a volatility index and the volatility premium in...
TRANSCRIPT
A Volatility Index and the Volatility Premium in Brazil
April 6, 2015
Abstract
In this paper we �rst construct an implied volatility index for Brazil (called "IVol-
BR"), based on daily market prices of options over IBOVESPA. As expected, we show
that the IVol-BR has signi�cant predictive power over future volatility of equity re-
turns. After that, we decompose the IVol-BR into (i) the expected volatility of stock
returns and (ii) the equity volatility premium. This is an interesting decomposition,
once the equity volatility premium should commove with the risk-aversion level of the
representative investor. Therefore, with such a decomposition, we produce a series that
can be seen as a risk-aversion index for the Brazilian investor. Finally, we show that
this measure of risk-aversion is able to predict future stock returns as theory suggests:
when risk-aversion is higher, expected returns are higher.
JEL Codes: G12, G13, G17
1
1 Introduction
This is the �rst article to propose and evaluate an implied volatility index for the Brazilianstock market1. This is our �rst contribution. Besides, this is the �rst article to propose andevaluate a model-free risk-aversion index for the Brazilian investor. This is our secondcontribution.
An implied volatility index is a very useful series for both researchers and practitioners,once it is a �fear gauge� for the �nancial market. The best known example is the VIX, �rstintroduced by the Chicago Board of Options Exchange (CBOE) in 1993. Methodologies onthe construction of implied volatility indices have been evolving and, in 2003, the CBOErevamped the calculation of the VIX. In 2004, the CBOE launched a new exchange, theChicago Futures Exchange, and started trading futures on the new volatility index (suchas volatility and variance swaps).2 In this paper, to compute the IVol-BR, we propose amethodology that combines the one used in the calculation of the �new� VIX with someadjustments that take into account some local aspects of the Brazilian stock market.
An implied volatility index re�ects the dynamics of two very important variables. The�rst relates to the level, or quantity, of risk that investors face: the expected future volatilityof the market portfolio. The second relates to the price of such risk: the risk-aversion of therepresentative investor.
To understand this, one should note that, since options' payo�s are asymmetric, the valueof any option (call or put) is increasing in the expected volatility of the underlying asset.Because of that, options are often used as a protection against changes in volatility. Sincethe typical risk-averse investor dislikes volatility, options are traded with a premium becauseof their insurance value. As a direct consequence, the implied volatility (which is computeddirectly from options prices) also has a premium with respect to the real expected volatility:it should always be higher than the real expected volatility. This is the so-called volatilitypremium.
The more the investor dislikes volatility, the more she is willing to pay for the insurancethat options provide. Therefore, the higher the risk aversion, the higher the volatility pre-mium. Thus, the volatility premium should positively commove with risk-aversion and canbe used as an indicator of how such a variable evolves in time.
To decompose the IVol-BR into (i) the expected future volatility conditional on theinformation set at time t and (ii) the volatility premium at time t, we search for the modelthat best forecasts volatility in Brazil. Following the recent literature on volatility forecasting(Chen and Ghysels (2011), Corsi (2009) and Corsi and Renò (2012)), we use high-frequencydata for this task. After that, from the estimation of this model, we produce a daily measureof expected volatility. Then, the di�erence between the IVol-BR and the estimated expectedvolatility gives us a daily measure of the volatility premium, which should commove withrisk-aversion. Such a methodology is based on Bekaert and Hoerova (2014).
The paper is divided as follows. Section 2 presents the IVol-BR. Section 3 decomposes
1Other related works include: Kapotas et al. (2004), that studied implied volatility of options over Telemarstocks, and Dario (2006) and Brostowicz Junior and Laurini (2010), that studied volatility indices for theBrazilian FX market.
2See CBOE (2009)
2
the IVol-BR into expected volatility and volatility premium, and documents the predictivepower of the last over future returns. Section 4 concludes.
2 Volatility Index for the Brazilian Stock Market
The �rst contribution of this article is to propose a methodology to compute an indexthat captures the forward-looking volatility in the Brazilian stock market. To the best ofour knowledge, this has not been done so far. A possible exception is the Chicago Board ofOptions Exchange (CBOE) volatility index called VXEWZ, which tracks the forward-lookingvolatility of a dollar-denominated index (EWZ) of the Brazilian stock market. However, aswe discuss in section 2.2, VXEWZ contains both the forward-looking volatility of equitiesand of the FX market (R$/US$). The measure we propose, in turn, is a clean measure ofthe forward-looking volatility of the Brazilian stock market only.
Forward-looking volatility measures are seen as market wide �fear gauges� and, given theirevident asset market implications, are of great interest for �nancial and macro economists.Both market and academic interest in equity-index volatility measures have grown rapidlyin the last 20 years.3
2.1 Methodology
The methodology we use to compute the forward-looking volatility in the Brazilian stockmarket (hereinafter, IVol-BR) is based on the one described in Carr and Wu (2009) - the oneused in the calculation of the �new� VIX. On the top of that, we propose some adjustmentsto take into account some local aspects of the Brazilian stock market.
Options over IBOVESPA, the main stock market index of Brazil, expire only on evenmonths: February, April, etc. Because of that, we compute the IVol-BR in order to referto 2-month ahead forward-looking volatilities. It is calculated as a weighted average oftwo volatility vertices: the near-term and next-term implied volatilities of options over theIBOVESPA spot. At a given date t, the near-term refers to the closest expiration date ofthe options over IBOVESPA, while the next-term refers to the expiration date immediatelyfollowing the near-term. For instance, on any day in January 2015, the near-term refersto the options that expire in February 2015, while the next-term refers to the options thatexpires in April 2015. 4
The formula for the near and next-term squared volatilities is the following:
σ2k (t) =
2
Tk − t∑i
4Ki
K2i
ert(Tk−t)Ot (Ki)−j
Tk − t
[F (t, Tk)
K0
− 1
]2
(1)
where
3The best known example is the publication of the so-called volatility index, or VIX, �rst introduced bythe CBOE in 1993. Methodologies on this subject have been evolving and, in 2003, the CBOE revampedthe calculation of the VIX. In 2004, the CBOE launched a new exchange, the Chicago Futures Exchange,and started trading futures on the new volatility index.
4All options expire on the Wednesday closest to the 15th day of the expiration month.
3
• k = 1 if the formula uses the near-term options and k = 2 is the formula uses thenext-term options � that is, σ2
1 (t) and σ22 (t) are, respectively, the squared near- and
the next-term implied volatilities on day t.
• F (t, Tk) is the settlement price on day t of the IBOVESPA futures contract whichexpires on day Tk
• K0 is the option strike value which is closest to F (t, Tk)
• Ki is the strike of the i-th out-of-the-money option: a call if Ki > K0, a put if Ki < K0
and both if Ki = K0
• ∆Ki = 12
(Ki+1 −Ki−1)
• rt is risk-free rate from day t to day Tk, obtained from the daily settlement price of thefutures interbank rate (DI)
• Ot (Ki) is the market price on day t of option with strike Ki
• j is an adjustment factor that can take the values 0, 1 or 2
The adjustment factor j is necessary because of the following. As K0 is de�ned as thestrike closest to F (t, Tk), K0 can be in-the-money. In this case, we need to transform in-the-money call (put) into its counterpart out-of-the-money put (call). For that, we use theput-call parity. This is done by the adjustment factor j, as explained in Carr and Wu (2006).5
After calculating both the near- and next-term implied volatilities using equation (1), wethen aggregate these into a weighted average that corresponds to the 2-month (42 businessdays) implied volatility, as follows:
IV olt = 100×
√[(T1 − t)σ2
1
(NT2 −N42
NT2 −NT1
)+ (T2 − t)σ2
2
(N42 −NT1
NT2 −NT1
)]× N252
N42
(2)
where IV olt is the IVol-BR in percentage points and annualized at time t, NT1 is the numberof minutes from 5 pm of day t until 5 pm of the near-term expiration date T1, NT2 is thenumber of minutes from 5 pm of day t until 5 pm of the next-term expiration date T2, N42
is the number of minutes in 42 business days (42× 1440) and N252 is the number of minutesin 252 business days (252× 1440).
To adapt the methodology above to particular features of the Brazilian market, we dothe following:
5If K0 < F (t, Tk) and there are both a call and a put with such a strike, we set j = 1; if K0 < F (t, Tk)and there is only a call, we set j = 2; if K0 < F (t, Tk) and there is only a put, we set j = 0; if K0 > F (t, Tk)and there are both a call and a put, we set j = 1; if K0 > F (t, Tk) and there is only a call, we set j = 0;�nally, if K0 > F (t, Tk) and there is only a put , we set j = 2. In fact, Carr and Wu (2006) consider only thecase when there are both a call and a put and K0 < F (t, Tk). The other 5 cases, are necessary because (i) wede�ne K0 as the option strike value which is closest to F (t, Tk) - therefore we may have either K0 > F (t, Tk)or K0 < F (t, Tk) and (ii) we can have days with only a call, only a put, or both.
4
• we create the cases where j = 0 and j = 2 in Equation (1) � Carr and Wu (2006)only consider j = 1. This is necessary because (i) we de�ne K0 as the option strikevalue which is closest to F (t, Tk) � therefore we may have either K0 > F (t, Tk) orK0 < F (t, Tk) � and (ii) we can have days with only a call, only a put, or both.
• we restrict the set of options that is used to calculate the IVol-Br index to those tradedbetween 3 p.m. and 6 p.m. - in order to improve the synchronization of the data withthe settlement price of the IBOVESPA futures;
• we use only the last trade that took place in the above-mentioned time interval foreach option ticker;
• we only calculate σ21(t) and σ2
2(t) if, for each vertex, there are at least 2 trades involvingOTM call options at di�erent strikes and 2 trades involving OTM put options also atdi�erent strikes - this is done in order to circumvent errors associated with lack ofliquidity in the options market. If on a given day only one volatility vertex can becalculated, we suppose that the volatility surface is �at and the IVol-Br is set equal tothe computed vertex. If both near- and next-term volatilities cannot be calculated, wereport the index for that day as missing;
• on days when the weight of the second term of equation (2) is negative, we ignore thenext-term volatility (thus, the IVol-Br index equals the near-term volatility);
• the rollover of maturities occurs when the near-term options expire (we have testedrolling-over the vertices a couple of days prior to near-term expiration to avoid mi-crostructure e�ects, but the results do not change).
The volatility index calculated according to equations (1) and (2) could be biased becauseit considers only traded options at a �nite (often small) number of strikes. To assess thepossible loss in the accuracy of the integral calculated with a small number of points, wealso re�ned the grid of options via a linear interpolation (using 2,000 points) of the volatilitysmile that can be obtained from the traded options (based on the procedure suggested byCarr and Wu 2008).6 Results did not change signi�cantly.
6The `coarse' volatility smile for both near and next-term is retrieved from the options market values andthe Black-76 formula. We then `re�ne' the grid of strike prices Ki using the implied volatilities and implieddeltas of the options with the formula:
Ki = F (t, T ) exp[−wσi
√T − tN−1
(|4i|+ 1
2σ2i (T − t)
)]where w = 1 for calls and w = −1 for puts; N−1 (·) is the inverse of the standard normal cumulative
density function. To simplify the process of retrieving Ki , we transform all traded options in calls (viaput-call parity) and create a smile in the (Δcall, σ) space. We then interpolate linearly in the interval oftrading deltas [∆max,∆min] this smile and generate 2,000 points.
5
2.2 The IVol-BR
Figure (1) plots the IVol-Br for the period between August 2011 and February 2015,comprising 804 daily observations. The series spikes around events that caused �nancialdistress in Brazil, such as the Euro Area debt crisis (2011), the meltdown of oil companyOGX (2012), the Brazilian protests of 2013, the second election of Mrs. Rousse� (2014) andthe corruption and �nancial crisis in Petrobras (2015).
[Figure 1 about here]
It is interesting to compare the IVol-Br with the VXEWZ, the CBOE's index that tracksthe implied volatility of a dollar-denominated index (EWZ) of the Brazilian stock market.Figure (2) shows the evolution of both series.
[Figure 2 about here]
As Figure (2) presents, the VXEWZ is often higher than IVol-Br. This happens becausethe VXEWZ, which is constructed using options over the EWZ index (that tracks the level indollars of the Brazilian stock market), embeds directly the exchange rate volatility. In turn,the IVol-Br is constructed using options over the IBOVESPA itself and, hence, re�ects onlythe stock market volatility. Actually, during the period depicted in Figure (2) there wereimportant changes in the exchange rate volatility that directly impacted the VXEWZ but notthe IVol-Br. Thus we consider that IVol-Br is better suited to describe the forward-lookingvolatility of the Brazilian stock market.
In the next section we use the IVol-BR series in a number of interesting empirical exercises.In special, we decompose the IVol-BR into (i) the conditional volatility of stock returns and(ii) the equity volatility premium, a proxy for the risk-aversion level of the representativeinvestor. We then examine the predictive power of the equity volatility premium for stockmarket returns.
3 Empirical Analysis using the IVol-BR
An implied volatility index re�ects the dynamics of two very important variables. The�rst relates to the level, or quantity, of risk that investors face: the expected future volatilityof the market portfolio. The second relates to the price of such risk: the risk-aversion of therepresentative investor.
To understand this, one should note that, since options' payo�s are asymmetric, the valueof any option (call or put) is increasing in the expected volatility of the underlying asset.Because of that, options are often used as a protection against changes in volatility. Sincethe typical risk-averse investor dislikes volatility, options are traded with a premium because
6
of their insurance value. As a direct consequence, the implied volatility (which is computeddirectly from options prices) also has a premium with respect to the real expected volatility:it should always be higher than the real expected volatility. This is the so-called volatilitypremium.
In this section we decompose the IVol-BR into (i) the expected volatility of stock returnsand (ii) the equity volatility premium, a proxy for the risk-aversion level of the representativeinvestor. Then, we show this proxy of risk-aversion is able to predict future stock returns astheory suggests: when risk-aversion is higher, expected returns are higher.
3.1 A Model for Expected Volatility
In this Section we search for the model that best forecasts realized volatility. As explainedbefore, since the expiration of option contracts on IBOVESPA takes place every two months,the IVol-BR measures the 2-month ahead implied volatility. Consistent with this, the relevantmeasure of (annualized) realized volatility to be predicted is computed by taking the squaredroot of the sum of squared 5-minute returns over two months (42 business days):
RV(42)t =
√√√√252
42×
[42/∆n]∑i=1
r2i
where ∆n = 5/425 is the 5-minute fraction of a full trading day with 7 hours includingthe opening observation, and ri = 100× [ln(Ibovi)− ln(Ibovi−i)] is a 5-minute log-return inpercentage form on the IBOVESPA index, except when i refers to the �rst price of the day,in which case ri corresponds to the opening/close log-return.
Following the recent literature on volatility forecasting (Chen and Ghysels (2011), Corsi(2009) and Corsi and Renò (2012)), we construct several explanatory variables from the5-minute returns data set7. First, we include in the set of explanatory variables lags ofrealized volatility at heterogeneous frequencies to account for the clustering feature of stockvolatility. In the spirit of Corsi's (2009) HAR model, bimonthly, monthly, weekly and daily
realized volatilities are included: RV(42)t−1 , RV
(21)t−1 , RV
(5)t−1 and RV
(1)t−1. Formally:
RV(k)t−1 =
√√√√252
k×
[k/∆n]∑i=1
r2i
for each k = 42, 21, 5, 1.One important feature of volatility is the asymmetric response to positive and negative
returns, commonly referred to as leverage e�ect. To take this into account, Corsi and Renò(2012) suggests the following leverage explanatory variables:
Lev(k)t−1 = −42
k
[k/∆n]∑i=1
min [ri, 0]
with k = 42, 21, 5, 1. For a convenient interpretation of the estimated parameters, we take
7We thank BM&FBovespa for providing the intraday data set.
7
the absolute value of the cumulative negative returns.Andersen et al. (2007) show that jumps help in predicting volatility. Following the theory
laid out by Barndor�-Nielsen and Shephard (2004), realized variance can be decomposedinto its continuous and jump components with the usage of the so-called bipower variation(BPV). As shown by these authors, under mild conditions the BPV is robust to jumps inprices while the realized variance is not. This insight allows one to identify jumps indirectlyby calculating the di�erence:
Jumpt−1 = max[RV 2
t−1 −BPVt−1, 0]
whereBPVt−1 = (252/42)×∑[42/∆n]
i=1 |ri| |ri−1|. The maximum operator is included to accountfor the situation when there are no jumps and the BPV is eventually higher than the realizedvariance. Since our regressions have realized volatility as the dependent variable, we takethe square root of the jump measure as an explanatory variable, Jt−1 =
√Jumpt−1.
The continuous component of the realized variance is de�ned as follows:
Contt−1 = RV 2t−1 − Jumpt−1
To be consistent with the dependent variable, we also use the square root of the continuousmeasure as an explanatory variable, Ct−1 =
√Contt−1.
Lagged variables of the continuous and jump components at other time frequencies arealso included. Using the same notation as before, the following eight variables are addedC
(k)t−1, J
(k)t−1 with k = 42, 21, 5, 1.
Finally, we follow Bekaert and Hoerova (2014) and include the lagged implied volatilityas explanatory variable, IV olt−1. As will be shown, this variable contains information aboutfuture realized volatility that is not contained in lagged realized volatility and other measuresbased on observed stock returns.
To �nd the best forecasting model, we apply the General-to-Speci�c selection methodproposed by David Hendry (see for instance Hendry et al. (2009)). The starting model,also called GUM or General Unrestricted Model, includes all the variables described above.Following an iterative process, the method searches for variables that improve the �t of themodel but penalizes for variables with statistically insigni�cant parameters. The regressionsare based on daily observations. Table (1) shows the estimates of the �nal model � the GETSmodel. Five variables remain in the GETS model: leverages at k = 42, 21, and 5 frequencies,the bi-month continuous component of realized volatility, C
(42)t−1 and, interestingly, the lagged
implied volatility, IV olt−1. This shows that, apart from containing information about marketwide risk-aversion, the implied volatility also possesses relevant predicting properties.
[Table 1 about here]
From the selected GETS model, we calculate a time-series of expected volatility. Wename the di�erence between implied volatility and this time-series of expected volatility asthe volatility premium:
V olatility Premiumt = IV olt − σ̂t (3)
8
where σ̂t = Et−1 [σt] is the conditional mean of the GETS model. Figure (3) shows bothseries and Figure (4) shows the volatility premium. We observe that the premium variesconsiderably. The 3-month moving average shown in Figure (4) suggests that the averagepremium varies and remains high for several months.
[Figure 4 about here]
3.2 The Volatility Premium and Future Returns
If the volatility premium does indeed relates to investors risk-aversion, it should alsopredict stock returns. As theory predicts, periods of high risk aversion should be followedby periods of high stock returns. In this Section we test this prediction by regressing futuremarket returns on the volatility premium.
Table (2) shows the results of our main regression. The dependent variable is the returnon the market portfolio 4 weeks ahead. To avoid working with overlapping regressions, wereduce the frequency of our data set from daily to weekly by keeping only the last observationof the week. Additionally, to account for possible serial correlation in the error term, thestandard errors are computed using Newey-West lags. Columns (1) and (2) show that impliedvolatility and, particularly, expected volatility are not very good predictors of future returns.However, when we include the volatility premium in Column (3), we see that both variablescombined, IV olt − σ̂t, strongly predict future returns at the 4-week horizon. The estimatedcoe�cient is positive, 0.472, and signi�cant at the 1% con�dence level.
[Table 2 about here]
The predictive power of the volatility premium remains after we include in the regressionthe divided yield log(Dt/Pt), another common predicting variable. Again, Columns (4)and (5) show that implied volatility and expected volatility alone are poor predictors ofreturns, but the volatility premium does predict future returns. Column (6) shows a positivecoe�cient for the volatility premium, 0.345, signi�cant at the 5% con�dence level.
In Columns (1) through (8) of Table (3) the regressions are the same as the one onColumn (6) of Table (2) except for the horizon of future returns. As the estimates show, thevolatility premium predictability is stronger at the 4-week horizon (Columns (7) and (8)).
[Table 3 about here]
9
The standard errors in the �rst eight Columns in Table (3) may be biased due to thepresence of a persistent explanatory variable such as the log dividend yield (see for instanceStambaugh (1999)). To correct for this, Columns (9) and (10) show the same regressionsof Columns (7) and (8) but based on non-overlapping 4-week returns. As we can see, thecoe�cients on the volatility premium remain positive and highly signi�cant.
3.3 Robustness
The actual expected volatility by market participants cannot be observed, and our mea-sure of expected volatility depends on the model chosen by the econometrician. In thisSection we assess to which extent our results depend on the chosen volatility model.
Table (4) and (5) show the estimates of several models. Table (4) brings the estimates ofCorsi's (2009) HAR model in Column (1), with the addition of a 42-day realized volatilitylag in accordance with the frequency of the dependent variable. In Columns (2), (3) and(4) we include the lagged implied volatility, IV olt−1, that was shown to contain importantpredictive information. Columns (3) and (4) include leverage variables to account for theasymmetric response of volatility to past negative returns.
[Table 4 about here]
In Table (5) we separate the realized volatility into its continuous and jump componentsand use these variables instead. Column (1) shows the estimates of the GUM model, thestarting model in the General-to-Speci�c selection method adopted in Section 3.1. TheGUM regression includes all the variables initially selected as candidate variables to forecastvolatility. Columns (2) through (4) are variants of this more general model.
[Table 5 about here]
As we can conclude by comparing the statistical properties of each regression in Tables(1), (4) and (5), the GETS model has the lowest information criterion, BIC, as the selectionmethod strongly penalizes the inclusion of variables and favors a more parsimonious model.In general, however, most models are reasonably successful in explaining around 25% to 35%of the variation in the dependent variable.
We now assess how sensitive is our predictive regression to the selection of the volatilitymodel. For each one of the regression models shown in Tables (4) and (5) we calculate avolatility premium as in equation (3). The results of the predictability regressions at the 4-week return horizon are shown in Table (6). In Column (1) we use a simple model to predictfuture volatility and set σ̂t = σt−1 following the de�nition of Bollerslev et al. (2009). Column(2) replicates our main regression that uses the GETS model to predict volatility. Columns(3) through (10) show the predictability regressions for each of the 8 models presented in
10
Tables (4) and (5). As can be seen, the results are largely robust to the selection of thevolatility model.
[Table 6 about here]
4 Conclusion
This paper aims to contribute in two dimensions. First, we propose and evaluate animplied volatility index for the Brazilian stock market. Second, we propose and evaluate amodel-free risk-aversion index for the Brazilian investor.
We call our implied volatility index IVol-BR. The methodology used combines the oneused in the calculation of the �new� VIX with some adjustments that take into account somelocal aspects of the Brazilian stock market. The index is publicly available and can be usedfor future research by academics and practitioners.
We decompose the IVol-BR into (i) the expected future volatility and (ii) the volatilitypremium. Using high-frequency data, we run a number of models to predict volatility. Wecalculate the volatility premium as the di�erence between the IVol-BR and the estimatedexpected volatility.
As theory suggests, the volatility premium should proxy for investor risk-aversion. Wevalidate this claim by showing the volatility premium is able to predict future market returns.
11
References
Andersen, T. G., T. Bollerslev, and F. X. Diebold (2007, October). Roughing it up: Includ-
ing jump components in the measurement, modeling, and forecasting of return volatility.
Review of Economics and Statistics 89 (4), 701�720.
Barndor�-Nielsen, O. E. and N. Shephard (2004, January). Power and bipower variation
with stochastic volatility and jumps. Journal of Financial Econometrics 2 (1), 1�37.
Bekaert, G. and M. Hoerova (2014, April). The VIX, the variance premium and stock market
volatility. SSRN Scholarly Paper ID 2431301, Social Science Research Network, Rochester,
NY.
Bollerslev, T., G. Tauchen, and H. Zhou (2009, November). Expected stock returns and
variance risk premia. Review of Financial Studies 22 (11), 4463�4492.
Brostowicz Junior, R. J. and M. P. Laurini (2010). Variance swaps in BM&f: Pricing and
viability of hedge. Brazilian Review of Finance 8 (2), 197�228.
Carr, P. and L. Wu (2006, January). A tale of two indices. The Journal of Derivatives 13 (3),
13�29.
Carr, P. and L. Wu (2009, March). Variance risk premiums. Review of Financial Stud-
ies 22 (3), 1311�1341.
CBOE (2009). The CBOE volatility index�VIX. White Paper .
Chen, X. and E. Ghysels (2011, January). News�good or bad�and its impact on volatility
predictions over multiple horizons. Review of Financial Studies 24 (1), 46�81.
Corsi, F. (2009, March). A simple approximate long-memory model of realized volatility.
Journal of Financial Econometrics 7 (2), 174�196.
12
Corsi, F. and R. Renò (2012, February). Discrete-time volatility forecasting with persistent
leverage e�ect and the link with continuous-time volatility modeling. Journal of Business
& Economic Statistics 30 (3), 368�380.
Dario, A. (2006). Apreçamento de ativos referenciados em volatilidade. Revista Brasileira
de Finanças 4 (2), pp�203.
Hendry, D. F., J. Castle, and N. Shephard (2009). The methodology and practice of econo-
metrics: a festschrift in honour of David F. Hendry. Oxford; New York: Oxford University
Press.
Kapotas, J. C., P. P. Schirmer, and S. M. Manteiga (2004, June). Forward volatility contract
pricing in the brazilian market. Brazilian Review of Finance 2 (1), 1�21.
Stambaugh, R. F. (1999, December). Predictive regressions. Journal of Financial Eco-
nomics 54 (3), 375�421.
13
A Tables and Figures
Figure 1: Implied Volatility in Brazil - the IVol-BR
This Figure shows the daily time-series of the IVol-Br in percentage points and annualized.
14
Figure 2: Comparing IVol-BR and VXEWZ
This Figure shows the daily time-series of the IVol-Br and the VXEWZ. Both series are inpercentage points and annualized. VXEWZ is the implied volatility index on the Brazilianstocks ETF EWZ and is calculated by CBOE.
15
Figure 3: IVol-Br and Expected Volatility
This Figure shows the weekly time-series of the IVol-Br (Implied volatility) and the estimatedexpected volatility. The model for expected volatility is the GETS model shown on Table(1). Both series are in percentage points and annualized.
1020
3040
01jan2012
01jan2013
01jan2014
01jan2015
data
Implied Volatility Expected Volatility
16
Figure 4: Volatility Premium: A Proxy for Risk Aversion
This Figure shows the weekly time-series of the volatility premium calculated by the dif-ference of the IVol-Br and the expected volatility, predicted by the GETS model shown onTable (1). Both series are in percentage points and annualized.
-50
510
1520
01jan2012
01jan2013
01jan2014
01jan2015
data
Ex-Ante Volatility Premium 3-Month MA
17
Table 1: General-to-Speci�c Best Model
The Table shows the estimates for the best model following the General-to-Speci�c selection method. The
starting model, also called GUM or General Unrestricted Model, comprises of all independent variables.
The standard errors reported in parenthesis are robust to heteroskedasticity. Regressions are based on daily
observations. The corresponding p-values are denoted by * if p < 0.10, ** if p < 0.05 and *** if p < 0.01.
(1)
IV olt�1 0.202***(0.046)
C(42)t�1 1.598***
(0.209)
Lev(42)t�1 -20.406***
(2.064)
Lev(21)t�1 10.620***
(1.627)
Lev(5)t�1 14.357***
(3.925)
Constant 14.243***(0.869)
Number of Obs. 741R2 0.352Adjusted R2 0.348RMQE 3.798BIC 4114.3
18
Table2:PredictabilityRegressions
TheTable
show
stheestimatesofpredictabilityregressions.
Thedependentvariable
isthereturn
onthemarket
portfolio4-weeksahead.The
explanatory
variablesare:i)σ̂t,theexpectedvolatility
onthenext8weeksestimatedbybestmodel
follow
ingtheGeneral-to-Speci�cselection
method,ii)IVol
t,theexpectedim
plied
volatility
onthenext8weeksestimatedfrom
pricesofoptionscontractsattimet,iii)IVol
t−σ̂t,thevolatility
premium,andiv)log
(Dt/P
t),thelogdividendyield.Regressionsare
basedonweekly
observations.
Toaccountforerrorcorrelation,thestandard
errors
are
computedusingNew
ey-W
estlags.
Thestandard
errors
are
reported
inparenthesis.Thecorrespondingp-values
are
denotedby*ifp<
0.10,**ifp<
0.05and***ifp<
0.01.
4Weeks
(1)
(2)
(3)
(4)
(5)
(6)
σ̂t
0.125
-0.001
(0.233)
(0.199)
IVol
t0.234*
0.144
(0.119)
(0.104)
IVol
t�σ̂t
0.472***
0.345**
(0.142)
(0.148)
log(D/P
)14.552**
12.381*
11.137*
(5.659)
(6.285)
(6.189)
Constant
-2.214
-5.039*
-0.677
45.839**
35.689*
34.363*
(4.960)
(2.785)
(0.616)
(18.614)
(20.866)
(19.464)
Number
ofObs.
175
175
175
175
175
175
R2
0.004
0.050
0.085
0.093
0.110
0.134
Adjusted
R2
-0.002
0.045
0.080
0.083
0.100
0.124
19
Table3:PredictabilityRegressionsatDi�erentHorizons
TheTableshow
sregressionshavingfuture
marketreturns1-weekahead,2-,3-and4-weeksaheadasthedependentvariable.Theexplanatory
variables
are:i)IVol
t−σ̂tistheex-antevolatility
premium,andii)log
(Dt/Pt)isthelogdividendyield.Regressionsin
Columns(1)through(8)are
based
onweekly
observations.
Regressionsin
Columns(9)and(10)are
non-overlappingonthedependentvariableandare
basedonmonthly
observations.
Toaccountforerrorcorrelation,standard
errors
inColumns(3)through(8)are
computedusingNew
ey-W
estlags.
Thestandard
errors
reported
in
parenthesis.Thecorrespondingp-values
are
denotedby*ifp<
0.10,**ifp<
0.05and***ifp<
0.01.
1Week
2Weeks
3Weeks
4Weeks
4WeeksN-O
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
IVol
t�σ̂t
0.144*
0.112
0.210*
0.142
0.330**
0.244*
0.472***
0.345**
0.692***
0.591**
(0.075)
(0.080)
(0.111)
(0.111)
(0.129)
(0.131)
(0.142)
(0.148)
(0.224)
(0.275)
log(D/P
)3.022
5.914
7.533
11.137*
6.703
(2.283)
(3.606)
(5.339)
(6.189)
(8.413)
Constant
-0.231
9.274
-0.268
18.338
-0.475
23.228
-0.677
34.363*
-0.905
20.173
(0.226)
(7.193)
(0.371)
(11.321)
(0.486)
(16.810)
(0.616)
(19.464)
(0.771)
(26.464)
Number
ofObs.
178
178
177
177
176
176
175
175
4141
R2
0.032
0.044
0.032
0.057
0.054
0.083
0.085
0.134
0.153
0.172
Adjusted
R2
0.026
0.034
0.026
0.046
0.049
0.072
0.080
0.124
0.131
0.128
20
Table 4: Robustness - Volatility Models
The Table shows the estimates of di�erent models of expected realized volatility. The dependent variable is
the realized volatility over the following 8-weeks, calculated from 5-minute returns on the Ibovespa portfolio.
The explanatory variables are: i) IV olt−1 is the expected implied volatility on the next 8 weeks estimated
from prices of options contracts at time t − 1, ii) RV(k)t−1 is the realized volatility on the following k days
at time t − 1, where k = 42, 21, 5, 1, computed iii) Lev(k)t−1 is the cumulative negative 5-minute returns
continuous component of the realized volatility on the following k days at time t− 1, where k = 42, 21, 5, 1.
The standard errors reported in parenthesis. The corresponding p-values are denoted by * if p < 0.10, ** if
p < 0.05 and *** if p < 0.01.
M1 M2 M3 M4
RV(42)
t�1 -0.240*** -0.278*** 1.274***
RV(21)
t�1 0.365*** 0.347*** -0.502***
RV(5)
t�1 0.184*** 0.113** 0.025
RV(1)
t�1 0.036 0.015 -0.014
IV olt�1 0.232*** 0.254*** 0.203***
Lev(42)t�1 -5.830*** -21.502***
Lev(21)t�1 11.813*** 22.504***
Lev(5)t�1 10.691** 9.723
Lev(1)t�1 13.852 23.824
Constant 13.888*** 11.574*** 12.338*** 14.008***
Number of Obs. 762 741 741 741R2 0.261 0.274 0.301 0.363Adjusted R2 0.257 0.270 0.297 0.355RMQE 4.054 4.019 3.944 3.776BIC 4323.7 4198.1 4170.0 4128.1
21
Table 5: Robustness - Volatility Models (Cont.)
The Table shows the estimates of di�erent models of expected realized volatility. The dependent variable is
the realized volatility over the following 8-weeks, calculated from 5-minute returns on the Ibovespa portfolio.
The explanatory variables are: i) IV olt−1 is the expected implied volatility on the next 8 weeks estimated
from prices of options contracts at time t− 1, ii) C(k)t−1 is the continuous component of the realized volatility
during the following k days at time t−1, where k = 42, 21, 5, 1, iii) J(k)t−1 is the jump component of the realized
volatility during the following k days at time t− 1, where k = 42, 21, 5, 1 and iv) Lev(k)t−1 is the absolute of
the sum 5-minute negative returns during the following k days at time t − 1, where k = 42, 21, 5, 1. The
standard errors reported in parenthesis. The regressions are based on daily observations. The corresponding
p-values are denoted by * if p < 0.10, ** if p < 0.05 and *** if p < 0.01.
M5 M6 M7 M8
IV olt�1 0.217*** 0.181*** 0.229*** 0.291***
C(42)t�1 1.147** 2.269*** -1.616***
C(21)t�1 0.440 -0.528* 1.305***
C(5)t�1 -0.208 -0.101 0.085
C(1)t�1 -0.055 -0.038 0.001
J(42)t�1 0.826*** 1.129*** 1.173***
J(21)t�1 -0.727*** -0.545*** -0.839***
J(5)t�1 0.131 0.066 0.101
J(1)t�1 0.013 -0.006 0.009
Lev(42)t�1 -23.182*** -26.424*** -15.184***
Lev(21)t�1 14.121** 20.467*** 19.210***
Lev(5)t�1 18.698** 17.180** 8.706
Lev(1)t�1 31.593* 32.756* 17.142
Constant 14.145*** 14.545*** 13.311*** 11.371***
Number of Obs. 741 741 741 741R2 0.375 0.362 0.350 0.315Adjusted R2 0.364 0.354 0.342 0.306RMQE 3.750 3.781 3.813 3.917BIC 4140.2 4129.7 4142.5 4182.3
22
Table6:Robustness
-PredictabilityRegression
TheTable
show
stheestimatesofpredictabilityregressionsusingvariousmeasuresofvolatility
premium.Each
measure
ofvolatility
premium
is
calculatedwithadi�erentmodel
forexpectedvolatility.Column(1)usesasimple
model
forexpectedvolatility:σ̂t≡E
t−1
[σt]
=σt−
1,andwas
proposedbyBollerslev
etal.(2009).
Column(2)replicatesourmain
regressionthatusestheGETSmodelto
forecastvolatility.Columns(3)through
(10)varies
theexpectedvolatility
model
fromM
1to
M10.log
(Dt/Pt)isthelogdividendyield.Regressionsare
basedonweekly
observations.
Toaccountforerrorcorrelation,thestandard
errors
are
computedusingNew
ey-W
estlags.
Thestandard
errors
are
reported
inparenthesis.The
correspondingp-values
are
denotedby*ifp<
0.10,**ifp<
0.05and***ifp<
0.01.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Constant
52.429***34.363*
36.962**
36.606*
35.563*
31.272
32.980*
33.661*
31.366
34.391*
(17.788)
(19.464)
(18.663)
(19.025)
(19.519)
(19.447)
(19.469)
(19.571)
(19.371)
(19.216)
log(D/P
)16.836***11.137*
11.932**
11.843*
11.482*
10.200
10.713*
10.916*
10.228*
11.162*
(5.706)
(6.189)
(5.955)
(6.057)
(6.207)
(6.184)
(6.188)
(6.219)
(6.162)
(6.113)
IVol
t�σt�
10.261**
(0.109)
IVol
t�σ̂t
0.345**
(0.148)
IVol
t�σ̂t(M
1)0.304**
(0.121)
IVol
t�σ̂t(M
2)0.342**
(0.149)
IVol
t�σ̂t(M
3)0.297**
(0.148)
IVol
t�σ̂t(M
4)0.410***
(0.146)
IVol
t�σ̂t(M
5)0.365**
(0.145)
IVol
t�σ̂t(M
6)0.345**
(0.152)
IVol
t�σ̂t(M
7)0.412***
(0.145)
IVol
t�σ̂t(M
8)0.378**
(0.148)
Number
ofObs.
175
175
175
175
175
175
175
175
175
175
R2
0.146
0.134
0.135
0.127
0.120
0.155
0.143
0.135
0.155
0.141
Adjusted
R2
0.137
0.124
0.125
0.117
0.110
0.145
0.133
0.125
0.145
0.131
23