Modelling financial returns

Transcription

1 Financial Econometrics Lecture 1 Modelling financial returns Roberto Renò Università di Siena February 22, Stylized facts A 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: ds t = µs t dt + σs t dw t which implies a lognormal distribution for the price. Option payoff (European) at maturity T : φ(s T ) Denote the option value by F(t,S) Replication arguments: F t + rs F S σ 2 S 2 2 F S 2 = rf F(T,S) = φ(s) Random walk 1.2 The distribution of returns Roberto Renò, 2011 c 1

2 According to the lognormal model, we have, for every > 0: S t+ = S t exp ((µ σ 2 ) ) + σw( ) 2 and then log S(t + ) S(t) N In discrete time, we may write: ( ) (µ σ 2 /2),σ 2 r t = µ + σε t where ε t is iid noise and µ = µ σ 2 / Moments of a distribution Given a random variable X, we define its k th (centered) moment as: M 1 = E[X] Characteristic function: M k = E[(X E[X]) k ], k 2 φ(u) = E[e iux ] has the property, when E[X] = 0, that: M k = 1 k φ i n u k Mean = E[X] Variance = M 2 M 3 Skewness = (M 2 ) 1.5 Kurtosis = M 4 (M 2 ) 2 u=0 Normalizations makes skewness and kurtosis pure numbers, independent from the units of X. 1.4 The moment-generating function is close to the characteristic function, and it is defined as: M X (t) = E[e tx ], Roberto Renò, 2011 c 2

3 with t R. To see that it is linked (as the characteristic function) to moments, just consider the Taylor expansion: which implies, taking expectations, e tx = 1 +tx t2 X t3 X M X (t) = 1 +tm t2 M t3 M Immediately, we get: M k = k M X (t) t k t=0 Sample moments Given a sample X 1,...,X n, we define its k th centered sample moment, for k 2 as: where ˆM k = 1 n n i=1 ˆµ = 1 n (X i ˆµ) k n X i i=1 sample skewness: ŝ = ˆM 3 ˆM sample kurtosis: ˆk = ˆM 4 ˆM 2 2 Check the following concepts: 1. Sample moments are random variables 2. What is the relation between sample moments and the moments introduced 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 ŝ = 0 Distributions with skewness different from 0 are not symmetric. What is the sign of the skewness of the lognormal distribution? Roberto Renò, 2011 c 3

4 2. Check if ˆk = 3 Distributions with kurtosis above 3 are called leptokurtic or fattailed. Distributions with kurtosis less than 3 are called platokurtic. EXERCISE: PROVE THAT THE KURTOSIS OF THE NORMAL DISTRIBUTION IS Solution of the exercise. It is harmless to work with a standard normal random variable X, for which is E[X 2 ] = 1. We then have: kurtosis = E[X 4 ] = = x 4 1 e x2 2 } 2π {{} normal density dx ) dx x 3( x 1 2π e x2 2 }{{} integration by parts = x 3( 1 2π e x2 2 = 3. primitive o f 1 2π e x2 2 ) dx + + ( ) 1 3x 2 e x2 2 dx 2π Mix the two above in the Jarque-Bera test: JB = n 6ŝ2 + n 24 (ˆk 3) 2, which, under the null hyphothesis of normality, is distributed as a χ 2 with 2 degrees of freedom. Famous studies on the non-normality of returns Benoit Mandelbrot (1963), the father of the theory of fractals, using cotton prices suggested the possibility of infinite variance (Lévy flights). 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) Roberto Renò, 2011 c 4

5 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 independent from each other. Roberto Renò, 2011 c 5

6 Test: compute the autocorrelation function of returns Recall the definition of the autocorrelation function: ρ X (h) = n h i=1 (X i ˆµ)(X i+h ˆµ) n i=1 (X i ˆµ) 2, 1.12 The Efficient Market Hypothesys The 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 concept of invisible hand for the prices of financial securities. It can be stated in three forms, weak, semi-strong and strong. Definition 1 (The Efficient Market Hypothesis). A financial market for a given securities is said to be informationally efficient if the price of the security 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) Believe it or not, the weak form has been extensively tested and it works very well. It implies that you cannot make money using technical analysis Beyond linear independence We have a strong theoretical argument for a null autocorrelation function. We also tested it empirically. Is this the end of the story? Please notice that, while two independent random variables are certainly uncorrelated, this does not imply that two uncorrelated random variables 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 Scholes model Roberto Renò, 2011 c 6

7 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 clustering. 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. Stylized facts about financial returns Summarizing, financial returns are characterized by: fat tails The skewness of returns is close to zero, their kurtosis is larger than 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 Can BSM price options, at least? The failures of the BSM model make its use questionable for all the applications 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 maturity 1.17 The smile effect Roberto Renò, 2011 c 7

8 1.18 Smiles and smirks Source: Bakshi, Cao and Chen (1997) 1.19 The volatility surface Discrete-time models for returns Black-Scholes-Merton and homoskedastic models The discretization of the BSM model, would imply the following homoskedastic model for returns: r t = c + σε t Roberto Renò, 2011 c 8

9 with the ε t 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 autocorrelation function (efficent markets, random walk hypothesis) We need the ARCH-GARCH family ARCH model ARCH models We need to introduce heteroskedasticity ans leptokurtosis in our model. The first contribution in this sense has been given by Robert Engle (1982, Econometrica) with the introduction of the ARCH (Auto- Regressive Conditional Heteroskedasticity) model. The ARCH(p) model: r t = h t ε t h t = ω + p i=1 α ir 2 t i where ε t is iid noise with E[ε 2 t ] = 1, ω > 0, α i 0, i = 1,..., p 1, α p > ARCH(1) Tha ARCH(1) model: r t = E[r t F t 1 ] + ε t Var[r t F t 1 ] = ω + αr 2 t 1 The conditional mean and conditional variance are allowed to vary with F t 1 The unconditional mean and unconditional variance are instead constant (as it should be for a stationary process) Thus also the k-step ahead forecasts depend on F t 1, unless k Roberto Renò, 2011 c 9

10 Properties of the ARCH model To ensure second-order stationarity, we need (why?): p α i < 1 i=1 Unconditional mean and variance: Conditional mean and variance: E[r t ] = 0 E[rt 2 ω ] = Var(r t ) = 1 p i=1 α i E[r t r t 1 ] = 0 E[r 2 t r 2 t 1 ] = h t The autocorrelation function of returns is then zero (as it should be), while h t is the conditional variance Leptokurtosis in ARCH models Can the ARCH(p) model reproduce the leptokurtosis? We can answer from a theoretical standpoint. Since E[r t ] = 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(r t ) = E[r4 t ] E[r 2 t ] 2 = 3 1 α2 1 3α 2 It is equal to 3 iff α = 0. Condition for the existence of kurtosis: Strong stationarity condition: α < , E[log(αε 2 t )] < Roberto Renò, 2011 c 10

11 Heteroskedasticity in the ARCH models We can rewrite: r 2 t = ω + p i=1 α i r 2 t 1 + (r2 t h t ) }{{} η t Setting r 2 t h t = η t, we can see that E[η t ] = 0. Thus the above model is similar to an AR(p) model for squared returns. This is exactly the type of heteroskedasticity we are looking for (note that the η t s are not actual residuals: why?) In particular, for the ARCH(1) model we have that the first term of the autocorrelation function is α GARCH model GARCH models The GARCH model, introduced in Bollerslev (1986) is an extension of 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: r t = h t ε t h t = ω + p i=1 α ir 2 t i + q i=1 β ih t i where ω > 0, α i 0, i = 1,..., p 1, β i 0, i = 1,...,q 1, α p > 0, β q > Properties of the GARCH model To ensure stationarity of the GARCH(1,1) model, we need: E[αεt 2 + β] < 0 It is sufficient that α + β < 1 Unconditional mean and variance: E[r t ] = 0 E[rt 2 ω ] = Var(r t ) = 1 p i=1 α i q i=1 β i Roberto Renò, 2011 c 11

12 Conditional mean and variance: E[r t r t 1 ] = 0 E[r 2 t r 2 t 1 ] = h t 1.28 GARCH(1,1) representations ARMA(1,1) representation of the GARCH(1,1) model: rt 2 = ω + (α + β)rt (r2 t h t ) }{{} η t βη t 1 ARCH( ) representation of the GARCH(1,1) model: h t = ω + αrt βh t 1 = = ω + αrt β(ω + αr2 t 2 + βh t 2) = = ω 1 β + α(r2 t 1 + βr2 t 2 + β 2 rt ) 1.29 Why the GARCH(1,1) model can reproduce the long memory of squared returns, while the ARCH(1) model cannot? To see this, remind that the autocorrelation function of rt 2 in the ARCH(1) model (using its AR(1) representation) is ρ h = α h. This implies that the characteristic decay time of ρ h is τ = 1/logα. τ will be long (in daily terms, we need τ 10 2 ) if α is close to one, but this is prevented by the stationarity condition of the kurtosis: 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 ρ h (α + β) h. The precise calculation gives: ρ 1 = α(1 β 2 αβ) 1 β 2 2αβ ρ h = (α + β)ρ h 1 which is very close to the guess. Thus, the characteristic decay time is τ 1/log(α + β), which now can be arbitrarely large since the stationarity of kurtosis is guaranteed by a weaker condition. Indeed, on real financial returns, we always estimate α + β 1. Roberto Renò, 2011 c 12

13 Leptokurtosis of GARCH models In practice, only the GARCH(1,1) model is used. We will stick to this good tradition and write α = α 1, β = β 1. We then have: The kurtosis is given by: E[r 2 t ] = E[h t ] = ω 1 α β k(r t ) = k ε 1 (α + β) 2 1 (α + β) 2 α 2 (k ε 1), and it is equal to k ε iff α = 0 (absence of ARCH effects). If ε is standard Normal, the kurtosis will be finite if: 3α 2 + 2αβ + β 2 = (α + β) 2 + 2α 2 < Existence of kurtosis in GARCH(models) From Francq and Zakoian (2010) Serial correlation in the GARCH process It can easily be computed from the ARMA(1,1) representation. We have, if the kurtosis exists, Corr(rt 2,rt h 2 α(1 β(α + β)) ) = 1 (α + β) 2 + α2(α + β)h 1 and they are all positive Roberto Renò, 2011 c 13

14 Estimation of the GARCH(1,1) model Maximum likelihood estimation. Assume ε t N (0,1). Then: P(r t r t 1 ) = 1 ( ) exp r2 t 2πht 2h t We have at disposal T observations ˆr 1,..., ˆr T. The log-likelihood is given by: logl(ω,α,β) = 1 T = 1 T T t=2 T t=2 logp(ˆr t ˆr t 1 ) ( 1 ) 2 log2πh t ˆr2 t 2h t The value of h t is found recursively. The total likelihood is: logl(ω,α,β) = 1 T T ( 1 i=2 2 log2π(ω + α ˆr2 t 1 + βh t 1) ) 2(ω + α ˆr t βh t 1) ˆr 2 t Other models in the GARCH family The GARCH-M model and the risk-return tradeoff In the GARCH-M (Garch-in-Mean) we introduce the dependence on returns on conditional variance which is postulated by the economic theory (e.g., by the CAPM). The positive correlation between returns and volatility is named risk-return tradeoff. The specification of the model is: r t = µ + γh t + h t ε t h t = ω + αr 2 t 1 + βh t 1 Given the inherent noise of financial returns, the estimates of γ are typically very difficult and require very long time series to turn out positive and significant Roberto Renò, 2011 c 14

15 Asymmetric GARCH models The ARCH and GARCH model analyzed so far cannot account for the leverage effect, that is the tendency of negative returns to impact more on future variance than positive returns. The Glosten, Jagannathan and Runkle (1989) model: r t = h t ε t h t = ω + αε 2 t 1 + βh t 1 + γε 2 t 1 I {ε t 1 <0}, The leverage effect materializes in a positive estimate of γ The EGARCH model The Exponential-GARCH model was introduced by Nelson (1991): r t = h t ε t logh t = ω + + p i=1 q i=1 β i logh t 1 α i [ϑε t i + γ ( ε t i E ε t i )] By being specified in logs, it is better suited to the data. 2.4 Stochastic volatility models Is conditional variance deterministic? In the models of the ARCH-GARCH family, the variance at time t is completely 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 a stochastic volatility model Roberto Renò, 2011 c 15

16 A discrete-time stochastic volatility model A stochastic volatility model has the structure: r t = σ t ε t σ t is a random variable, which is positive by the log-transformation. Conditionally to the information up to time t 1, σ t is now a random variable. It is easy to check that, if σ t is indpendent of ε t, these models automatically generate leptokurtosis. Indeed: k = E[r4 t ] E[rt 2 ] 2 = k εe[σt 4 ] E[σt 2 ] 2, where k ε is the kurtosis ε t. By Jensen inequality, E[σ 4 t ] E[σ 2 t ] 2 where the equality sign holds iff σ t is deterministic. Also notice that the serial correlation of r t is still null, while the serial correlation of r 2 t will inherit the serial correlation of σ t Volatility distribution In agreement with empirical observations, it is reasonable to assume that a lognormal distribution for σ t. This is done in the model proposed by Taylor (1989). In this model, logσ N (α,β 2 ). This implies E[σ t ] = exp(α β 2 ). Higher moments can be computed with the formula: E[σ n t ] = exp (nα + 12 n2 β 2 ) The kurtosis of this model is k = 3e 4β 2. Serial correlation in r 2 t can be introduced with an AR(1) process for logσ t : logσ t = α + ϕ(logσ t 1 α) + η t where η t is iid Normal noise which might be correlated with ε t (to allow for leverage effects) and with a variance of β 2 (1 ϕ 2 ). Big problem with discrete-time models: how do we find option prices? 1.39 Roberto Renò, 2011 c 16

17 3 Realized variance The definition of volatility may vary wildly around the idea of the standard deviation of price movements Let X t be a stochastic process (log-price or interest rate) Representation theorem + smoothness conditions : Ito s semimartingales (see Jacod and Shiraev, Theorem I.4.18) dx t = µ t dt + σ t dw t + dj t W t is a standard Brownian motion J t is a jump process with jump measure µ(dx,dt) and compensator ν(dt) Volatility is the stochastic process σ t 1.40 Quadratic variation The quadratic variation of two semimartingales X and Y is defined by the following process: We also write: t t [X,Y ] t := X t Y t X 0 Y 0 X s dy s Y s dx s 0 0 [X] t := [X,X] t = X 2 t X t 0 X s dx s 1.41 Quadratic variation: an alternative definition A sequence τ n,m,m [ N of adapted] subdivisions is called a Riemann sequence if lim τn,m+1 τ n,m = 0 for all t R+. sup n m N Let X and Y be two semimartingales. Then for every Riemann sequence τ n,m of adapted subdivisions, the process S τn (X,Y ) defined by: ( )( S τn (X,Y ) t = Xτn,m+1 t X τn,m t Yτn,m+1 t Y τn,m t) m 1 converges, for m, to the process [X,Y ] t, in measure and uniformly on every compact interval Roberto Renò, 2011 c 17

18 Properties of quadratic variation [X,Y ] 0 = 0 [X,Y ] t = [X X 0,Y Y 0 ] t [X,Y ] t = 1 4 ([X +Y ] t [X Y ] t ) (polarization) [X,Y ] t is of finite variation. [X,X] t is increasing in t. [X,Y ] t = X t + Y t, where X = X t X t if X or Y is continuous, then [X,Y ] is continuous as well. We can estimate σ t using infill asymptotics: µ t dt is of order if J t is a Lèvy process, dj t is of order (stochastic continuity) σ t dw t is of order σ t is the leading order as 0. We can estimate it in a fixed time span [0,T ] with observations at times sup i (t i+1 t i ) 0 We need high-frequency data From the above discussion, it is clear that estimating σ t is similar to the problem of numerical differentiation A possible solution is concentrating on integrated volatility: T 0 σ 2 s ds Integrated volatility plays a fundamental role in: option pricing portfolio selection volatility forecasting estimating stochastic volatility models 1.45 Roberto Renò, 2011 c 18

19 Evenly sampled returns Consider an interval [0,T ] with T fixed. Consider a real number δ = T /n. We define the evenly sampled returns as: j X = X jδ X ( j 1)δ, j = 1,...,n The case with unevenly sampled returns is similar (even if times are stochastic, but not in the case they are not independent from price) 1.46 Realized variance Realized variance is defined as: RV δ (X) t = [t/δ] ( j X ) 2 j=1 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 ( t RV δ (X) t σs 2 ds 0 ) t L 2 0 σ 2 s dw s where W is a Brownian motion uncorrelated with W 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 performance 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 Roberto Renò, 2011 c 19

20 Sketch of the proof When dj = 0, it follows from Ito s lemma that: thus t Xt 2 = [X] t + 2 X s dx s 0 ( j X ) 2 = (Xδ j X δ( j 1) ) 2 = [X] δ j [X] δ( j 1) +2 δ j which implies: and δ 1 2 (RVδ (X) t [X] t ) = 2δ 1 2 δ( j 1) [t/δ] δ j (X s X δ( j 1) )dx s j=1 δ( j 1) ( j X ) δ[t/δ] 2 = 2δ 1 2 (X s X δ[s/δ] )dx s As δ 0, the right term converges to: δ 1 2 t 0 0 (X s X δ[s/δ] )dx s 1 2 t 0 σ 2 s dw s, (X s X δ( j 1) )dx s Intuitively: δ j δ( j 1) (X s X δ( j 1) )dx s σ 2 δ( j 1) δ j δ( j 1) (W s W δ( j 1) ) dw s 1.49 Power Variation The realized power variation (see, e.g., Barndorff-Nielsen and Shephard 2004) of order γ is defined as: PV (X) γ t = [t/ ] 1 γ/2 j=1 j X γ the normalization term 1 γ/2 disappears when γ = 2, goes to infinity when γ > 2 and goes to zero when γ < 2. when γ = 2 PV coincides with RV Roberto Renò, 2011 c 20

21 Power variation of continuous semimartingales When J = 0 and σ, µ are independent of W, power variation converges to the integrated power-volatility: where t (J = 0) p lim PV (X) t γ = µ γ σs γ ds 0 0 µ γ = E( u γ ) = 2 u N (0,1) p+1 γ/2γ( 2 ) Γ(1/2) Feasible confidence intervals for RV We can then rewrite: (J = 0) δ 1 2 (RV δ (X) t [X] t ) 2QVδ (X) t N (0,1) On moderate sample, the performance of these confidence intervals is quite poor; this is mostly due to the non-normality of the distribution of realized volatility, better results can be obtained passing to logs, (J = 0) δ 1 2 (logrv δ (X) t log[x] t ) N (0,1) 2 QV δ (X) t (RV δ (X) t ) Financial Returns Intraday Time Series SPX500 index, one day of transactions, frequency: all data 1.53 Roberto Renò, 2011 c 21

22 Estimating Volatility with Microstructure Noise Consider q non-overlapping subsamples. Two scale estimator (Zhang et al., 2005): V ts = 1 q V subsamples }{{} q σ 2 s ds+ξ q 1 all data V q }{{} Ξ q σ 2 s ds The rate of convergence is n 1 6. See also the kernel estimator of Barndorff-Nielsen et al (2008). Both are not robust to jumps Combining both: preaveraged estimator Preaveraging (Jacod et al., 2009) is a two-step procedure for getting rid of microstructure noise First step: compute averaged returns: r i = n k n ( ) i g r i i=1 n with k n but k n /n 0 Second step: apply bipower variation Optimal rate of convergence: n 1/ Preaveraging SPX500 index, one day of transactions, frequency: all data 1.56 Roberto Renò, 2011 c 22

23 References Aït-Sahalia, Y. and J. Jacod (2008). Estimating the degree of activity of jumps in high frequency data. Annals of Statistics. Forthcoming. Andersen, T. and T. Bollerslev (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39, Andersen, T., T. Bollerslev, F. Diebold, and P. Labys (2003). Modeling and forecasting realized volatility. Econometrica 71, Barndorff-Nielsen, O., P. Hansen, A. Lunde, and N. Shephard (2008). Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76(6), Barndorff-Nielsen, O. E. and N. Shephard (2002). Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, Series B 64, Barndorff-Nielsen, O. E. and N. Shephard (2004). Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2, Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, Engle, R. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica 50, Fama, E. (1965). The behavior of stock market prices. Journal of Business 38, Fama, E. (1970). Efficient capital markets: a review of theory and empirical work. Journal of Finance 25, Jacod, J., Y. Li, P. Mykland, M. Podolskij, and M. Vetter (2009). Microstructure noise in the continuous case: the pre-averaging approach. Stochastic Processes and their Applications 119(7), Jacod, J. and P. Protter (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Annals of Probability 26, Jacod, J. and A. N. Shiryaev (2003). Limit Theorems for Stochastic Processes. Springer. Mantegna, R. and E. Stanley (1995). Scaling behaviour in the dynamics of an economic index. Nature 376, McAleer, M. and M. Medeiros (2008). Realized volatility: a review. Econometric Reviews 27(1), Merton, R. (1980). On estimating the expected return on the market: an exploratory investigation. Journal of Financial Economics 8, Schwert, G. W. (1989). Why does stock market volatility change over time? Journal of Finance 44(5), Zhang, L., P. A. Mykland, and Y. Aït-Sahalia (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association 100, Roberto Renò, 2011 c 23