Implied volatility as a forecast estimator for
realized volatility
Author: Sam Zwaan
Student number: 10553207 Date: June 2018
Place: Amsterdam
Thesis supervisor: Dr. L. Zou
Universiteit van Amsterdam, Amsterdam Business School MSc Finance – Quantitative Finance
Abstract
Prior research studied mainly relatively stable markets and found mixed evidence about the ability of implied volatility to predict the realized volatility for different periods of uncertainty. Besides, new insights suggest that historical volatility could perform similarly as estimator when using high-frequency data, using different time spans and separating historical volatility into two parts: the continuous and jump component. This research took these insights into account and analyzed the implied volatility of the American (SPY) and Brazilian stock market (EWZ) in the period from 2007 till 2017. With this comparison, the volatility of a less stable market is analyzed and previous results can be verified. The outcome shows that the implied volatility contains more relevant information to predict the realized volatility, performs better in the stable market, outperforms the historical volatility in all cases and incorporates both historical components highly. Generally, the implied volatility of the SPY is close to efficiency, while the implied volatility of the EWZ overestimates the movements of the realized volatility. In contrast to most former findings, the performance for both markets decreases during years of relatively low volatility.
Statement of originality
This document is written by Sam Zwaan, who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents
Table of contents
1 Introduction ... 4
2 Literature review ... 7
2.1 Previous findings... 7
2.2 Volatility and relevant theories ... 11
2.2.1 Implied volatility ... 11 2.2.2 Realized volatility ... 13 2.3 Theoretical hypotheses ... 16 3 Methodology ... 17 3.1 Procedure... 17 3.2 Statistical hypotheses ... 20
4 Data and descriptive statistics ... 21
4.1 Data ... 21 4.2 Descriptive statistics ... 25 5 Results ... 30 5.1 Main outcomes ... 30 5.2 Other outcomes ... 37 5.3 Robustness ... 40 5.3.1 Variables ... 40
5.3.2 Proxy for unstable market ... 43
5.3.3 Time span ... 45
6 Conclusion and discussion ... 47
6.1 Conclusion ... 47
6.2 Discussion ... 49
1 Introduction
Both in financial theory as in practice, predicting volatility is a far-reaching theme on the grounds that it is an essential element for risk management and the pricing of financial securities (Busch, Christensen & Nielsen, 2011, p.48; Muzzioli, 2010, p.561). As stated by Poon and Granger (2003, p.507), volatility can be predicted by different models. They reported that models based on implied volatility, which results from the Black-Scholes model, outperforms other models and incorporates most data about forthcoming volatility. According to Berk and DeMarzo (2014, p.752), an option its implied volatility represents the expected volatility of the underlying security until the expiration date. Since it is the missing variable in the Black-Scholes model and gets derived from option pricing, trading in options is usually seen as trading in volatility (Poon & Granger, 2003, p.486). Subsequently, implied volatility is reasonably a good forecast estimator for realized volatility as long as the option market operates efficiently and the Black-Scholes model is accurate (Christensen & Prabhala, 1998, p.1126).
Nevertheless, there are critics who have shown that implied volatility is biased when predicting volatility (e.g. Muzzioli, 2010, p.581; Poon & Granger, 2003, p.503). These studies contradict each other however with respect to conclusions whether the performance of the implied volatility improves during periods of relatively mainstream or extreme volatilities. New insights also suggest that historical-based models that rely on high-frequency data could achieve similar or even better results when forecasting volatility (e.g. Martens & Zein 2004, p.1005; Koopman, Jungbacker, & Hol, 2005, p.472). Moreover, Andersen, Bollerslev, Diebold and Labys (2003, p.619) mentioned that realized volatility is divided into a continuous and jump component and that refined forecasting models need to frame both components exclusively. Busch et al. (2011, p.49) stated in addition that using different time frames for historical volatility variables could enhance the ability to predict the realized volatility, since short-term volatilities could incorporate more information than longer-term volatilities. At last, most studies have investigated relatively stable markets such as the S&P500 index (e.g. Giot & Laurent, 2007, p.348; Muzzioli, 2010, p.566). As a result, there is less information about relatively unstable markets. Figure 1 shows however that volatility for a unstable market (Brazilian stock market) seems incomparable with a stable market (American stock market), seeing that the realized volatility is relatively higher at all times and experiences
heavier spikes. Busch et al. (2011, p.49) also provided that jumps appear more often and excessive for unstable markets.
This thesis wants to add value to this area of research by managing the issues with reference to implied volatility as a forecast estimator for the realized volatility. While previous research found mixed evidence regarding contrasting periods of volatility and primarily focused on stable stock markets, this study will also evaluate if implied volatility performs similar for markets that experience different degrees of volatilities. The additional insights will also be covered by using high-frequency data, including the continuous and jump components and using different time spans for the historical volatilities.
Figure 1 Monthly realized volatility of the American and Brazilian stock market
This figure represents the 30-days realized volatility from a sample that includes observations from January 1, 2001 to December 31, 2017 for the American stock market (SPY is a proxy for this market) and the Brazilian stock market (EWZ is a proxy for this market). The data is obtained from Bloomberg and the realized volatility (standard deviation) is based on the closing prices per trading day. The volatilities are expressed in annual terms. The x-axis contains the dates in years and the y-axis contains the realized volatility.
The main objective of this research is to investigate if the fundamental theory of implied volatility, which is efficiently pricing the expected movements, is valid for both stock markets that experience various levels of volatility as different periods of fluctuations. In other words, the effect of uncertainty on the efficiency of implied volatility as a forecast estimator will be investigated, considering that higher volatility generally means higher uncertainty (Poon & Granger, 2003, p.478). As Muzzioli (2010, p.582) showed, the answer on
0 20 40 60 80 100 120 140 160 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 EWZ SPY
this research question is important for option traders, because predicting volatility is decisive for making trading decisions, risk management, and eventually for their trading profits. He also mentioned that it could be relevant for policy-makers, as predicting uncertainty in the long term is essential for reflecting financial strength and composing monetary policy. In addition, Andersen and Bollerslev (1998, p.885) showed that volatility serves as a leading factor to price financial assets and derivatives. In conclusion, precise realized volatility predictions are crucial and decisive to implement and assess monetary policies, manage risk, construct investing strategies and value securities.
The combination of the described issues and the stakeholders that could benefit from this research question validates the contribution of this research to the continuous debate of the estimating power of implied volatility. By studying a new and less stable market and by making comparisons between different markets and periods, this research could get enhance the overall insight in the ability of implied volatility to predict future fluctuations. This can be relevant for option traders and policy-makers by knowing if the implied volatility is a precise estimator for future realized volatility for both stable as non-stable markets. Besides, the forecast model could be more precise by acquiring volatility via high-frequency data and including both components and different time spans. This could optimize trading strategies and risk management. Finally, by knowing if the implied volatility performs well for different markets could show if these securities are fairly priced.
The following data and methodology will help to reach this paper its objective. The dataset contains the realized volatility as dependent variable, the implied volatility as independent variables, and the first lag of realized volatility (historical volatility) and its components as control variables for the American stock market (SPY) and the Brazilian stock market (EWZ) between the 1st of January 2007 and the 31st of December 2017. By using these variables in different regression compositions it is possible to investigate if the implied volatility of an option holds relevant information about the realized volatility and incorporates historical information for both markets. Besides, this method could evaluate previous contradicting results as regards to biases. At last, Newey-West standard errors will be used to manage overlapping data, which is slightly different from previous research.
The remainder of this research has the following structure. Section 2 provides relevant theories and empirical researches to take a deeper look into previous thoughts, methodologies, results and conclusions regarding the ability of implied volatility to forecast
volatility. The background literature that has been obtained will be analyzed and compared to find any connections and/or misalignments to form theoretical hypotheses. On the basis of the foregoing, a new research method will be defined with corresponding statistical hypotheses in section 3. After describing the relevant data and descriptive statistics in section 4, section 5 will highlight the outcomes and robustness tests. Next, a conclusion is drawn in section 6 and the research question will be answered: Is implied volatility a better forecast
estimator for the realized volatility of a stock market that experience relatively less uncertainty than for a stock market that experience relatively more uncertainty? This last section also
presents the paper its limitations and the implications for further research. 2 Literature review
The ability of the implied volatility to predict the realized volatility is an ongoing discussion, but the results are contradicting and new insights and techniques arise. Researchers and academics find this subject therefore appealing. This results in enough existing literature to find. In order to realize a new study, the main and relevant studies will be discussed in this chapter. First, prior outcomes are described to detect which theories will be used in this research to add value. Afterwards, these theories will be extensively explained. The relation between the empirical evidence and theories will also be illustrated during these two sub-sections. Finally, the theoretical hypotheses will be formulated based on the foregoing.
2.1 Previous findings
As reported in the introduction, realized volatility can be predicted by a plurality of models. Poon and Granger (2003, p.478) have compared these different models by reviewing and analyzing the outcomes of prior studies. They mentioned that forecast models could be based on volatility derived from option prices or historical time series. According to Christensen and Prabhala (1998, p.125), volatilities from option-based models are derived by backward induction from current option prices. They provided that these models want to reveal the expectation about the volatility until the expiration date of the option, which is the so-called implied volatility. Meanwhile, time series models are using historical volatility to forecast future volatility and these models can be GARCH-based or history-based (Muzzioli, 2010, p.561). Poon and Granger (2003, p.507) found that option-based models outperformed
GARCH-based models in 94% of the papers and achieved better outcomes as history-based models in 76% of the studies.
That implied volatility could predict the realized volatility and outperforms other models seems in accordance with the main theories. Jorion (1995, p.507) mentioned that it is likely that implied volatility is a good forecast estimator for the realized volatility, because traders could otherwise accomplish profits by mispriced options. Besides, Ederington and Guan (2002, p.30) stated that option prices integrate all information in case of an efficient option market and a proper pricing model, including the past data of a security. Implied volatility is therefore superior to for instance historical volatility. Shu and Zhang (2003, p.83) also mentioned that implied volatility is likely a better forecast estimator than historical volatility, since option traders are largely institutional investors and they have access to more information. Despite previous outcomes and theories are in favor of option-based models, results are mixed when examine the ability of these kind of models to forecast the realized volatility without any bias. Besides, outcomes are also contradicting during different levels of volatility and between different option-based models.
In contrast to Poon and Granger (2003, p.507), who have reviewed the model that is based on the Black-Scholes model, Muzzioli (2010, p.561) mentioned that implied volatility could be determined by another procedure as well. The outcome of this procedure is the model-free implied volatility. The implied volatility of the Black-Scholes model depends on the Black-Scholes formula, whereas the model free version does not hinge on an option model. Simply put, the model free volatility can directly be derived from a whole range of option prices. Muzzioli (2010, p.581) found however that the Black-Scholes model contains more information for the realized volatility of the DAX index between 2001 and 2005, while the calculations of the model-free volatility are more complicated. This result is similar to Becker, Clements and White (2007, p.2548), who have completed a similar kind of research for the S&P 500 index between 1990 and 2003. At last, Andersen and Bondarenko (2007, p.17) found that the Black-Scholes model performed better as forecast estimator for the S&P 500 futures during the period 1990 till 2006 than the model-free model. They (2007, p.1) mentioned in addition that the model-free model does not exclusively predict volatility, because this model tries to assess the expected volatility by assuming a risk-neutral environment. Volatility is however stochastic. Since most studies and also the more recent researches are supportive for the Black-Scholes model, this model will be used in this study.
As previously mentioned, many researchers found positive results. Christensen and Prabhala (1998, p.148) studied the S&P 100 index for the period between 1983 and 1995 and they stated that implied volatility does predict realized volatility. This is in line with Busch et al. (2011, p.56). They described that implied volatility incorporates information about the future volatility for the bond market and is even unbiased for the forex and stock market. Blair, Poon and Taylor (2001, p.24) found that the VIX index, which represents the implied volatility of the S&P 500 index, incorporates all relevant data and therefore is the most accurate forecast estimator.
Nevertheless, some studies acknowledged that implied volatility is a good estimator, but remarked that it is biased. Jorion (1995, p. 527) and Fleming (1998, p.341) agreed that implied volatility contains relevant data for predicting volatility, but indicated that it has an upward bias. The bias is not substantial however for option traders to enhance profits. Besides, Christensen and Prabhala (1998, p.147) acknowledged that a higher implied volatility will decrease the forecast ability due to noise. Also Ederington and Guan (2002, p. 44) stated that the prediction ability of implied volatility is sensitive to uncertain periods, like a crisis. Furthermore, Poon and Granger (2003, p.503) mentioned that implied volatility overestimate (underestimate) realized volatility when volatility is high (low). On the other hand, Muzzioli (2010, p.581) concluded that option-based models are performing better when volatility is extremely high. As there are mixed evidence regarding different levels of volatility, this study will analyze periods and markets that experience comparatively low and high volatility. In addition, most studies have investigated relatively stable markets such as the S&P 500 index and the DAX index. The S&P 500 will be included, but also an emerging market will be added to detect the estimating power of implied volatility for more extreme and unstable markets. Besides, in this way it is possible to compare both stable as unstable markets,
Poon and Granger (2003, p.486) found that implied volatility is usually a better forecast estimator than historical volatility. However, whereas for example Koopman et al. (2005, p.472) already incidentally showed that historical volatility performed better as forecast estimator for the realized volatility, other studies showed that historical volatility measures based on high-frequency returns could achieve similar or even better results as implied volatility models on a more regular basis. Noh and Kim (2006, p.413) found this for predicting the realized volatility of FTSE 100 futures between 1994 and 1999, which is in contrast to the S&P 500 futures in their research. Moreover, Martens and Zein (2004, p.1005)
described that historical volatility produced similar results as implied volatility for predicting the realized volatility for the S&P 500, YEN/USD and Light Sweet Crude Oil (p.1005). According to Martens and Zein (2004, p.1006), this can be due to the fact that option models are inefficient or that the study was not able to extract the expectation of the market regarding the volatility. At last, Andersen and Bollerslev (1998, p.900) demonstrated that volatilities based on 5-minutes returns decreased the error term, increased the predictive power and showed that these intervals not suffer from autocorrelation.
Next to the addition of high-frequency data, there is also another extension. Andersen et al. (2003, p.619) stated that realized/historical volatility is divided into a continuous and jump component. The jump component is not constant and scarcely correlates with the realized volatility, while the opposite is true for the continuous flow. Therefore, they declared that models that includes historical volatility need to frame both components exclusively. In addition, they suggested that implied volatility could barely incorporate the jump component due to their inconsistency. Giot and Laurent (2007, p.357) included the historical components as control variables when predicting future volatility and found that implied volatility does incorporate information of the jump component partly. Furthermore, Busch et al. (2011, p.56) tried to predict both components and described that implied volatility could predict the future continuous components almost entirely and the jump components partly for the forex, bond and stock market. Both findings are in contrast to the theory about jumps. As a lot of findings are supportive for intraday data and the two different components, both insights will be inserted into the methodology. As a result, it would be possible to make comparisons with the implied volatility and find out if implied volatility subsumes historical information.
At last, recent articles also used different time spans for the historical independent variables in their regressions. Giot and Laurent (2007, p.338) used monthly volatility variables or shorter because the expiration of options is generally monthly. According to Busch et al. (2011, p.49), forecasting the realized volatility using various time frames is based on the concern if implied volatility incorporates information from the historical volatility and/or its components from one month before, and if monthly historical volatilities contain the same amount of information as volatilities from yesterday or a couple of days ago for predicting the realized volatility. This thesis will therefore also investigate different time spans. Busch et al (2011, p.53) found for example that the daily historical continuous component contains more information than the monthly estimator for predicting the realized volatility of the S&P 500
for the coming month. The daily and monthly jump component as explanatory variables are in addition negative in their research, which in contrast to the weekly component. Nevertheless, the implied volatility incorporates almost all historical and relevant information as its coefficient is close to one and the historical coefficients are close to zero. On the other hand, Giot and Laurent (2007, p.354) found that the coefficient of the implied volatility declined from approximately 1 to 0.60 for predicting the realized volatility 22 trading days ahead of the S&P 100 when the two components are included. Especially the coefficients for the weekly and monthly trading days continuous flows are different from zero and significant, which is in contrast to the jump coefficients. As stated by Giot and Laurent (2007, p.355), it implies that the implied volatility does not subsume the historical continuous flow entirely. Since the effects of the different time spans are diverge and in some cases significant, the different time spans will also be added to the methodology.
2.2 Volatility and relevant theories
Volatility is another word for the degree of movement of a financial instrument and shows the return its standard deviation (Berk & DeMarzo, 2014, p.318). For a stable market, such as the American stock market, the volatility is relatively low because the price fluctuations of this index are generally small. On the other hand, a relatively unstable market is characterized by higher price results and therefore a higher volatility. This is the case for the Brazilian stock market. The differences between both markets are shown in Figure 1. Two different types of volatility can be distinguished: implied volatility and realized volatility. Realized volatility refers to the actual movement in the past (Berk & DeMarzo, 2014, p.323), while implied volatility relates to the expected movement in the future (Berk & DeMarzo, 2014, p.752). In the next two sub-sections, both volatilities will be broadly defined in combination with relevant theories and information for this thesis.
2.2.1 Implied volatility
The Black-Scholes model will be used to derive the implied volatility. Black and Scholes (1976, p.640) mentioned that their model is suitable to price options. They stated in their research (1976, p.637) that there exist two different types of options: a call option and a put option. A call option gives an owner the right to buy shares on the expiration date (in some cases also up to this date) of the option at a predetermined price (the exercise price). The reverse applies to a put option. A put option gives the owner the right to sell shares at
the exercise price on the maturity date (in some cases also up to this date). Another important note is that various styles of options are traded on the regular derivatives markets: American and European style options (Black and Scholes, 1976, p.637). In contrast to European options, American options give the owner the right and opportunity to early exercise its option contract. This means that the holder of an option can exercise its option at any time before the expiration date and can therefore request/deliver the shares/cash prematurely.
Black and Scholes (1976, p.637) defined a methodology to price European call options (CBS) and European put options (PBS). The Black-Scholes model contains in total six different
factors: the underlying financial instrument its price (Pu), the period to the expiration date (T),
the exercise price (S), the risk free interest rate (R), the cumulative normal distribution (Ø(di))
and the annual volatility (σv). According to Berk and DeMarzo (2014, p.752), only the last
variable is not straightforwardly detectable. They described that it is possible to retrieve the volatility that corresponds to the Black-Scholes model by using the current option prices and the other given factors, which is called the implied volatility. The implied volatility therefore reveals the expectation about the volatility until the expiration date of the option (Berk & DeMarzo, 2014, p.752), as it is derived from option pricing by backward induction (Poon & Granger, 2003, p.486). The correct formulas for calculating the theoretical call and put option prices are as follows (Berk & DeMarzo, 2014, pp.747-749):
CBS = Pu × ∅(d1) − e−T × R× S × ∅(d2) (1) PBS = e−T × R× S × ∅(1 − d2) − Pu × ∅(1 − d1) (2) where d1= ln( Pu / e−T × R× S) σv√T + σv / √T 2 ; and d2 = d1 − σv√T (3)
The option prices, and thus the implied volatility, are nowadays automatically calculated by computers based on mostly a variant of this formula. This will also be the case when evaluating American style options and when dividends will be paid. Berk and DeMarzo (2014, pp.750-751) described that the Black-Scholes formulas for stocks that pay dividend are similar to equation (1), (2) and (3), with the exception that the present value of the dividend will be discounted from the price of the underlying financial instrument. Furthermore, their
book (2014, p.748) explained that the value of an American call option is similar to a European call option when there is no payment of dividend. However, an early exercise can be beneficial for owners of American call and put options if the underlying instrument pays dividend. Therefore, the price of an American option exceeds the price of a European option. Berk and DeMarzo (2014, p.725) described for instance that American options will never fall below the intrinsic value. However, when the time value of a European call option is negative due to a high present value of the dividend payout, the option price falls below the intrinsic value. They also described that early exercising an American call option will be optimum just before the ex-dividend date. In this research American options will be studies that pay out dividend. Section 4.1 illustrates the procedure, and thus the variation on the Black-Scholes model, that will be used to define the implied volatility.
According to Blair et al. (2001, p.8), using the Black-Scholes model and the corresponding factors can comprise inaccuracy in the implied volatility. First, they stated that using an incorrect model or factor create errors in the retrieved volatility. This can be for example due to using a model for calculating European options instead of American options or using a wrong dividend level and schedule. They also mentioned that biases can appear when stock and option exchanges markets do not have similar regular closing hours when using closing prices. Besides, they stated that implied volatility can suffer from a negative serial correlation when using bid and ask prices of options. Jorion (1995, p.508) also mentioned that sparse trading could ensure biases. As mentioned by Blair et al. (2001, p.8), prior researches suffered from these errors, among which Christensen and Prabhala (1998, p.131) who used an European model for American options. For this research it is therefore important to prevent such errors.
2.2.2 Realized volatility
As described in section 2.1, prior studies showed that implied volatility has the ability to predict realized volatility entirely or partly and with or without a bias when performing regressions. In these regressions, the first lag of the realized volatility is generally included as independent variable to find out if the implied volatility incorporates historical information. According to Andersen et al. (2003, p.619), realized volatility consists however of a continuous and a jump component and using high-frequency returns makes it possible to detect both components. By detaching these components from the first lag of the realized
volatility, it will be achievable to find out if the implied volatility could subsume both components.Andersen et al. (2007, p.701) illustrated namely that there is a large difference between both components when predicting the realized volatility. The continuous flow experiences high autocorrelation, whereas the jump component theoretically seems not predictable because it is less constant.
Barndorff-Nielsen and Shephard (2004, p.7) developed an econometric approach to split both parts of the realized volatility and started by assuming that that the logarithmic price (pt,j) of a financial instrument ensues the continuous-time jump diffusion process, which
is the following equation:
dp(t) = γ(t)dt + σ(t)dω(t) + δ(t)d(t), t ≥ 0 (4)
In this formula (t) denotes an unceasing locally confined variation process, σ is a non-negative volatility process, (t) is a regular Brownian movement, d(t) is similar to one if a jump is noticed and zero if there is no jump, and (t) refers to the size of the corresponding jump.
Furthermore, Barndorff-Nielsen and Shephard (2004, p.2) showed that the intraday returns (rt,j) for period t are calculated as follows:
rt,j= pt,j− pt,j−1, t = 1, … , T; j = 1, … , ∆ (5)
where the total amount of returns per day is stated by . In accordance with Andersen et al. (2007, p.702), the realized volatility per day is subsequently calculated by the formula below: σrvt = ∑ rt,j
2 ∆
j=1 , t = 1, … , T (6)
Next, Andersen, Bollerslev and Diebold (2007, p.703) showed in formula (7) that, when the jump component doesn’t exist and ∞, the right-side of the equation is equal to zero and realized volatility is a constant estimator of the integrated volatility.
σrvt → ∫ σ
2(s)ds + ∑ δ2(s) t<s≤t t
t (7)
Another estimator will be used however when the jump component does exist. In addition to prior literature, Barndorff-Nielsen and Shephard (2004, p.4) described namely that the bi-power variation method is suitable to obtain a constant prediction of the integrated volatility for this scenario. Huang and Tauchen (2005, p.485) specified this as follows:
σbvt = (μ)−2 ∆
∆ −(𝑘+1)∑ |rt,j|
∆
j=k+2 − |rt,j−k−1|, t = 1, … , T; j = 1, … , ∆ (8)
where μ represents √2⁄ . As mentioned by Busch et al. (2011, p.50), a higher amount of
intraday returns () will increase the accuracy of estimating the bi-power variation, but the affection by market microstructure noise is growing. Next, Huang and Tauchen (2005, p.484) described that the staggered version (k greater than zero) of the bi-power variation ensures a lower bias in relation to the non-staggered version (k equal to zero) because it prevents multiplicating returns that are using the similar price in formula (5) to calculate the relevant returns. Besides, Barndorff-Nielsen and Shephard (2004, p.4) showed that equation (9) results from equation (8) when ∞.
σbvt → ∫ 𝜎2 𝑡+1
𝑡 (s)ds (9)
To find out if the jump component is significant, Andersen et al. (2007, p.713) developed the formula below.
Zt=
ln(σrvt)−ln(σbvt) √∆(μ−4+2μ−2−5)TQ
tBVt−2
> φ (10)
So, in case when Zt is less than , which is the quantile of the standard normal distribution,
the jump is not significant. To execute this formula, the staggered realized tri-power quarticity (TQt) needs to be calculated and Busch et al. (2011, p.50) reported this formula.
TQt= ∆2 ∆−2(k+1) υ4/3 −3 ∑ |r t,j| 4/3 |rt,j−k−1| 4/3 |rt,j−2k−2| 4/3 1/∆ j=2k+3 (11)
where υ4/3 is equal to 22/3Γ(7/6)/Γ(1/2), which is approximately 0.83.
With the calculations above and knowing that the difference between the bi-power variation and the realized volatility is the jump component, it is possible to calculate the continuous and jump parts. Following Giot and Laurent (2007, p.342), the jump component (σrvjt) equals the formula below:
σrvjt = max [θ(Zt>)(σrvt− σbvt), 0] , t = 1, … , T (12)
In this equation is similar to one if a jump is significant and otherwise it is zero. Consequently, the continuous component is equal to the realized volatility if the jump
component is not significant, because the continuous component (σrvct) is calculated as
follows (Busch et al., 2011, p.50):
σrvct = σrvt− σrvjt, t = 1, … , T (13)
2.3 Theoretical hypotheses
Given the previous researches and theories, the main hypothesis is that implied volatility incorporates more information for the stock market that experience relatively less uncertainty than for the stock market that has more uncertainty. Nevertheless, the expectation is that the both implied volatility estimators will be biased. Despite there are ambiguous findings about forecasting relatively low, general and high volatilities, the hypothesis is based on the fact that the jump component in a more volatility market will occur more and heavier. Since it seems theoretically difficult to forecast the jump component, the expectation of this article is that implied volatility performs better as a forecast estimator for the SPY its realized volatility. This jump component will nonetheless cause that the estimators for both markets will be biased, because it seems unreasonable that using a daily implied volatility can fully predict unexpected shocks. A similar reasoning applies to the different sub periods. Since a relatively more unstable period will experience more jumps, the prediction ability of the implied volatility will perform less during this period.
Considering the results regarding historical components as control variables, it seems likely that implied volatility includes the historical jump component partly and the historical continuous component mainly as it is easier to predict. Overall, the outlook is that the implied volatility incorporates more information than these components as the implied volatility is forward looking and the components are based on historical data. Therefore, the prognosis is also that the implied volatility outperforms the historical regressions in univariate regressions. Though, a benefit of the historical volatility is that is used more data points by using high-frequency returns. Also, the historical continuous flow had a significant effect on the realized volatility in previous researches, which can result in an even more decreasing coefficient of the implied volatility in the multivariate regressions.
As there are contradictory results with respect to the different time spans for the historical components, it is hard to form an unambiguous expectation. Based on prior studies and on the fact that information will be incorporated quickly, the expectation is that the daily and weekly coefficients for both the components have a higher information content than the
monthly components. In line with the theory, the predicting ability of the continuous flow should be certainly higher than the jump component. Based on the foregoing, the implied volatility will incorporate more information from the monthly components than the shorter time spans.
3 Methodology
The start of this section contains the method that will be used to find out to which extent the implied volatility can predict volatility. Then, the theoretical hypotheses will be translated into statistical hypotheses in order to clarify when the research question can be accepted or rejected.
3.1 Procedure
The main purpose of this research is to investigate if it is possible to predict realized volatility with the help of implied volatility for different markets. In order to make a distinction between different markets in terms of volatility, the stock markets that will be studied are the US stock market (SPY) and the Brazilian stock market (EWZ) between January 1, 2007 and the December 31, 2017. This period will also be divided into a relatively stable and unstable period. Besides, the realized volatilities (σrvti), historical volatilities (σ
hvti) and
implied volatilities (σivti) are collected, whereas the suggestions for high-frequency data and
the continuous (σhvcti) and jump (σ
hvjti) components are taken into account. Section 4.1
provides a comprehensive explanation of the entire dataset.
Testing the statistical relationship between the implied volatility and the realized volatility is generally done by univariate and multivariate regressions (Muzzioli, 2010, p.570; Giot & Laurent, 2007, p.345). To start, the univariate regression will be executed. Blair et al. (2001, p.15) mentioned that the univariate regression estimates the ability of implied volatility and historical volatility to predict the dependent variable singularly. This thesis handles the regressions that include only the different components also as univariate regressions. According to Blair et al. (2001, p.15), there are normally no models that composite different forecast models as independent variables, such as option-based and history-based models, to predict the realized volatility straightforwardly. This is in contrast with models that include one forecast model. However, these multivariate regressions are used to find out if the implied volatility as independent variable incorporates input from past data and hence historical volatility and its components.
The variables in the regressions are all translated into their logarithmic form. As Muzzioli mentioned (2010, p.575), there are various benefits by doing this. First, the logarithms of volatilities will less suffer from outliers. Next, the logarithms will increase the divergence among volatilities because ln(σ 2 ) = 2 × ln(σ). This makes it easier to find differences between the independent variables. In addition, the density of log volatilities is roughly normally distributed, which make it more reliable when making conclusions. Andersen, Bollerslev, Diebold and Ebens (2001, p.54) mentioned that also the error term is almost normally distributed. Besides, they stated that the regressions and residuals will suffer less from heterescedasticity.
In this research six Ordinary Least Squares regressions (with Newey-West standard deviations considering overlapping data) will be executed on both markets. The regression models that will be used are as follows:
ln (σrvt,t+fi ) = β0 + β1 × ln (σivti) + εti (R1) ln (σrvt,t+fi ) = β0 + β2 × ln (σhvt,t−m+1i ) + εti (R2) ln (σrvt,t+fi ) = β0 + θ1 × ln (σhvct,t−m+1i ) + θ2 × ln (σhvct,t−w+1i ) + θ3 × ln (σhvcti) + π1 × ln (σhvjt,t−m+1i ) + π2 × ln (σhvjt,t−w+1i ) + π3 × ln (σhvjti) + εti (R3) ln (σrvt,t+fi ) = β0+ β1 × ln (σivti) + θ1× ln (σhvct,t−m+1i ) + θ2 × ln (σhvct,t−w+1i )+ θ3× ln (σhvcti) + π1 × ln (σhvjt,t−m+1i ) + π2 × ln (σhvjt,t−w+1i ) + π3× ln (σhvjti) + εti (R4) ln (σrvt,t+fi ) = β0 + β1 × ln (σivti) + π1 × ln (σhvjt,t−m+1i ) + π2 × ln (σhvjt,t−w+1i ) + π3 × ln (σhvjti) + εti (R5) ln (σrvt,t+fi ) = β0 + β1 × ln (σivti) + θ1 × ln (σhvct,t−m+1i ) + θ2 × ln (σhvct,t−w+1i ) + θ3 × ln (σhvcti) + εti (R6)
In these equations i represents SPX and EWX, t represents the observed value on day t, f represents 22 trading days ahead, w represents weekly and m represents monthly.
The first regression contains the univariate version by testing the forecast ability of implied volatility with respect to the realized volatility. This regression helps to find potential estimator biases and differences between the two markets. In the second regression implied
volatility is replaced by historical volatility to find out if the recent critics are appropriate and past information could explain the realized volatility as good as the implied volatility or even better. In the third regression the different historical components will be taken into account to find out if both components have different impacts on the realized volatility. The last three models will enclose both forecasting models to discover if implied volatility includes historical information. Initially, both the continuous as the jump components are included. In this way, it is also possible to test if there are differences between both components for different markets. Then, in the fifth regression the continuous component is excluded to measure distinctly if the implied volatility includes the jump component. As mentioned by Andersen et al. (2007, p.701), it is more difficult to predict the jump component and therefore it is theoretically harder for the implied volatility to include this component. At last, in the sixth regression the continuous component is included next to the implied volatility because Andersen et al. (2003, p.580) provided that this component reveals most information about the pattern of the volatility in the long-term.
These regressions are almost similar to previous literature (Christensen & Prabhala, 1998, p.134; Muzzioli, 2010, p.570), with the exception that the historical volatility is divided into its two components and its different time frames. Giot and Laurent (2007, p.346) mentioned that using the entire historical volatility would be too extensive, since it underestimates the differences between the two components and indicates that stakeholders are only using a t-number of days for their forecast analysis. By also using the two components and different time frames these issues are managed. This can be essential for declaring the ability of the implied volatility to forecast the realized volatility in case there are significant differences between the Brazilian and American stock market. Also Busch, et al. (2011, p.49) described that the jump component could ensure differences between the predicting performance for different markets in terms of the volatility level, since jumps occur more frequently and heavier in uncertain markets. In conclusion, the historical components are important control variables that can influence the estimating power of the implied volatility.
Muzzioli (2010, p.571), and Busch et al. (2011, p.52) used the Durbin Watson measure to test for heteroscedasticity in the error term of the regressions. It is however not feasible to perform this test when using the Newey-West standard deviations. Therefore, the methodology of Giot and Laurent (p.347) will be followed. They performed similar regressions
as the main regressions, except that the dependent variable is replaced by the squared error term that results from the previously executed regression. Then, a (joint) test will be executed where the null-hypothesis states that all independent variables are similar to zero. They mentioned that this test shows if the accuracy of the error term is altered by the independent variables. Besides, the interpretation is suitable in a financial foundation for the multivariate regressions, since a long term of jumps result in higher uncertainty that could subsequently harm the predicting ability of the model. Christensen and Prabhala (1998, p.134) performed a similar method, which they called errors-in-variables. For convenience purposes, this term will be used when these tests are discussed in the remainder of this thesis.
3.2 Statistical hypotheses
To find out if the implied volatility contain relevant information for the realized volatility, its coefficient β1 needs to be significantly different from zero. In this research, the
expectation is however that the coefficient is approaching one. Translating the main theoretical hypothesis therefore into a statistical hypothesis is twofold. First, the hypothesis is formed in terms of the main independent variable of regression (R1), (R4), (R5) and (R6) which is the coefficient of the implied volatility: H0: β1 = 1 and H1: β1 ≠ 1. So, β1 will answer a
part of the research question and will tell us if the implied volatility has been an excellent estimator for both markets. Second, the constants of the regressions are needed to define the potential bias. β0 = 0 and β1 = 1 imply that option traders do not overestimate or
underestimate implied volatility. If β0 >(<) 0 and β1 =1 or β0 = 0 and β1 >(<) 1 1 the implied
volatility will overestimate (underestimate) the realized volatility. Besides, in case when β0
>(<) 0 and β1 <(>) 1 the implied volatility underestimates (overestimates) moderate volatility
and overestimates (underestimates) excessive volatility. So, the biasness will be tested by a joint hypothesis (β0 = 0 and β1 = 1). Muzzioli (2010, p.570) performed similar tests. The
constants and coefficients of both markets will eventually be compared to make any conclusions and find out if the estimator of the SPY performs better. Finally, Christensen and Prabhala (1998, p.134) mentioned that, next to being unbiased, the error term needs to be heteroscedastic to declare that the implied volatility is efficient. The errors-in-variables regressions will be used to evaluate this.
Besides, β1 will be compared with β2 in regression (R2) and the different components
the historical volatility and its components for both markets. In fact, similar tests will be executed for regression (R2) and (R3) as the tests described above, only the coefficient of the implied volatility will be replaced by the coefficients of the historical volatility and its components.
To find out if the implied volatility includes historical information completely, the following joint hypotheses will be executed for regressions (R4): β0 = 0, β1 = 1 and 1 + 2 + 3
+ π1 + π2 + π3 = 0, (R5): β0 = 0, β1 = 1 and π1 + π2 + π3 = 0 and (R6): β0 = 0, β1 = 1 and 1 + 2 +
3 = 0. These joint hypotheses are in agreement with Busch et al. (2011, p.56). It seems
unreasonable however to expect that implied volatility incorporates all historical data. Therefore, it already would be a good sign if the coefficients of both components are, in contrast to the implied volatility, not significant. At last, in accordance with Muzzioli (2010, p.570), the constant term will be disregarded. So, if the joint hypothesis is accepted for a multivariate regression, the prediction have no bias after modifying the intercept.
With all tests above, it is possible to investigate if the implied volatility contains more information about the realized volatility than the historical volatility for both markets and incorporates historical information. Besides, since all tests will performed for both a stable and an unstable market as a relatively stable and unstable period, the effect of uncertainty can be measured by comparing the results.
4 Data and descriptive statistics
In this chapter the data and data sources will be listed. Besides, the descriptive statistics will be presented.
4.1 Data
In order to make a distinction between different markets in terms of volatility, the stock markets that will be studied are the US stock market and the Brazilian stock market between the 1st of January, 2007 and the 31nd of December, 2017. SPY options will be used as a proxy for the US stock market. SPY covers the market that experience relatively less uncertainty as it tracks the movements of the S&P 500 index, which contains 500 large-cap companies in a relatively stable country. The options on EWZ, which is a tracker that contains approximately 55 large and middle-sized companies in Brazil and reflect the market of this country (iShares, 2018a), will act as a proxy for a market that experience relatively more uncertainty. In relation to other emerging markets and products, this tracker contains
relatively long-term (option) data of Brazil (Fang & Refalo, 2013, p.76). Furthermore, the realized volatility is higher than the SPY during the entire time span (see Figure 1) and Bhuyan, Robbani, Talukdar and Jain (2016, p.180) stated that Brazil is an emerging market. They (2016, p.183) also mentioned that the regular trading hours of the American market are almost equal to the trading hours of the Brazilian stock market, with the exception of 30 minutes. This ensures that EWZ, which is offered on an American exchange, reflects real time and directly the changes on the Brazilian market. Starting point of the time span is the 1st of January, 2007 because the data for all options and intraday prices are available since then. To identify biases during relatively high and low periods of volatility, the observed years will also be divided into two sub periods. As the financial crises started in 2007, this year will be taken as beginning for the uncertain period. Figure 1 shows that the most volatile period is around 2008 and 2009 and stabilize entirely in 2010. Therefore, December 31, 2010 will be used as end point. Subsequently, the period between January 1, 2011 and December 31, 2018 indicates a relatively stable period. The separation of the entire period is similar to Christensen and Prabhala (1998, p.127), who did this for the stock market crash in 1987. The implied volatilities will be obtained from Bloomberg and are based on a variation of the Black-Scholes model, as the options on the SPY and the EWZ are American and several stocks that are listed on these underlying indices are paying dividend. Bloomberg (2014) uses the Bloomberg Live Implied Volatility Engine to calculate the implied volatility. This advanced and technical algorithm uses the options that are closest to 22 trading days out to expiry to define the monthly implied volatility. The underlying model, which is a variation of the Black-Scholes model (CBSBB), is as follows (Bloomberg, 2008):
CBSBB = Fimpl× e−R × T × ∅(d1) − e−R × T× S × ∅(d2) (14) where d1= ln( Fimpl / S) + 1 2 × σv × T σv√T and d2= ln( Fimpl / S) − 1 2 × σv × T σv√T (15)
Therefore, the implied forward (Fimpl) for the appropriate maturity needs to be calculated,
which can theoretically can be done by the following formula:
Where CA and PA are the actual observed call and put prices, D represents the dividend yield that have been estimated by another function within Bloomberg and the risk-free rate is based on America its 10-year treasury bond and will also be adjusted in the event of potential future movements. All variables are determined for the relevant exercise price and maturity per option. However, the implied forward calculation is in practice more complicated due to bid-ask spreads within the observed call and put prices. Bloomberg subsequently calculates the implied forward by taking the median of the matching mid prices (bid price+ask price2 ) of three forwards. These forwards are based on the options with the strike prices that is closest to 100% moneyness and the two strike prices around it.
In case of European options, the formulas above can be used. However, there is a more state-of-the-art method needed to calculate American options. It requires to identify the implied forward, dividend and volatility synchronously. These three numbers will first trigger the price of the European options. Subsequently, the formulas will be repeated over and over again by the algorithm to refine the three values up to the point that the inputs will equal the American option prices. In this research, the implied volatility will be obtained at the end of the trading day and from the 100% moneyness call options. Besides, the closing hours of the option markets are similar to the closing of the underlying markets. All volatilities will be shown in annual terms, as the volatility of a financial instrument is generally expressed as a percentage on annual basis.
The realized volatilities and its components will be obtained following the theory from Section 2.2. Starting point is to acquire the 5-minutes returns for both markets and the corresponding period. For this purpose, the Trade and Quote (TAQ) database of the New York Stock Exchange (NYSE) has been used, just like Rahman, Lee, and Ang (2002, p.157) did to calculate their realized volatility. The TAQ databank shows every transaction and transaction price up to milliseconds for the SPY and EWZ as they are listed on the NYSE. Besides, only data within the regular trading hours will be used to ensure liquidity, so from 09:30 EST until 16:00 EST. Then, it is possible to modify this data to 5 minutes market prices. Thereafter, the market prices will be turned into high-frequency returns. The first two returns will be taken out the data set, since Stoll and Whaley (1990, p.447) mentioned that these returns could be reflections of stale prices from the previous closing. With the help of this data, the entire process of calculating the realized volatilities and its components can be followed. In line with
the study of Muzzioli (2010, p.567), the historical volatility will then be the first lag of the realized volatility per frequency. At last, for calculating the jump component it is needed to choose an alpha that represents the significance level. Andersen et al. (2007, p.704) will be followed for this. They used an alpha of 0.999%.
The implied volatilities are stated in annual terms and the realized volatilities, historical volatilities and its components in daily terms. Therefore, all variables need to be expressed in a similar period to perform the regressions. The most convenience matter is to transform the implied volatility into a daily basis. This can be done by the following formula for the implied volatility: √2521 × 𝜎𝑖𝑣𝑡𝑖. Besides, this research will use different time spans for
the realized volatility, historical volatility and its components. The realized volatility for the f advanced trading days can be obtained by accumulate the daily volatilities for the period of interest and modify it into a daily basis. The exact formula is as follows: σrvt,t+fi =∑t+ft+1(∑∆j=1rt,j2)
f :.
For the historical volatility in regression (R2) the realized volatilities in the past month (m) will be used: σhvt,t−m+1i =∑t−m+1t (∑∆j=1rt,j2)
m . Finally, the jump and continuous volatilities need to be
calculated for the last day, week and month. Therefore, the following formulas will be used: σhvct,t−q+1i =∑t−q+1t (σhvcti)
q and σhvjt,t−q
i =∑t−q+1t (σhvjti)
q where q is 1, 5 or 22 trading days. These
formulas are corresponding to Giot and Laurent (2007, p.344). Eventually, the logarithmic values of all volatilities will be used in the regressions.
Christensen and Prabhala (1998, p.127) noticed however that the conclusions regarding implied and realized volatility can suffer from inaccuracy due to overlapping data when the dependent variable is sampled daily. This will be the case when using prediction horizons longer than 1 trading day, with a result the error term will be affected by autocorrelation. The method of Andersen et al. (2007, p.707) will be followed to manage this problem. They mentioned that overlapping data does not influence the coefficient its consistency. Nevertheless, it is important it is important to modify the standard errors using the Newey-West method considering heteroscedasticity and autocorrelation in the residual. According to Britten-Jones, Neuberger and Nolte (2011, p.663), this method is straightforward, trustworthy and consistent when dealing with overlapping data. Besides, they (2011, p.675) mentioned that this method have better outcomes than more advanced methodologies.
4.2 Descriptive statistics
Because the implied volatility generally predicts the realized volatility for approximately the coming month, the summary statistics and figures that are presented in this section for all volatilities are based on 22 trading days. The implied volatility for the entire observed period is based on a total of 2,769 daily observations, which is shown in Sample A of Table 1. The final sums of observations for the other volatilities are equal. These numbers result however from 218,589 intraday 5-minutes returns. A trading day generally consists of 79 intraday data points, including the first two dropped returns. The daily jump component is however noteworthy. It was in total 1,581 and 1,706 times significant for respectively the American and Brazilian stock market. Therefore, it can be stated that this component is substantial for both markets. That the Brazilian market experience more jumps is in accordance with the theory. Busch et al. (2011, p. 51) found also that the jump component is highly present in the bond market, the S&P 500, and the forex market between 1990 and 2002. The descriptive statistics are also divided in the two sub periods that present more information for different periods of uncertainty. Sample B in Table 1 shows the relatively uncertain period and Sample C the relatively certain period.
First, to visualize the various monthly volatilities, Figure 2 and 3 display graphical representations. At first sight, both charts follow similar patterns, except that the overall EWZ volatilities are higher and have heavier spikes. Furthermore, the volatilities in the first sub period for both markets are more extreme than the second period. During the first period and particularly before the largest spike in 2008, the implied volatility seems generally equal to or even less than the realized volatility, whereas the implied volatility for the second period seems higher than the realized volatility. Overall however, the implied volatility follows the realized volatility nearly, which corresponds to the theory and the charts of Giot and Laurent (2007, p.348) for the S&P 100 and the S&P 500. Moreover, the continuous component accompanies the entire realized volatility closely. The differences between both only grow during a large upward movement. This is also reflected in the jump component its line, which rises during such movements.
Table 1 confirms the inferences stated from the figures mostly. It is found that the implied volatility is relatively lower than the realized volatility (23.72 and 24.29) during the first period and vice versa for the second period (13.84 and 12.86) for the SPY. The overall mean for the implied volatility is nevertheless close to the realized volatility, 17.43 versus
Figure 2 Various monthly volatility measures of the American and Brazilian stock market
This figure represents the monthly implied volatility, the realized volatility and its continuous and jump components from a sample that includes observations from January 1, 2007 to December 31, 2017 for the American stock market (SPY is a proxy for this market) and the Brazilian stock market (EWZ is a proxy for this market). The data for the implied volatility is obtained from Bloomberg and is derived by using the Black-Scholes model. The monthly realized volatility and its components are calculated using 5-minutes intraday returns and are obtained from the Trade and Quote database. All volatilities are expressed in annual terms. The x-axis contains the dates in years and the y-axis contains the volatility.
Product: SPY Product: EWZ 0 20 40 60 80 100 120 140 160 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Implied Volatility Realized Volatility Continuous Component Jump Component 0 10 20 30 40 50 60 70 80 90 100 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Implied Volatility Realized Volatility Continuous Component Jump Component
17.05. Meanwhile, the average implied volatility for the EWZ is during all periods higher than the realized volatility. Also for this market however, the difference between the implied volatility and realized volatility in the second period (30.46 and 27.45) is relatively and absolutely higher than the difference in the first period (44.54 and 42.41). As shown in Sample A, the implied volatility over the entire period is 35.58 against a realized volatility of 32.93. According to Busch et al. (2011, p.51), a higher implied volatility can imply an option premium due to volatility risk or the possibility of early exercising since investors are risk averse. Muzzioli (2010, p.568) described that costs due to hedging also could ensure a higher implied volatility. These costs arouse in replicating the option by investing in the underlying instrument. Furthermore, the jump component its mean is highest for both markets during the crisis period, which corresponds with the highest spike. Besides, the jump its mean for the EWZ is higher than the SPY for all periods. Both findings are equal to the theory of jumps. Busch et al. (2011, p.51) expected that the unconditional averages of the realized volatility and the implied volatility are similar, but that the conditional deviation of the realized volatility is higher. They assumed this because implied volatility is based on the forecast of the realized volatility in the future, whereas the realized volatility is based on a smaller time span (daily data), which could enlarge the difference between relatively lower and higher volatilities. This is the case for both markets during all samples, with the exception of EWZ between 2011 and 2017. The largest discrepancies exist during the first period, where the standard deviation of the realized volatility are 14.87 and 22.94 for the American and Brazilian market against the implied volatility its deviation of 10.78 and 19.18. This is also the case for the continuous and jump components. Further, the maximums and minimums of the realized volatility over the entire period are more extreme for both markets than the implied volatility. All maximums are set during October 2008.
In the data section was stated that overlapping data is used and that therefore serial correlation could be a threat. Table 1 confirms that this is the case for all volatilities during all periods. The minimum LjungBox statistic, which evaluates for autocorrelation up to the 10th lag, is namely 3,941. Therefore, it is validated that Newey-West standard deviations need to be used. In the research of Andersen et al. (2007, p.704), the LjungBox statistics for the jump component was lower than for the other variables. It means that the serial correlation is smaller for this variable, which is also the case in this research during all periods. They mentioned that this also indicates that the jump and continuous component need to be
Table 1 Descriptive statistics of various monthly volatilities measures during different periods This table shows various summary statistics for the monthly implied volatility (σiv), the realized volatility (σrv) and its
continuous (σrvc) and jump (σrvj) components for the American stock market (SPY is a proxy for this market) and the Brazilian
stock market (EWZ is a proxy for this market). The logarithmic values of the implied and realized volatility are also showed. Sample A includes observations from January 1, 2007 to December 31, 2017. Sample B includes a subsample from January 1, 2007 to December 31, 2010 and Sample C includes a subsample from January 1, 2011 to December 31, 2017. The data for the implied volatility is obtained from Bloomberg and is derived by using the Black-Scholes model. The monthly realized volatility and its components are calculated using 5-minutes intraday returns and are obtained from the Trade and Quote database. All volatilities are expressed in annual terms. LB10 represents the statistic for the Ljung-Box test, which evaluates
for autocorrelation up to the 10th lag.
Descriptive statistics: various monthly volatilities
SPY EWZ
Statistic σiv σrv σrvc σrvj ln(σiv) ln(σrv) σiv σrv σrvc σrvj ln(σiv) ln(σrv) Sample A: Period between January 1, 2007 and December 31, 2017
Mean 17.43 17.05 15.62 1.43 -1.85 -1.91 35.58 32.93 30.39 2.55 -1.10 -1.19 Max 71.47 89.05 84.37 5.32 -0.34 -0.12 144.29 146.66 139.88 8.25 0.37 0.38 Min 6.40 4.92 4.47 0.39 -2.75 -3.01 14.96 14.51 12.96 0.59 -1.90 -1.93 St. Dev. 9.03 11.44 11.00 0.71 0.43 0.50 15.17 16.66 15.80 1.14 0.34 0.37 Skewness 2.18 2.96 3.00 1.57 0.63 0.87 2.67 3.48 3.54 1.65 0.89 1.23 Kurtosis 9.45 14.60 14.82 6.36 3.38 3.91 13.52 19.27 19.87 6.85 4.56 5.61 N 2,769 2,769 2,769 2,769 2,769 2,769 2,769 2,769 2,769 2,769 2,769 2,769 LB10 11,906 12,950 12,943 10,422 11,891 12,699 11,941 12,962 12,971 10,609 11,992 12,663
Sample B: Period between January 1, 2007 and December 31, 2010
Mean 23.72 24.29 22.39 1.90 -1.52 -1.55 44.54 42.41 39.26 3.15 -0.88 -0.96 Max 71.47 89.05 84.37 5.32 -0.34 -0.12 144.29 146.66 139.88 8.25 0.37 0.38 Min 9.17 8.53 7.62 0.67 -2.39 -2.46 23.55 18.04 15.59 1.00 -1.45 -1.71 St. Dev. 10.78 14.87 14.36 0.84 0.40 0.50 19.18 22.94 21.88 1.39 0.34 0.42 Skewness 1.69 2.17 2.21 1.06 0.45 0.68 2.23 2.43 2.45 1.26 1.14 1.02 Kurtosis 6.21 8.11 8.21 4.27 3.29 3.38 8.63 9.47 9.67 4.70 4.19 4.17 N 1,008 1,008 1,008 1,008 1,008 1,008 1,008 1,008 1,008 1,008 1,008 1,008 LB10 4,249 4,757 4,758 3,608 4,252 4,598 4,349 4,785 4,785 3,941 4,363 4,660
Sample C: Period between January 1, 2011 and December 31, 2017
Mean 13.84 12.86 11.71 1.15 -2.03 -2.12 30.46 27.45 25.25 2.20 -1.22 -1.33 Max 41.81 39.95 38.96 2.95 -0.87 -0.92 78.81 48.54 44.79 6.04 -0.24 -0.72 Min 6.40 4.91 4.47 0.39 -2.75 -3.01 14.96 14.51 12.96 0.59 -1.90 -1.93 St. Dev. 5.13 5.56 5.48 0.44 0.32 0.36 8.89 7.17 6.71 0.80 0.27 0.25 Skewness 1.85 2.05 2.11 1.00 0.58 0.72 1.37 0.76 0.69 1.20 0.41 0.26 Kurtosis 7.72 8.03 8.34 3.95 3.71 3.96 5.97 2.90 2.74 5.10 3.30 2.45 N 1,761 1,761 1,761 1,761 1,761 1,761 1,761 1,761 1,761 1,761 1,761 1,761 LB10 6,963 7,938 7,931 5,712 6,974 7,871 6,981 7,846 7,851 5,913 7,217 7,840
