Top PDF Distribution of Historic Market Data – Implied and Realized Volatility

Distribution of Historic Market Data – Implied and Realized Volatility

Distribution of Historic Market Data – Implied and Realized Volatility

would be expected to approach a normal or a stable distribution by central and generalized central limit theorem respectively, depending on whether the variance of the daily PDF exists or not. Single-day returns seem to be better described by power-law-tailed distributions (Fuentes, Gerig & Vicente, 2009; Gerig, Vicente & Fuentes, 2009; Ma & Serota, 2014) with existing variance, while intra-day data seem to point to very long tails with a diverging variance (Behfar, 2016) (with a usual caveat that the tail behavior is hard to pinpoint, especially with smaller data sets; for multi-day returns, see (Dashti Moghaddam & Serota, 2018; Liu et al., 2019)). Our own work (Dashti Moghaddam, Liu & Serota, 2019a) indicates that correlations fall off quickly, as a power law, over a period of about five days and then persist to slowly decay exponentially. Fig. 1 indicates a tailed distribution for RV 2 which saturates to its final shape over about five days as well. As per our current results (Dashti Moghaddam et al., 2019a), it is best fitted -- and with high precision -- by Generalized Beta Prime distribution – a generalization of BP – and Beta Prime distribution. Conversely, while VIX 2 and VXO 2 are best fitted by these two distributions as well, the precision is considerably worse, which may be another indicator of their deficiencies.
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Realized Volatility Risk

Realized Volatility Risk

To keep our discussion concise we have left out some important issues that can be explored in future work. We select two examples. First, in practice advances in realized volatility model- ing may not be translated so neatly into improvements in modeling the conditional distribution of returns. Two aspects of the link between realized volatility and returns should be studied more carefully. The assumption that returns standardized by realized volatility are approximately normal and independent seems to be inadequate for some series. Is there a role for jumps in ad- justing the distribution? Do the problems in measuring realized volatility make this relation less straightforward? We have also only considered a simple model for the dependence between return and volatility innovations. Second, we have mostly analyzed the performance of different models in one day ahead applications. Because financial quantities are so persistent many incongruent models are misleadingly competitive at very short horizons. More emphasis should be placed in investigating whether different models are consistent with a realistic longer horizon dynamics. Our analysis suggest that to do so we may need a more solid understanding of asymmetric effects.
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"Realized Volatility Risk"

"Realized Volatility Risk"

Despite the sizable forecasting gains made possible by volatility models based on high frequency data, our descriptive results can be directly related to the failure of GARCH volatility models to completely account for the excess kurtosis of returns (see for example Malmsten and Ter¨asvirta, 2004, Carnero et al., 2004). The researcher or practitioner interested in evaluating the density of returns from the perspective of a time series model still lives in a fat tailed world and purely predictive models of volatility may have little to say about it. In this paper, we do not interpret those facts as evidence against those models, but as a consequence of two factors: volatility risk (which causes excess kurtosis in the ex ante distribution of returns) and volatility feedback (or intraday leverage) effects (causing negative skewness). In the following section, we will argue that an adequate volatility model for return density forecasting and risk management in this setting should illuminate the dynamics of the higher moments. To pursue this objective we will turn to the idea of studying the time series volatility of realized volatility following Corsi et al. (2008).
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Realized Volatility Risk

Realized Volatility Risk

We will present several arguments for this emphasis. First, the presence of high and time-varying volatility risk brings substantially more uncertainty to the tails of the distribution of asset returns. In a standard stochastic volatility setting where returns given volatility follow a gaussian distribution the degree of volatility risk is the main determinant of the size of the tails of the ex-ante distribution of returns. If future realized volatility is relatively unpredictable a focus on predictive models will be insufficient for obtaining a good grasp of the tails of the return distribution, which in many cases (e.g., in risk management applications) is the main objective of the econometrician. Intuitively, when there is substantial volatility risk the ex-post realized volatility will frequently turn out to be much higher than the forecasted values: tail returns that would be virtually impossible with the distribution based on the point forecast (sometimes used implicitly or explicitly as an approximation) may be observed. Even though returns standardized by ex-post quadratic variation measures are nearly gaussian, returns standardized by fitted or predicted values of time series volatility models are far from normal. Given the uncertainty in volatility this is expected and should not be seen as evidence against those models; explicitly modeling the higher moment is necessary.
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Dynamic Estimation of Volatility Risk Premia and Investor Risk Aversion from Option-Implied and Realized Volatilities

Dynamic Estimation of Volatility Risk Premia and Investor Risk Aversion from Option-Implied and Realized Volatilities

Tables 1-3 summarize the parameter estimation for the volatility risk premium. The use of model-free implied volatility (MFIV) achieves a similar root-mean-squared error (RMSE) and convergence rate as the true infeasible risk-neutral implied volatility (RNIV). On the other hand, the misspecified Black-Scholes implied volatility (BSIV) shows slow convergence in estimating the volatility risk premium. Also, using realized volatility from five-minute returns (over a monthly horizon) has virtually the same small bias and high efficiency as the estimates based on the (infeasible) integrated volatility. In contrast, using the realized volatility from daily returns generally results in larger bias and significantly lower efficiency. Figures 1-3 report the Wald test for the risk premium parameter, which should be asymp- totically X 2 (1) distributed. In the cases of (infeasible) integrated volatility and five-minute realized volatility, the test statistics for the MFIV and RNIV measures are generally indis- tinguishable and closely approximated by the asymptotic distribution, the only exception being the high volatility persistence scenario (b) for which the MFIV measure results in slight over-rejection. In contrast, the (misspecified) BSIV measure shows clear evidence of over-rejection for all of the different scenarios. When the realized volatility is constructed from daily squared returns, the Wald test systematically loses power to detect any misspeci- fication, and the RNIV and MFIV measures now both show some under-rejection bias, while the over-rejection bias for the BSIV measure is somewhat mitigated. 9
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The properties of realized volatility and realized correlation: evidence from the Indian stock market

The properties of realized volatility and realized correlation: evidence from the Indian stock market

Table 1A reports descriptive statistics for volatility series. Average values of returns and volatilities are compatible between all Indian stock indices. Skewness values are around 8 and kurtosis values at around 70. All statistic values are extreme; indicating lack of normality. So, returns are far from being normal; as expected. Fat tails are also evident. The kurtosis of returns is much higher than that of a normal distribution at intraday frequency and tends to decrease as the return length increases. Thus, the probability density functions (pdf) of returns are leptokurtic with shapes depending on the time scale and presenting a very slow convergence of the Central Limit Theorem to the normal distribution. These results are consistent with Jarque-Bera (JB) test; in which, normality is rejected in all series of returns. However, normality is not rejected by the Cramer-von Mises (CVM) test on returns. Moreover, the skewness and kurtosis values for volatilities are close to 3 and between 11 and 13, respectively; indicating distributions not strongly adverse to normality. However, normality is rejected by the CVM test and JB test on volatilities of all Indian stock indices.
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Realized volatility risk

Realized volatility risk

To keep our discussion concise we have left out some important issues that can be explored in future work. We select two examples. First, in practice advances in realized volatility model- ing may not be translated so neatly into improvements in modeling the conditional distribution of returns. Two aspects of the link between realized volatility and returns should be studied more carefully. The assumption that returns standardized by realized volatility are approximately normal and independent seems to be inadequate for some series. Is there a role for jumps in ad- justing the distribution? Do the problems in measuring realized volatility make this relation less straightforward? We have also only considered a simple model for the dependence between return and volatility innovations. Second, we have mostly analyzed the performance of different models in one day ahead applications. Because financial quantities are so persistent many incongruent models are misleadingly competitive at very short horizons. More emphasis should be placed in investigating whether different models are consistent with a realistic longer horizon dynamics. Our analysis suggest that to do so we may need a more solid understanding of asymmetric effects.
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Long memory and the relation between implied and realized volatility

Long memory and the relation between implied and realized volatility

In order to assess the statistical signiÞcance of the regression estimates, we rely on subsam- pling (see Politis, Romano, and Wolf (1999) for a complete discussion). We prefer subsampling over the usual bootstrap because of its wider applicability. The only requirements for its valid- ity are the existence of a limiting distribution and some (rather mild) conditions limiting the dependence of either the data or the subsampled statistics. For example, subsampling is ap- plicable to the case of an autoregression with a unit root, while the standard bootstrap is not. Another advantage of subsampling over the bootstrap is that the rate of convergence to the asymptotic distribution does not have to be known and can be estimated (see Bertail, Politis, and Romano (1999)). This property is particularly attractive given that there is uncertainty as to whether we are in the stationary range or not and the convergence rates depend on the long memory parameters of both the regressors and the errors. Moreover, it is likely that the rates of convergence of the constant and slope estimators are different. Our subsampling approach can estimate these different rates consistently.
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Realized volatility, jump and beta: evidence from Canadian stock market

Realized volatility, jump and beta: evidence from Canadian stock market

Our results reveal that Canadian stock market experiences signi�cant price discontinues (jumps) and average jump arrival rate is about 0.17 jumps per day. We �nd that about 55% of jumps are due to the overnight returns and about 90% of jumps occur within 30 minutes of the market opening for trading – providing a strong evidence of jump clustering. While looking at the jump intensi- ties, our results show an asymmetric distribution of positive versus negative jumps for intraday returns but such asymmetry disappears when we include overnight returns in our analysis. Berk- man et al. (2012) and Lou et al. (2018) suggest that institutional investors tend to trade relatively more during the day and individual investors trade relatively more overnight. Such di�erences in jump charecteristics in intraday versus overnight returns potentially re�ecting the corresponding clientele e�ects. Therefore, it is important to incorporate overnight returns in jump risk analysis. 1 In our paper, we further show that although the e�ect of jump component in volatility fore- casting is statistically signi�cant, its economic signi�cance is very nominal - large portion of re- alized volatility is coming from the continuous component. When we examine e�ect of market
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The pricing of idiosyncratic risk: evidence from the implied volatility distribution

The pricing of idiosyncratic risk: evidence from the implied volatility distribution

we need the best possible predictor for the future return covariance matrix. Here, we have different choices for the estimation of the return volatility vector, as well as dif- ferent estimators for the correlation matrix. Roughly speaking, volatility estimators can be grouped into estimators based on historical time-series observations or implied volatility estimators. Reliance on risk-neutral estimates requires careful consideration of several points. At least two opposing effects come into play. On the one hand, im- plied estimators are based on empirically observable market prices. Therefore, they may be considered purely forward-looking variables with a high ability to reflect changing market conditions in a timely way. Thus, implied return moments are less prone to the statistical inertia of sample return time series. On the other hand, estima- tors under the risk-neutral measure reflect investor sentiment at the time of portfolio construction. As such, they can substantially differ from the realized values in the future. It is well-known that implied volatility typically overestimates future realiza- tions. This phenomenon entered the financial literature as volatility risk. 3 In addition, liquidity effects contained in option prices may distort implied moment estimators. To investigate the relative importance of the two opposing effects, we implement esti- mators on the historical, as well as the implied probability measures. Summarizing, it is not clear ex ante whether estimators relying on implied volatility outperform those relying on the sample time series. 4
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The Behaviour of Volatility Ratio: Implied vs. Realized Volatility of the S&P 500 and DAX Stock Indices

The Behaviour of Volatility Ratio: Implied vs. Realized Volatility of the S&P 500 and DAX Stock Indices

Christensen and Prabhala (1998) set out to check out the results presented by Canina et al. (1993). Time period for the S&P 100 index options price data used in the study over- laps with Canina et al. (1993) as the data is for 139 months period from November 1983 to May 1995. Options included in the data are at-the-money call options. They found that the implied volatility predicts the future volatility better than the realized volatility when forecasting. There are three reasons given for the different results when compared to the results presented by Canina et al. (1993). The first reason is that Christensen et al. have longer time period for the data to use in the study. The second reason is the data sampling as authors use a monthly data for the option and index prices, and options are those that are expiring just before the next sample date. The sampling made in this way gives the results some robustness against the autocorrelation in a daily returns. The third reason is that according to the authors the October 1987 stock market crash caused a shift in both implied and realized volatility levels. After the crash the explanatory value of the implied volatility for the future volatility is significantly better than before the crash. The result of Christensen et al. is supported by Gwilym and Buckle (1999) on one-month-forward forecasts.
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Implied volatility forecasting in the options market: a survey

Implied volatility forecasting in the options market: a survey

Filis (2009) analyzed the relationship between implied volatility and realized volatility in the Greek derivatives market using call and put option implied volatility. Daily data from January 2000 to January 2003, obtained from the Athens Derivatives Exchange and Athens Stock Exchange, were used to calculate the at-the-money call and put option implied volatilities. The results indicated that the implied volatility was a biased and inefficient predictor for the realized volatility, reflecting that the Greek derivatives market was inefficient. A unique study in relation to the informational content of implied volatility was carried out by Viteva et al. (2014) using the carbon futures traded on the European Climate Exchange (ECX), for the period from January 2008 to December 2010. The ECX is the world’s largest carbon derivatives exchange, on which carbon dioxide emissions are traded between companies that have installed climate-friendly technology and reduced their carbon emissions by more than is required and companies that exceed their emission limits. They found implied volatility to be highly informative about future volatility but a biased forecast of future volatility over the remaining life of the option. However, the forecast provided by the implied volatility was found to be statistically significant in predicting future volatility changes. Another recent and distinct study came from Birkelund et al. (2015), who developed an implied volatility index for the Nordic power forwards market, an electricity-linked implied volatility index. They used high-frequency data (every 30 minutes) to develop their daily observations, with a sample period from October 2005 to September 2011. The results suggested that there was a risk premium in the option contract price and that the volatility index was a biased estimator of the future realized volatility. The authors claimed that their findings were similar to those of Ederington and Guan (2002b).
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The Implied-Realized Volatility Relation with Jumps in Underlying Asset Prices

The Implied-Realized Volatility Relation with Jumps in Underlying Asset Prices

The analysis of nancial market volatility is of the utmost importance to asset pricing, derivative pricing, hedging, and risk management. Several di erent sources of information may be invoked in generating forecasts of unknown future volatility. Besides measurements based on historical return records, observed derivative prices are known from nance theory to be highly sensitive to and hence informative about future volatility. It is therefore natural to consider data on both asset prices and associated derivatives when measuring, modelling and forecasting volatility. Earlier literature has shown that implied volatility backed out from option prices provides a better volatility forecast than sample volatility based on past daily returns, but more recent literature shows that volatility forecasting based on past returns may be improved dramatically by using high-frequency (e.g., 5-minute) returns, and explicitly allowing for jumps in asset prices when computing forecasts. The important question addressed in the present paper is whether implied volatility from option prices continues to be the dominant volatility forecast, even when comparing to these new improved return based alternatives, using high-frequency data and accommodating a jump component in asset prices.
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The Role of Implied Volatility in Forecasting Future Realized Volatility and Jumps in Foreign Exchange, Stock, and Bond Markets

The Role of Implied Volatility in Forecasting Future Realized Volatility and Jumps in Foreign Exchange, Stock, and Bond Markets

C is for bond data and the results in the …rst line show that only monthly RV is signi…cant. In row two of each panel of Table 1, x = (C; J ), so this is a monthly frequency HAR- RV-CJ model (Andersen et al. (2007)). The conclusions for C are similar to those for RV in the …rst row, except that the monthly and weekly components become insigni…cant in the stock market. The jump components are insigni…cant, except daily J in the stock and bond markets. Adjusted R 2 improves when moving from …rst to second line of each panel of Ta- ble 1, thus con…rming the enhanced in-sample (Mincer-Zarnowitz) forecasting performance obtained by splitting RV into its separate components also found by Andersen et al. (2007). Out-of-sample forecasting performance (MAFE) improves in the stock market when sepa- rating C and J , but remains unchanged in the bond market, and actually deteriorates in the currency market, hence showing the relevance of including this criterion in the analysis. Next, implied volatility is added to the information set at time t in the HAR regressions. When RV is included together with IV , fourth row, all the realized volatility coe¢ cients turn insigni…cant in the foreign exchange and bond markets, and only daily RV remains sig- ni…cant in the stock market. Indeed, IV gets t-statistics of 6:15, 6:84, and 4:46 in the three markets, providing clear evidence of the relevance of IV in forecasting future volatility. The last row of each panel shows the results when including C and J together with IV , i.e., the full HAR-RV-CJIV model (14). In the foreign exchange market, IV completely subsumes the information content of both C and J at all frequencies. Adjusted R 2 is about equally high in the third line of the panel, where IV is the sole regressor and where also MAFE takes the best (lowest) value in the panel. In the stock market, both daily components of RV remain signi…cant, and the adjusted R 2 increases from 62% to 68% relative to having IV as the sole regressor, but again MAFE points to the speci…cation with only IV included as the best forecast. In the bond market, IV gets the highest t-statistic, as in the other two markets. In this case, the monthly jump component J t 22;t is also signi…cant and adjusted
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Measuring High-Frequency Causality Between Returns, Realized Volatility and Implied Volatility

Measuring High-Frequency Causality Between Returns, Realized Volatility and Implied Volatility

Other empirical studies on the link between returns and volatility are based on lower-frequency data or model-based measures of volatility; see Christie (1982), French et al. (1987), Schwert (1989), Turner, Startz and Nelson (1989), Nelson (1991), Glosten, Jagannathan and Runkle (1993) and Campbell and Hentschel (1992), Bekaert and Wu (2000), Whaley (2000), Ghysels, Santa-Clara and Valkanov (2004), Giot (2005), Ludvigson and Ng (2005), Dennis, Mayhew and Stivers (2006), and Guo and Savickas (2006) among others. On the relationship and the relative importance of the leverage and volatility feedback effects, the results of this literature are often ambiguous, if not contradictory. In particular, studies focusing on the leverage hypothesis conclude that the latter cannot completely account for changes in volatility; see Christie (1982) and Schwert (1989). However, for the volatility feedback effect, empirical findings conflict. French et al. (1987), Campbell and Hentschel (1992) and Ghysels et al. (2004) find a positive relation between volatility and expected returns, while Turner et al. (1989), Glosten et al. (1993) and Nelson (1991) find a negative relation. From individual-firm data, Bekaert and Wu (2000) conclude that the volatility feedback effect dominates the leverage effect empirically. The coefficient linking volatility to returns is often not statisti- cally significant. Ludvigson and Ng (2005) find a strong positive contemporaneous relation between the conditional mean and conditional volatility and a strong negative lag-volatility-in-mean effect. Guo and Savickas (2006) conclude that the stock market risk-return relation is positive, as stipulated by the CAPM; however, idiosyncratic volatility is negatively related to future stock market returns. Giot (2005) and Dennis et al. (2006) use lower frequency data (such as, daily data) to study the relationship between returns and implied volatility. Giot (2005) uses the S&P 100 index and an implied volatility index (VIX) to show that there is a contemporaneous asymmetric relationship between S&P 100 index returns and VIX: negative S&P 100 index returns yield bigger changes in VIX than do positive returns [see Whaley (2000)]. Dennis et al. (2006), using daily stock returns and innovations in option-derived implied volatilities, show that the rela- tion between stock returns and innovations in systematic volatility (idiosyncratic volatility) is substantially negative (near zero).
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Forecasting Realized Volatility of Russian stocks using Google Trends and Implied Volatility

Forecasting Realized Volatility of Russian stocks using Google Trends and Implied Volatility

variables. [10] evaluated the role of the online search activity for forecasting realized volatility of financial markets and commodity markets using models that also include market-based variables. They found that Google search data play a minor role in pre- dicting the realized volatility once implied volatility is included in the set of regressors. Therefore, they suggested that there might exist a common component between implied volatility and Internet search activity: in this regard, they found that most of the predictive information about realized volatility contained in Google Trends data is also included in implied volatility, whereas implied volatility has additional predictive content that is not captured by Google data.
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Realized volatility

Realized volatility

a stochastic volatility (SV) model estimated with data at daily or lower fre- quency. An alternative approach is to invoke option pricing models to invert observed derivatives prices into market-based forecasts of “implied volatility” over a fixed future horizon. Such procedures remain model-dependent and further incorporate a potentially time-varying volatility risk premium in the measure so they generally do not provide unbiased forecasts of the volatility of the underlying asset. Finally, some studies rely on “historical” volatility measures that employ a backward looking rolling sample return standard de- viation, typically computed using one to six months of daily returns, as a proxy for the current and future volatility level. Since volatility is persistent such measures do provide information but volatility is also clearly mean re- verting, implying that such unit root type forecasts of future volatility are far from optimal and, in fact, conditionally biased given the history of the past returns. In sum, while actual returns may be measured with minimal (measurement) error and may be analyzed directly via standard time series methods, volatility modeling has traditionally relied on more complex econo- metric procedures in order to accommodate the inherent latent character of volatility.
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Option data and modeling BSM implied volatility

Option data and modeling BSM implied volatility

In Figure 2, we plot the time series of 1Y at-the-money IV of DAX index options (left axis, black line) together with DAX closing prices (right axis, gray line). An option is called at-the-money (ATM) when the exercise price is equal or close to the spot (or to the forward). The index options were traded at the EUREX, Frankfurt (Germany), from 2000 to 2008. As is visible IV is subject to considerable variations. Average DAX index IV was about 22% with significantly higher levels in times of market stress, such as after the World Trade Center attacks 2001, during the pronounced bear market 2002-2003 and the financial crisis end of 2008.
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Forecasting global stock market implied volatility indices

Forecasting global stock market implied volatility indices

We use daily data from the 1 st of February, 2001 up to the 9 th of July, 2013 (i.e. 3132 trading days) from eight implied volatility indices. The implied volatilities are the following: VIX (S&P500 Volatility Index – US), VXN (Nasdaq-100 Volatility Index – US), VXD (Dow Jones Volatility Index – US), VSTOXX (Euro Stoxx 50 Volatility Index – Europe), VFTSE (FTSE 100 Volatility Index – UK), VDAX (DAX 30 Volatility Index – Germany), VCAC (CAC 40 Volatility Index – France) and VXJ (Japanese Volatility Index - Japan). The stock markets under consideration represent six out of the ten most important stock markets internationally, in terms of capitalization. In addition, these markets are among the most liquid markets of the world. Thus, we maintain that their implied volatility indices are representative of the world’s stock m arket uncertainty. The data were extracted from Datastream ® . As we aim for a common sample of the aforementioned implied volatility indices, the starting data of the sample period were dictated by the availability of the data of the VXN index.
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Active Trading in the Stock Market Using Implied Volatility

Active Trading in the Stock Market Using Implied Volatility

list. First, we have the indicator section which keeps the indicators used to predict the stocks by (default is only VIX). Second, we have the watch list section which contains the stocks, equities, or the securities that the user wants to keep track. The indicators, as discussed in the technology section, uses a two-week time interval to predict how the stocks in the watch list are going to perform, based on our models for the future 7 days. The user after seeing our prediction can then go on to do more research using other indicators such as the stochastic oscillator, MACD, or the momentum for the stock, to determine a more technical analysis. The user using our software can judge the future performance of their portfolio using strictly the Volatility Index and moving average flags as indicated by the volatility index or another indicator added. Figures 11 and 12 below show the regression tool component of our web application and a bit more detailed page about the stock in the watch list, respectively.
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