Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

We study the mass at the origin in the uncorrelated stochastic alpha, beta, rho stochastic volatility model and derive several tractable expressions, in particular when time becomes small or large. As an application—in fact the original motivation for this paper—we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.     

The implied volatility smirk

This paper provides an industry standard on how to quantify the shape of the implied volatility smirk in the equity index options market. Our local expansion method uses a second-order polynomial to describe the implied volatility–moneyness function and relates the coefficients of the polynomial to the properties of the implied risk-neutral distribution of the equity index return. We present a formal, two-way representation of the link between the level, slope and curvature of the implied volatility smirk and the risk-neutral standard deviation, skewness and excess kurtosis. We then propose a new semi-analytical method to calibrate option-pricing models based on the quantified implied volatility smirk, and investigate the applicability of two option-pricing models.     

Dynamics of foreign exchange implied volatility and implied correlation surfaces

The prices of currency options expressed in terms of their implied volatilities and the implied correlations between foreign exchange rates at a given point in time depend on option delta and time to maturity. Implied volatilities and implied correlations likewise may thus be represented as a surface. It is well known that these surfaces exhibit both skew/smile features and term structure effects and their shapes fluctuate substantially over time. Using implied volatilities on three currency pairs as well as historical implied correlation values between them, we study the nature of these fluctuations by applying a Karhunen-Loève decomposition that is a generalization of a principal component analysis. We demonstrate that the largest share in the dynamics of these surfaces' fluctuations may be explained by exactly the same three factors, providing evidence of strong interdependences between implied correlation and implied volatility of global currency pairs.     

What is the correct meaning of implied volatility?

This paper presents a closed-form solution for the valuation of European options under the assumption that the excess returns of an underlying asset follow a diffusion process. In light of our model, the implied volatility computed from the Black–Scholes formula should be viewed as the volatility of excess returns rather than as the volatility of gross returns. Using the SPX and the OMX options data, we test whether implied volatility obtained from Black-Scholes option price explains the volatilities of excess returns better than gross returns, even though the result is not statistically significant.     

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.

A PDE method for estimation of implied volatility

In this paper it is proved that the Black–Scholes implied volatility satisfies a second order non-linear partial differential equation. The obtained PDE is then used to construct an algorithm for fast and accurate polynomial approximation for Black–Scholes implied volatility that improves on the existing numerical schemes from literature, both in speed and parallelizability. We also show that the method is applicable to other problems, such as approximation of implied Bachelier volatility.     

The bias in Black-Scholes/Black implied volatility: An analysis of equity and energy markets

In this paper we examine the extent of the bias between Black and Scholes (1973)/Black (1976) implied volatility and realized term volatility in the equity and energy markets. Explicitly modeling a market price of volatility risk, we extend previous work by demonstrating that Black-Scholes is an upward-biased predictor of future realized volatility in S&P 500/S&P 100 stock-market indices. Turning to the Black options-on-futures formula, we apply our methodology to options on energy contracts, a market in which crises are characterized by a positive correlation between price-returns and volatilities: After controlling for both term-structure and seasonality effects, our theoretical and empirical findings suggest a similar upward bias in the volatility implied in energy options contracts. We show the bias in both Black-Scholes/Black implied volatilities to be related to a negative market price of volatility risk. JEL Classification G12 · G13     

Using implied volatility jumps for realized volatility forecasting: Evidence from the Chinese market

This study examines the Chinese implied volatility index (iVIX) to determine whether jump information from the index is useful for volatility forecasting of the Shanghai Stock Exchange 50ETF. Specifically, we consider the jump sizes and intensities of the 50ETF and iVIX as well as cojumps. The findings show that both the jump size and intensity of the 50ETF can improve the forecasting accuracy of the 50ETF volatility. Moreover, we find that the jump size and intensity of the iVIX provide no significant predictive ability in any forecasting horizon. The cojump intensity of the 50ETF and iVIX is a powerful predictor for volatility forecasting of the 50ETF in all forecasting horizons, and the cojump size is helpful for forecasting in short forecasting horizon. In addition, for a one-day forecasting horizon, the iVIX jump size in the cojump is more predictive of future volatility than that of the 50ETF when simultaneous jumps occur. Our empirical results are robust and consistent. This work provides new insights into predicting asset volatility with greater accuracy.     

Striking the implied volatility of US drone companies

This study analyzes the impact of drone strikes on the implied volatility of US drone companies. We find evidence of an overall increase in the implied volatility the day after the strike. We subset drone strikes according to countries targeted and US president in office, finding a more significant impact for strikes in Afghanistan and Pakistan, and under the Bush or Obama’s administration. We find that drone strikes are also associated with next day decreasing stock returns of the drone companies. A possible increasing geopolitical risk concern and resiliency rationale may explain our findings.     

On the seasonality in the implied volatility of electricity options

Seasonality is an important topic in electricity markets, as both supply and demand are dependent on the time of the year. Clearly, the level of prices shows a seasonal behaviour, but not only this. Also, the price fluctuations are typically seasonal. In this paper, we study empirically the implied volatility of options on electricity futures, investigate whether seasonality is present and we aim at quantifying its structure. Although typically futures prices can be well described through multi-factor models including exponentially decreasing components, we do not find evidence of exponential behaviour in our data set. Generally, a simple linear shape reflects the squared volatilities very well as a curve depending on the time to maturity. Moreover, we find that the level of volatility exhibits clear seasonal patterns that depend on the delivery month of the futures. Furthermore, in an out-of-sample analysis we compare the performance of several implementations of seasonality in the one-factor framework.     

S&P 500 implied volatility and monetary policy announcements

While many studies have investigated the link between macroeconomic events and equity market volatility, few have considered the impact on option implied volatilities. Given the recent focus on trading in implied volatility, in the context of the S&P 500 VIX index, this paper examines how the VIX index behaves around US monetary policy announcements. It is revealed that the VIX index falls significantly on the day of Federal Open Market Committee meetings.     

The asymmetric relationship between returns and implied volatility: Evidence from global stock markets

We investigate the asymmetric relationship between returns and implied volatility for 20 developed and emerging international markets. In particular we examine how the sign and size of return innovations affect the expectations of daily changes in volatility. Our empirical findings indicate that the conditional contemporaneous return-volatility relationship varies not only based on the sign of the expected returns but also upon their magnitude, according to recent results from the behavioral finance literature. We find evidence of an asymmetric and reverse return-volatility relationship in many advanced, Asian, Latin-American, European and South African markets. We show that the US market displays the highest reaction to price falls, Asian markets present the lowest sensitivity to volatility expectations, while the Euro area is characterized by a homogeneous response both in terms of direction and impact. These results may be safely attributed to cultural and societal characteristics. An extensive quantile regression analysis demonstrates that the detected asymmetric pattern varies particularly across the extreme distribution tails i.e., in the highest/lowest quantile ranges. Indeed, the classical feedback and leverage hypotheses appear not plausible, whilst behavioral theories emerge as the new paradigm in real-world applications.     

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.     

Jump risk,stock returns,and slope of implied volatility smile

In the presence of jump risk, expected stock return is a function of the average jump size, which can be proxied by the slope of option implied volatility smile. This implies a negative predictive relation between the slope of implied volatility smile and stock return. For more than four thousand stocks ranked by slope during 1996–2005, the difference between the risk-adjusted average returns of the lowest and highest quintile portfolios is 1.9% per month. Although both the systematic and idiosyncratic components of slope are priced, the idiosyncratic component dominates the systematic component in explaining the return predictability of slope. The findings are robust after controlling for stock characteristics such as size, book-to-market, leverage, volatility, skewness, and volume. Furthermore, the results cannot be explained by alternative measures of steepness of implied volatility smile in previous studies.     

The information content of CDS implied volatility and associated trading strategies

Using the theoretical link between put options and credit default swaps (CDS) in a very general setting, we develop a robust measure of CDS implied volatility (CIV) that captures the information content of CDS markets. Specifically, we use the unit recovery claim to bridge CDS and deep out-of-the-money put options of the same firm and then back out CIV via the binomial tree. Our CIV measure strongly co-moves with the option implied volatility (OIV), with a correlation coefficient of 0.8. Based on the standardized difference between CIV and OIV, we construct CDS and option trading strategies. Without taking transaction costs into account, the long–short CDS trading strategy achieves an annualized return of 58.29% and a Sharpe ratio of 2.97, which cannot be explained by non-parametric skewness and volatility risk.     

An empirical comparison of continuous-time models of implied volatility indices

We explore the ability of alternative popular continuous-time diffusion and jump-diffusion processes to capture the dynamics of implied volatility indices over time. The performance of the various models is assessed under both econometric and financial metrics. To this end, data are employed from major European and American implied volatility indices and the rapidly growing CBOE volatility futures market. We find that the addition of jumps is necessary to capture the evolution of implied volatility indices under both metrics. Mean reversion is of second-order importance though. The results are consistent across the various metrics, markets, and construction methodologies.     

News implied volatility and disaster concerns

We construct a text-based measure of uncertainty starting in 1890 using front-page articles of the Wall Street Journal. News implied volatility (NVIX) peaks during stock market crashes, times of policy-related uncertainty, world wars, and financial crises. In US postwar data, periods when NVIX is high are followed by periods of above average stock returns, even after controlling for contemporaneous and forward-looking measures of stock market volatility. News coverage related to wars and government policy explains most of the time variation in risk premia our measure identifies. Over the longer 1890–2009 sample that includes the Great Depression and two world wars, high NVIX predicts high future returns in normal times and rises just before transitions into economic disasters. The evidence is consistent with recent theories emphasizing time variation in rare disaster risk as a source of aggregate asset prices fluctuations.     

The relation between implied and realised volatility in the Danish option and equity markets

We show that the conclusions to be drawn concerning the informational efficiency of illiquid options markets depend critically on whether one carefully recognises and appropriately deals with the econometrics of the errors‐in‐variables problem. This paper examines the information content of options on the Danish KFX share index. We consider the relation between the volatility implied in an option's price and the subsequently realised index return volatility. Since these options are traded infrequently and in low volumes, the errors‐in‐variables problem is potentially large. We address the problem directly using instrumental variables techniques. We find that when measurement errors are controlled for, call option prices even in this very illiquid market contain information about future realised volatility over and above the information contained in historical volatility.     

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.     

The impact of central bank digital currency variation on firm's implied volatility

The paper investigates whether a firm's implied volatility is affected by the volatility of central bank digital currencies. Our sample covers 2853 listed companies in the United States from 2014 to 2018. First, we find the variation of central bank digital currency has a positive impact on a firm’s implied volatility. Second, the healthier firms’ conditions can reduce the relationship between central bank digital currency variation and a firm’s implied volatility. Third, the positive relation between central bank digital currency and firm’s implied volatility still exists in investment-grade, speculative-grade, and unrated firms. Finally, to eliminate the endogeneity problem, we adopt simultaneous equation models (SEM) and find our results are still robust after excluding endogenous concerns. Our research provides a reminder for corporate managers and new implications for policymakers.