VOLATILITY FORECASTS,TRADING VOLUME,AND THE ARCH VERSUS OPTION‐IMPLIED VOLATILITY TRADE‐OFF

We investigate empirically the role of trading volume (1) in predicting the relative informativeness of volatility forecasts produced by autoregressive conditional heteroskedasticity (ARCH) models versus the volatility forecasts derived from option prices, and (2) in improving volatility forecasts produced by ARCH and option models and combinations of models. Daily and monthly data are explored. We find that if trading volume was low during period t?1 relative to the recent past, ARCH is at least as important as options for forecasting future stock market volatility. Conversely, if volume was high during period t?1 relative to the recent past, option‐implied volatility is much more important than ARCH for forecasting future volatility. Considering relative trading volume as a proxy for changes in the set of information available to investors, our findings reveal an important switching role for trading volume between a volatility forecast that reflects relatively stale information (the historical ARCH estimate) and the option‐implied forward‐looking estimate.     

The information content of implied volatilities and model-free volatility expectations: Evidence from options written on individual stocks

We measure the volatility information content of stock options for individual firms using option prices for 149 US firms and the S&P 100 index. We use ARCH and regression models to compare volatility forecasts defined by historical stock returns, at-the-money implied volatilities and model-free volatility expectations for every firm. For 1-day-ahead estimation, a historical ARCH model outperforms both of the volatility estimates extracted from option prices for 36% of the firms, but the option forecasts are nearly always more informative for those firms that have the more actively traded options. When the prediction horizon extends until the expiry date of the options, the option forecasts are more informative than the historical volatility for 85% of the firms. However, at-the-money implied volatilities generally outperform the model-free volatility expectations.     

Forecasting currency volatility: A comparison of implied volatilities and AR(FI)MA models

We compare forecasts of the realized volatility of the pound, mark and yen exchange rates against the dollar, calculated from intraday rates, over horizons ranging from one day to three months. Our forecasts are obtained from a short memory ARMA model, a long memory ARFIMA model, a GARCH model and option implied volatilities. We find intraday rates provide the most accurate forecasts for the one-day and one-week forecast horizons while implied volatilities are at least as accurate as the historical forecasts for the one-month and three-month horizons. The superior accuracy of the historical forecasts, relative to implied volatilities, comes from the use of high frequency returns, and not from a long memory specification. We find significant incremental information in historical forecasts, beyond the implied volatility information, for forecast horizons up to one week.     

The incremental volatility information in one million foreign exchange quotations

The volatility information found in high-frequency exchange rate quotations and in implied volatilities is compared by estimating ARCH models for DM/$ returns. Reuters quotations are used to calculate five-minute returns and hence hourly and daily estimates of realised volatility that can be included in equations for the conditional variances of hourly and daily returns. The ARCH results show that there is a significant amount of information in five-minute returns that is incremental to options information when estimating hourly variances. The same conclusion is obtained by an out-of-sample comparison of forecasts of hourly realised volatility.     

Forecasting exchange rate volatility: The superior performance of conditional combinations of time series and option implied forecasts

This paper provides empirical evidence that combinations of option implied and time series volatility forecasts that are conditional on current information are statistically superior to individual models, unconditional combinations, and hybrid forecasts. Superior forecasting performance is achieved by both, taking into account the conditional expected performance of each model given current information, and combining individual forecasts. The method used in this paper to produce conditional combinations extends the application of conditional predictive ability tests to select forecast combinations. The application is for volatility forecasts of the Mexican peso–US dollar exchange rate, where realized volatility calculated using intraday data is used as a proxy for the (latent) daily volatility.     

Correlated implied volatility with jump and cross section of stock returns

I derive the option‐implied volatility allowing for nonzero correlation between price jump and diffusive risk to examine the information content of implied diffusive, jump risks and their implied covariance in the cross‐sectional variation of future returns. This study documents a strong predictive power of realized volatility and correlated implied volatility spread (RV ? IV_C) in the cross section of stock returns. The difference of realized volatility with the implied diffusive volatility (RV ? σ_C), jump risk (RV ? γ_C) and covariance (RV ? ICov) can forecast future returns. These RV ? σ_C and RV ? γ_C anomalies are robustly persistent even after controlling for market, size, book‐to‐market value, momentum and liquidity factors.     

The quality of market volatility forecasts implied by S&P 100 index option prices

This study examines the performance of the S&P 100 implied volatility as a forecast of future stock market volatility. The results indicate that the implied volatility is an upward biased forecast, but also that it contains relevant information regarding future volatility. The implied volatility dominates the historical volatility rate in terms of ex ante forecasting power, and its forecast error is orthogonal to parameters frequently linked to conditional volatility, including those employed in various ARCH specifications. These findings suggest that a linear model which corrects for the implied volatility's bias can provide a useful market-based estimator of conditional volatility.     

Estimation and Prediction of a Non-Constant Volatility

In this paper we study volatility functions. Our main assumption is that the volatility is a function of time and is either deterministic, or stochastic but driven by a Brownian motion independent of the stock. Our approach is based on estimation of an unknown function when it is observed in the presence of additive noise. The set up is that the prices are observed over a time interval [0, t], with no observations over (t, T), however there is a value for volatility at T. This value is may be inferred from options, or provided by an expert opinion. We propose a forecasting/interpolating method for such a situation. One of the main technical assumptions is that the volatility is a continuous function, with derivative satisfying some smoothness conditions. Depending on the degree of smoothness there are two estimates, called filters, the first one tracks the unknown volatility function and the second one tracks the volatility function and its derivative. Further, in the proposed model the price of option is given by the Black–Scholes formula with the averaged future volatility. This enables us to compare the implied volatility with the averaged estimated historical volatility. This comparison is done for three companies and has shown that the two estimates of volatility have a weak statistical relation.     

Volatility conditional on price trends

The influence of the past price behaviour on the realized volatility is investigated, showing that trending (driftless) prices lead to increased (decreased) realized volatility. This ‘volatility induced by trend’ constitutes a new stylized fact. The past price behaviour is measured by the product of two non-overlapping returns (of the form r × L[r] where L is the lag operator), and is different from the usual heteroskedasticity. The effect is studied empirically using USD/CHF foreign exchange data, in a large range of time horizons. On the modelling side, a set of ARCH based processes are modified in order to include the ‘volatility induced by trend’ effect, and their forecasting performances are compared. The aim is to understand the role and importance of the various terms that can be included in such a model. For a better forecast, it is shown that the main factor is the shape of the memory kernel (i.e. power law), and the next most important factor is the trend effect. The subtle role of mean reversion is also discussed.     

The forecasting ability of correlations implied in foreign exchange options

This paper evaluates the forecasting accuracy of correlations derived from implied volatilities in dollar-mark, dollar-yen, and mark-yen options from January 1989 to May 1995. As a forecast of realized correlation between the dollar-mark and dollar-yen, implied correlation is compared against three alternative forecasts based on time series data: historical correlation, RiskMetrics' exponentially-weighted moving average correlation, and correlation estimated using a bivariate GARCH(1,1) model. At the 1-month and 3-month forecast horizons, we find that implied correlation outperforms, often significantly, these alternative forecasts. In combinations, implied correlation always incrementally improves the performance of other forecasts, but not the converse; in certain cases, historically-based forecasts contribute no incremental information to implied forecasts. The superiority of the implied correlation forecast holds even when forecast errors are weighted by realized variances, reflecting correlation's contribution to the dollar variance of a multicurrency portfolio.     

Market imperfections and the information content of implied and realized volatility

The information content of option implied volatility and realized volatility under market imperfections are studied in the context of GARCH modeling and volatility forecasts of Taiwan stock market (TAIEX) returns. Consistent with most studies, we find that the Taiwan implied volatility index (TVIX) calculated from the TAIEX option prices contains most of the information, and that White's [White, H., 2000. A reality check for data snooping. Econometrica 68, 1097–1126] reality check test cannot reject the null hypothesis that the TVIX provides the best forecast. Possibly due to market imperfections, however, the incremental information content of realized volatility as well as daily returns cannot be ruled out. Finally, we also find that the information is found only in the most recent TVIX, indicating information is being efficiently impounded on the TAIEX option prices. This finding suggests that appropriately designed derivative products can alleviate the problems caused by market imperfections.     

Forecasting daily variability of the S&P 100 stock index using historical,realised and implied volatility measurements

The increasing availability of financial market data at intraday frequencies has not only led to the development of improved volatility measurements but has also inspired research into their potential value as an information source for volatility forecasting. In this paper, we explore the forecasting value of historical volatility (extracted from daily return series), of implied volatility (extracted from option pricing data) and of realised volatility (computed as the sum of squared high frequency returns within a day). First, we consider unobserved components (UC-RV) and long memory models for realised volatility which is regarded as an accurate estimator of volatility. The predictive abilities of realised volatility models are compared with those of stochastic volatility (SV) models and generalised autoregressive conditional heteroskedasticity (GARCH) models for daily return series. These historical volatility models are extended to include realised and implied volatility measures as explanatory variables for volatility. The main focus is on forecasting the daily variability of the Standard & Poor's 100 (S&P 100) stock index series for which trading data (tick by tick) of almost 7 years is analysed. The forecast assessment is based on the hypothesis of whether a forecast model is outperformed by alternative models. In particular, we will use superior predictive ability tests to investigate the relative forecast performances of some models. Since volatilities are not observed, realised volatility is taken as a proxy for actual volatility and is used for computing the forecast error. A stationary bootstrap procedure is required for computing the test statistic and its p-value. The empirical results show convincingly that realised volatility models produce far more accurate volatility forecasts compared to models based on daily returns. Long memory models seem to provide the most accurate forecasts.     

The jump component of S&P 500 volatility and the VIX index

Much research has investigated the differences between option implied volatilities and econometric model-based forecasts. Implied volatility is a market determined forecast, in contrast to model-based forecasts that employ some degree of smoothing of past volatility to generate forecasts. Implied volatility has the potential to reflect information that a model-based forecast could not. This paper considers two issues relating to the informational content of the S&P 500 VIX implied volatility index. First, whether it subsumes information on how historical jump activity contributed to the price volatility, followed by whether the VIX reflects any incremental information pertaining to future jump activity relative to model-based forecasts. It is found that the VIX index both subsumes information relating to past jump contributions to total volatility and reflects incremental information pertaining to future jump activity. This issue has not been examined previously and expands our understanding of how option markets form their volatility forecasts.     

Pricing under rough volatility

From an analysis of the time series of realized variance using recent high-frequency data, Gatheral et al. [Volatility is rough, 2014] previously showed that the logarithm of realized variance behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated variance. We analyse in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.     

Volatility timing: How best to forecast portfolio exposures

This paper investigates how best to forecast optimal portfolio weights in the context of a volatility timing strategy. It measures the economic value of a number of methods for forming optimal portfolios on the basis of realized volatility. These include the traditional econometric approach of forming portfolios from forecasts of the covariance matrix. Both naïve forecasts using simple historical averages, and those generated from econometric models are considered. A novel method, where a time series of optimal portfolio weights are constructed from observed realized volatility and direct forecast is also proposed. A number of naïve forecasts and the approach of directly forecasting portfolio weights show a great deal of merit. Resulting portfolios are of similar economic benefit to a number of competing approaches and are more stable across time. These findings have obvious implications for the manner in which volatility timing is undertaken in a portfolio allocation context.     

Evaluating the Forecast Accuracy of Emerging Market Stock Returns

This paper analyzes the forecast performance of emerging market stock returns using standard autoregressive moving average (ARMA) and more elaborated autoregressive conditional heteroskedasticity (ARCH) models. Our results indicate that the ARMA and ARCH specifications generally outperform random walk models. Models that allow for asymmetric shocks to volatility are better for in-sample estimation (threshold autoregressive conditional heteroskedasticity for daily returns and exponential generalized autoregressive conditional heteroskedasticity for longer periods), and ARMA models are better for out-of-sample forecasts. The results are valid using both U. S. dollar and domestic currencies. Overall, the forecast errors of each Latin American market can be explained by the forecasts of other Latin American markets and Asian markets. The forecast errors of each Asian market can be explained by the forecasts of other Asian markets, but not by Latin American markets. Our predictability results are economically significant and may be useful for portfolio managers to enter or leave the market.     

The relationship between implied and realized volatility: evidence from the Australian stock index option market

This paper examines the relationship between the volatility implied in option prices and the subsequently realized volatility by using the S&P/ASX 200 index options (XJO) traded on the Australian Stock Exchange (ASX) during a period of 5 years. Unlike stock index options such as the S&P 100 index options in the US market, the S&P/ASX 200 index options are traded infrequently and in low volumes, and have a long maturity cycle. Thus an errors-in-variables problem for measurement of implied volatility is more likely to exist. After accounting for this problem by instrumental variable method, it is found that both call and put implied volatilities are superior to historical volatility in forecasting future realized volatility. Moreover, implied call volatility is nearly an unbiased forecast of future volatility.

The Predictive Power of the French Market Volatility Index: A Multi Horizons Study

The main purpose of this paper is to examine empirically the time series properties of the French Market Volatility Index (VX1). We also examine the VX1's ability to forecast future realized market volatility and finds a strong relationship. More importantly, we show how the index can be used to generate volatility forecasts over different horizons and that these forecasts are reasonably accurate predictors of future realized volatility.     

Forecasting volatility of the U.S. oil market

We examine the information content of the CBOE Crude Oil Volatility Index (OVX) when forecasting realized volatility in the WTI futures market. Additionally, we study whether other market variables, such as volume, open interest, daily returns, bid-ask spread and the slope of the futures curve, contain predictive power beyond what is embedded in the implied volatility. In out-of-sample forecasting we find that econometric models based on realized volatility can be improved by including implied volatility and other variables. Our results show that including implied volatility significantly improves daily and weekly volatility forecasts; however, including other market variables significantly improves daily, weekly and monthly volatility forecasts.