

• Random walk1.2

The distribution of returns

Roberto Renò, 2011 c© 1


• According to the lognormal model, we have, for every ∆ > 0:

St+∆ = St exp((

µ− σ2

2

)∆+σW (∆)

)and then

logS(t +∆)

S(t)∼N

((µ−σ

2/2)∆,σ2∆

)• In discrete time, we may write:

rt = µ +σεt

where εt is iid noise and µ = µ−σ2/2.1.3

Moments of a distribution

• Given a random variable X , we define its k−th (centered) momentas:

M1 = E[X ]

Mk = E[(X−E[X ])k], k ≥ 2

• Characteristic function:

φ(u) = E[eiuX ]

has the property, when E[X ] = 0, that:

Mk =1in

∂ kφ

∂uk

∣∣∣∣u=0

• Mean = E[X ]• Variance = M2

• Skewness =M3

(M2)1.5

• Kurtosis =M4

(M2)2

Normalizations makes skewness and kurtosis pure numbers, independentfrom the units of X . 1.4

The moment-generating function is close to the characteristic function,and it is defined as:

MX(t) = E[etX ],



with t ∈ R. To see that it is linked (as the characteristic function) to mo-ments, just consider the Taylor expansion:

etX = 1+ tX +12

t2X2 +16

t3X3 + . . .

which implies, taking expectations,

MX(t) = 1+ tM1 +12

t2M2 +16

t3M3 + . . .

Immediately, we get:

Mk =∂ kMX(t)

∂ tk

∣∣∣∣t=0

Sample moments

• Given a sample X1, . . . ,Xn, we define its k−th centered sample mo-ment, for k ≥ 2 as:

Mk =1n

n

∑i=1

(Xi− µ)k

where

µ =1n

n

∑i=1

Xi

• sample skewness: s = M3M1.5

2

• sample kurtosis: k = M4M2

2

Check the following concepts:1. Sample moments are random variables2. What is the relation between sample moments and the moments in-

troduced before? (law of large numbers)3. are sample moments unbiased estimators of moments (check with

the variance)? How can we correct for this?1.5

Simple normality tests

1. Check if s = 0

• Distributions with skewness different from 0 are not symmet-ric.

• What is the sign of the skewness of the lognormal distribution?



2. Check if k = 3

• Distributions with kurtosis above 3 are called leptokurtic or fat-tailed.• Distributions with kurtosis less than 3 are called platokurtic.• EXERCISE: PROVE THAT THE KURTOSIS OF THE NORMAL

DISTRIBUTION IS 3. 1.6

Solution of the exercise. It is harmless to work with a standard normalrandom variable X , for which is E[X2] = 1. We then have:

kurtosis = E[X4] =∫

x4 1√2π

e−x22︸︷︷︸

normal density

dx

=∫−x3( −x

1√2π

e−x22︸︷︷︸

primitive o f 1√2π

e−x22

)dx

integration by parts= −x3( 1√

2πe−

x22)dx∣∣∣∣+∞

−∞

+∫

3x2(

1√2π

e−x22

)dx

= 3.

• Mix the two above in the Jarque-Bera test:

JB =n6

s2 +n24

(k−3)2,

which, under the null hyphothesis of normality, is distributed as a χ2

with 2 degrees of freedom.1.7

Famous studies on the non-normality of returns• Benoit Mandelbrot (1963), the father of the theory of fractals, us-

ing cotton prices suggested the possibility of infinite variance (Lévyflights).• Eugene Fama (1965), the father of the theory of efficient markets.• The econophysics community, starting with Mantegna and Stanley

(1995). They suggest finite variance (truncated Levy flights).• The subject is still under empirical investigation! See e.g. Ait-

Sahalia and Jacod (2010).1.8



Journal of Business, 1963:

1.9

Journal of Business, 1965:

1.10

Mantegna & Stanley, Nature, 1995:

1.11

Is the time series of return independent?

• According to the Black and Scholes model, returns should be inde-pendent from each other.



• Test: compute the autocorrelation function of returns• Recall the definition of the autocorrelation function:

ρX(h) =

n−h

∑i=1

(Xi− µ)(Xi+h− µ)

n

∑i=1

(Xi− µ)2,

1.12

The Efficient Market HypothesysThe observed autocorrelation function of returns goes well beyond the

properties of the Black and Scholes model.The EMH, formulated by Eugene Fama, is the translation of the con-

cept of invisible hand for the prices of financial securities. It can be statedin three forms, weak, semi-strong and strong.

Definition 1 (The Efficient Market Hypothesis). A financial market for agiven securities is said to be informationally efficient if the price of thesecurity at time t synthetizes all the information contained in:

1. all the observed prices at times less than t (weak form)2. all the public information available in the market (semi-strong form)3. all the information about the security, including the non-public one

(strong form)1.13

Believe it or not, the weak form has been extensively tested and itworks very well. It implies that you cannot make money using technicalanalysis.

Beyond linear independence

• We have a strong theoretical argument for a null autocorrelationfunction. We also tested it empirically. Is this the end of the story?• Please notice that, while two independent random variables are cer-

tainly uncorrelated, this does not imply that two uncorrelated randomvariables are independent! (mumble mumble)• In particular, we might want to check if transformations of returns

are uncorrelated, which is also a prediction of the Black and Scholesmodel.

1.14



Volatility persistence

• The autocorrelation function of squared returns is largely positive,and for a long time horizon.• Returns are uncorrelated, but not independent -> the BSM model

cannot reproduce this feature of returns.• In particular, this finding means that if the returns in the last week

have been large (small), in the next week they will likely be large(small) as well.• This phenomenon is called volatility persistence or volatility cluster-

ing.• It is often argued that this is a signature of long memory, something

which an AR(n) model cannot reproduce.• When the variance of a time series varies with time, we say that we

are in the presence of heteroskedasticity.1.15

Stylized facts about financial returnsSummarizing, financial returns are characterized by:

fat tails The skewness of returns is close to zero, their kurtosis is largerthan three.

random walk The autocorrelation function of returns is statistically null.volatility persistence The autocorrelation function of squared returns is

significantly positive, with a renge of several months.leverage effect Squared returns are negatively correlated with returns. This

phenomenon takes the name of leverage effect.1.16

Can BSM price options, at least?

• The failures of the BSM model make its use questionable for all theapplications in the natural probability (VaR, scenario simulations)• However, they might still be ininfluential for option pricing• Can we price options with the Black and Scholes model?• Test: plot the implied volatility as a function of moneyness and ma-

turity1.17

The smile effect



1.18

Smiles and smirks

Source: Bakshi, Cao and Chen (1997) 1.19

The volatility surface

1.20

2 Discrete-time models for returnsBlack-Scholes-Merton and homoskedastic models

• The discretization of the BSM model, would imply the followinghomoskedastic model for returns:

rt = c+σεt



with the εt being iid noise. This model is also called a random walk(or white noise).• We immediately rule out such a model• Adding ARMA(p,q) terms is prevented by the need of a null auto-

correlation function (efficent markets, random walk hypothesis)• We need the ARCH-GARCH family.

1.21

2.1 ARCH modelARCH models

• We need to introduce heteroskedasticity ans leptokurtosis in our model.• The first contribution in this sense has been given by Robert En-

gle (1982, Econometrica) with the introduction of the ARCH (Auto-Regressive Conditional Heteroskedasticity) model.• The ARCH(p) model:

rt =√

ht · εtht = ω +∑

pi=1 αir2

t−i

where εt is iid noise with E[ε2t ] = 1, ω > 0, αi ≥ 0, i = 1, . . . , p−1,

αp > 0.1.22

ARCH(1)

• Tha ARCH(1) model:

rt = E[rt |Ft−1]+ εt

Var[rt |Ft−1] = ω +αr2t−1

• The conditional mean and conditional variance are allowed to varywith Ft−1• The unconditional mean and unconditional variance are instead con-

stant (as it should be for a stationary process)• Thus also the k-step ahead forecasts depend on Ft−1, unless k→ ∞.

1.23



Properties of the ARCH model

• To ensure second-order stationarity, we need (why?):

p

∑i=1

αi < 1

• Unconditional mean and variance:

E[rt ] = 0

E[r2t ] =Var(rt) =

ω

1−∑pi=1 αi

• Conditional mean and variance:

E[rt |rt−1] = 0

E[r2t |r2

t−1] = ht

• The autocorrelation function of returns is then zero (as it should be),while ht is the conditional variance.

1.24

Leptokurtosis in ARCH models

• Can the ARCH(p) model reproduce the leptokurtosis?• We can answer from a theoretical standpoint. Since E[rt ] = 0, the

kurtosis is given by• Let us now compute it for the ARCH(1) and assuming ε iid Normal.

We can show that:

k(rt) =E[r4

t ]

E[r2t ]

2 = 31−α2

1−3α2

• It is equal to 3 iff α = 0.• Condition for the existence of kurtosis:

α <1√3' 0.577,

• Strong stationarity condition:

E[log(αε2t )]< 0

1.25



Heteroskedasticity in the ARCH models

• We can rewrite:

r2t = ω +

p

∑i=1

αir2t−1 +(r2

t −ht)︸︷︷︸ηt

• Setting r2t −ht =ηt , we can see that E[ηt ] = 0. Thus the above model

is similar to an AR(p) model for squared returns. This is exactly thetype of heteroskedasticity we are looking for (note that the ηt’s arenot actual residuals: why?)• In particular, for the ARCH(1) model we have that the first term of

the autocorrelation function is α .1.26

2.2 GARCH modelGARCH models

• The GARCH model, introduced in Bollerslev (1986) is an extensionof the ARCH model (G means “Generalized”).• It has been introduced to soften the drawbacks of the ARCH model.• The GARCH(p,q) model is defined as:

rt =√

ht · εtht = ω +∑

pi=1 αir2

t−i +∑qi=1 βiht−i

where ω > 0, αi ≥ 0, i = 1, . . . , p−1, βi ≥ 0, i = 1, . . . ,q−1, αp >0, βq > 0..

1.27

Properties of the GARCH model

• To ensure stationarity of the GARCH(1,1) model, we need:

E[αε2t +β ]< 0

• It is sufficient thatα +β < 1

• Unconditional mean and variance:

E[rt ] = 0

E[r2t ] =Var(rt) =

ω

1−∑pi=1 αi−∑

qi=1 βi



• Conditional mean and variance:

E[rt |rt−1] = 0

E[r2t |r2

t−1] = ht1.28

GARCH(1,1) representations

• “ARMA(1,1)” representation of the GARCH(1,1) model:

r2t = ω +(α +β )r2

t−1 +(r2t −ht)︸︷︷︸

ηt

−βηt−1

• ARCH(∞) representation of the GARCH(1,1) model:

ht = ω +αr2t−1 +βht−1 =

= ω +αr2t−1 +β (ω +αr2

t−2 +βht−2) == ω

1−β+α(r2

t−1 +β r2t−2 +β 2r2

t−3 + . . .)

1.29Why the GARCH(1,1) model can reproduce the long memory of squared

returns, while the ARCH(1) model cannot? To see this, remind that theautocorrelation function of r2

t in the ARCH(1) model (using its “AR(1)”representation) is ρh = αh. This implies that the characteristic decay timeof ρh is τ = 1/ logα . τ will be long (in daily terms, we need τ ∼ 102) if α

is close to one, but this is prevented by the stationarity condition of the kur-tosis: α < 1/

√3. Clearly this issue might be solved by using a ARCH(p)

model with p > 1.For the GARCH model instead, we might guess from the “ARMA(1,1)”

representation that ρh ' (α +β )h. The precise calculation gives:

ρ1 =α(1−β 2−αβ )

1−β 2−2αβ

ρh = (α +β )ρh−1

which is very close to the guess. Thus, the characteristic decay time isτ ' 1/ log(α +β ), which now can be arbitrarely large since the stationar-ity of kurtosis is guaranteed by a weaker condition. Indeed, on real finan-cial returns, we always estimate α +β ' 1.



Leptokurtosis of GARCH modelsIn practice, only the GARCH(1,1) model is used. We will stick to this

good tradition and write α = α1, β = β1. We then have:

E[r2t ] = E[ht ] =

ω

1−α−β

• The kurtosis is given by:

k(rt) = kε

1− (α +β )2

1− (α +β )2−α2(kε −1),

and it is equal to kε iff α = 0 (absence of ARCH effects).• If ε is standard Normal, the kurtosis will be finite if:

3α2 +2αβ +β

2 = (α +β )2 +2α2 < 1.

1.30

Existence of kurtosis in GARCH(models)From Francq and Zakoian (2010).

1.31

Serial correlation in the GARCH process

• It can easily be computed from the “ARMA(1,1)” representation.• We have, if the kurtosis exists,

Corr(r2t ,r

2t−h) =

α(1−β (α +β ))

1− (α +β )2 +α2 (α +β )h−1

and they are all positive.1.32



Estimation of the GARCH(1,1) model

• Maximum likelihood estimation. Assume εt ∼N (0,1). Then:

P(rt |rt−1) =1√

2πhtexp(− r2

t2ht

)• We have at disposal T observations r1, . . . , rT . The log-likelihood is

given by:

logL(ω,α,β ) =1T

T

∑t=2

logP(rt |rt−1)

=1T

T

∑t=2

(−1

2log2πht−

r2t

2ht

)• The value of ht is found recursively. The total likelihood is:

logL(ω,α,β ) =1T

T

∑i=2

(−1

2log2π(ω +α r2

t−1 +βht−1)

− r2t

2(ω +α r2t−1 +βht−1)

)1.33

2.3 Other models in the GARCH familyThe GARCH-M model and the risk-return tradeoff

• In the GARCH-M (Garch-in-Mean) we introduce the dependence onreturns on conditional variance which is postulated by the economictheory (e.g., by the CAPM). The positive correlation between returnsand volatility is named risk-return tradeoff.• The specification of the model is:

rt = µ + γht +√

htεtht = ω +αr2

t−1 +βht−1

• Given the inherent noise of financial returns, the estimates of γ aretypically very difficult and require very long time series to turn outpositive and significant.

1.34



Asymmetric GARCH models

• The ARCH and GARCH model analyzed so far cannot account forthe leverage effect, that is the tendency of negative returns to impactmore on future variance than positive returns.• The Glosten, Jagannathan and Runkle (1989) model:

rt =√

ht · εtht = ω +αε2

t−1 +βht−1 + γε2t−1I{εt−1<0},

• The leverage effect materializes in a positive estimate of γ .1.35

The EGARCH model

• The Exponential-GARCH model was introduced by Nelson (1991):

rt =√

ht · εt

loght = ω +p

∑i=1

βi loght−1

+q

∑i=1

αi [ϑεt−i + γ (|εt−i|−E|εt−i|)]

• By being specified in logs, it is better suited to the data.1.36

2.4 Stochastic volatility modelsIs conditional variance deterministic?

• In the models of the ARCH-GARCH family, the variance at time t iscompletely determined by the variance and the returns at time t−1:it is conditionally deterministic.• However, a possibility suggested by financial returns is that also the

variance is affected by an idiosyncratic noise term, exactly as returns.• When there is an additional noise (with respect to those for returns)

which affects the evolution of the variance, we say that we are in astochastic volatility model

1.37



A discrete-time stochastic volatility model

• A stochastic volatility model has the structure:

rt = σtεt

• σt is a random variable, which is positive by the log-transformation.• Conditionally to the information up to time t−1, σt is now a random

variable.• It is easy to check that, if σt is indpendent of εt , these models auto-

matically generate leptokurtosis. Indeed:

k =E[r4

t ]

E[r2t ]

2 =kεE[σ4

t ]

E[σ2t ]

2 ,

where kε is the kurtosis εt . By Jensen inequality,

E[σ4t ]≥ E[σ2

t ]2

where the equality sign holds iff σt is deterministic.• Also notice that the serial correlation of rt is still null, while the serial

correlation of r2t will inherit the serial correlation of σt .

1.38

Volatility distribution

• In agreement with empirical observations, it is reasonable to assumethat a lognormal distribution for σt . This is done in the model pro-posed by Taylor (1989).• In this model, logσ 'N (α,β 2).• This implies E[σt ] = exp(α + 1

2β 2). Higher moments can be com-puted with the formula:

E[σnt ] = exp

(nα +

12

n2β

2)

• The kurtosis of this model is k = 3e4β 2.

• Serial correlation in r2t can be introduced with an AR(1) process for

logσt :logσt = α +ϕ(logσt−1−α)+ηt

where ηt is iid Normal noise which might be correlated with εt (toallow for leverage effects) and with a variance of β 2(1−ϕ2).• Big problem with discrete-time models: how do we find option

prices?1.39



3 Realized variance• The definition of volatility may vary wildly around the idea of the

standard deviation of price movements• Let Xt be a stochastic process (log-price or interest rate)• Representation theorem + “smoothness conditions”: Ito’s semi-

martingales (see Jacod and Shiraev, Theorem I.4.18)

dXt = µtdt +σtdWt +dJt

• Wt is a standard Brownian motion• Jt is a jump process with jump measure µ(dx,dt) and compensator

ν(dt)• Volatility is the stochastic process σt

1.40

Quadratic variation

• The quadratic variation of two semimartingales X and Y is definedby the following process:

[X ,Y ]t := XtYt−X0Y0−∫ t

0Xs−dYs−

∫ t

0Ys−dXs

• We also write:

[X ]t := [X ,X ]t = X2t −X2

0 −2∫ t

0Xs−dXs

1.41

Quadratic variation: an alternative definition

• A sequence τn,m,m ∈ N of adapted subdivisions is called a Riemannsequence if lim

n→∞supm∈N

[τn,m+1− τn,m

]= 0 for all t ∈ R+.

• Let X and Y be two semimartingales. Then for every Riemann se-quence τn,m of adapted subdivisions, the process Sτn(X ,Y ) definedby:

Sτn(X ,Y )t = ∑m≥1

(Xτn,m+1∧t−Xτn,m∧t

)(Yτn,m+1∧t−Yτn,m∧t

)converges, for m→ ∞, to the process [X ,Y ]t , in measure and uni-formly on every compact interval.

1.42



Properties of quadratic variation

• [X ,Y ]0 = 0• [X ,Y ]t = [X−X0,Y −Y0]t

• [X ,Y ]t =14([X +Y ]t− [X−Y ]t) (polarization)

• [X ,Y ]t is of finite variation.• [X ,X ]t is increasing in t.• ∆[X ,Y ]t = ∆Xt +∆Yt , where ∆X = Xt−Xt−• if X or Y is continuous, then [X ,Y ] is continuous as well.

1.43

• We can estimate σt using infill asymptotics:

– µtdt is of order ∆

– if Jt is a Lèvy process, dJt is of order ∆ (stochastic continuity)

– σtdWt is of order√

∆

• σt is the leading order as ∆→ 0.• We can estimate it in a fixed time span [0,T ] with observations at

timessupi(ti+1− ti)→ 0

• We need high-frequency data1.44

• From the above discussion, it is clear that estimating σt is similar tothe problem of numerical differentiation• A possible solution is concentrating on integrated volatility:∫ T

0σ

2s ds

• Integrated volatility plays a fundamental role in:

– option pricing

– portfolio selection

– volatility forecasting

– estimating stochastic volatility models1.45



Evenly sampled returns

• Consider an interval [0,T ] with T fixed. Consider a real numberδ = T/n.• We define the evenly sampled returns as:

∆ jX = X jδ −X( j−1)δ , j = 1, . . . ,n

• The case with unevenly sampled returns is similar (even if times arestochastic, but not in the case they are not independent from price)

1.46

Realized variance

• Realized variance is defined as:

RVδ (X)t =[t/δ ]

∑j=1

(∆ jX

)2

• It is the cumulative sum of intraday returns• It depends on δ ; the larger the δ , the closer the approximation to

quadratic variation• For financial data, δ cannot be smaller than the average time between

transactions.• As δ → 0, if J = 0, we have:

δ−1

2

(RVδ (X)t−

∫ t

0σ

2s ds)−→L

√2∫ t

0σ

2s dW ′s

where W ′ is a Brownian motion uncorrelated with W .1.47

Early literature on realized variance

• All ideas are already in Merton (1980), see also Schwert (1989).• Andersen and Bollerslev (1998): show that the bad forecasting per-

formance of GARCH models is due to the volatility estimator• Andersen, Bollerslev, Diebold and Labys (2003).• Jacod and Protter (1998): central limit theorem for RV• Barndorff-Nielsen and Shephard (2002)• Reviews: McAleer and Medeiros (2008) and many others

1.48



Sketch of the proof

• When dJ = 0, it follows from Ito’s lemma that:

X2t = [X ]t +2

∫ t

0XsdXs

• thus(∆ jX

)2=(Xδ j−Xδ ( j−1))

2 = [X ]δ j−[X ]δ ( j−1)+2∫

δ j

δ ( j−1)(Xs−Xδ ( j−1))dXs

• which implies:

δ−1

2 (RVδ (X)t− [X ]t) = 2δ−1

2

[t/δ ]

∑j=1

∫δ j

δ ( j−1)(Xs−Xδ ( j−1))dXs

• and (∆ jX

)2= 2δ

−12

∫δ [t/δ ]

0(Xs−Xδ [s/δ ])dXs

• As δ → 0, the right term converges to:

δ−1

2

∫ t

0(Xs−Xδ [s/δ ])dXs −→

1√2

∫ t

0σ

2s dW ′s ,

• Intuitively:∫δ j

δ ( j−1)(Xs−Xδ ( j−1))dXs ' σ

2δ ( j−1)

∫δ j

δ ( j−1)

(Ws−Wδ ( j−1)

)dWs

1.49

Power Variation

• The realized power variation (see, e.g., Barndorff-Nielsen and Shep-hard 2004) of order γ is defined as:

PV∆(X)γ

t = ∆1−γ/2

[t/∆]

∑j=1|∆ jX |γ

• the normalization term ∆1−γ/2 disappears when γ = 2, goes to infin-ity when γ > 2 and goes to zero when γ < 2.• when γ = 2 PV coincides with RV.

1.50



Power variation of continuous semimartingalesWhen J = 0 and σ ,µ are independent of W , power variation converges

to the integrated power-volatility:

(J = 0) p− lim∆→0

PV∆(X)γ

t = µγ

∫ t

0σ

γs ds

where

µγ = E(|u|γ) = 2γ/2 Γ( p+12 )

Γ(1/2)

u'N (0,1)

. 1.51

Feasible confidence intervals for RV

• We can then rewrite:

(J = 0)δ−

12 (RVδ (X)t− [X ]t)√

2QVδ (X)t−→N (0,1)

• On moderate sample, the performance of these confidence intervalsis quite poor; this is mostly due to the non-normality of the distri-bution of realized volatility, better results can be obtained passing tologs,

(J = 0)δ−

12 (logRVδ (X)t− log[X ]t)√

2 QVδ (X)t

(RVδ (X)t)2

−→N (0,1)

1.52

Financial Returns Intraday Time SeriesSPX500 index, one day of transactions, frequency: all data

1.53



Estimating Volatility with Microstructure Noise

• Consider q non-overlapping subsamples.• Two scale estimator (Zhang et al., 2005):

V ts =1q

V subsamples︸︷︷︸−→q

∫σ2

s ds+Ξ

q

− 1q

V all data︸︷︷︸−→Ξ

q

−→∫

σ2s ds

• The rate of convergence is n−16 .

• See also the kernel estimator of Barndorff-Nielsen et al (2008). Bothare not robust to jumps.

1.54

Combining both: preaveraged estimator

• Preaveraging (Jacod et al., 2009) is a two-step procedure for gettingrid of microstructure noise• First step: compute averaged returns:

ri =n−kn

∑i=1

g(

in

)ri

with kn→ ∞ but kn/n→ 0• Second step: apply bipower variation• Optimal rate of convergence: n−1/4

1.55

PreaveragingSPX500 index, one day of transactions, frequency: all data

1.56



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