Asymptotics of implied volatility to arbitrary order

In a unified model-free framework that includes long-expiry, short-expiry, extreme-strike, and jointly-varying strike-expiry regimes, we generate implied volatility and implied variance approximations, with rigorous error estimates asymptotically smaller than any given power of L, where L denotes the exogenously given absolute log of an option price that approaches zero. Our results, therefore, sharpen to arbitrarily high order of accuracy (and, moreover, extend to general extreme regimes) the model-free asymptotics of implied volatility. We then apply these general formulas to particular examples: Heston (using a previously known L expansion) and Lévy (using saddlepoint methods to derive L expansions).     

Maturity cycles in implied volatility

The skew effect in market implied volatility can be reproduced by option pricing theory based on stochastic volatility models for the price of the underlying asset. Here we study the performance of the calibration of the S&P 500 implied volatility surface using the asymptotic pricing theory under fast mean-reverting stochastic volatility described in [8]. The time-variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. By extending the mathematical analysis to incorporate model parameters which are time-varying, we show this behaviour can be explained in a manner consistent with a large model class for the underlying price dynamics with time-periodic volatility coefficients.Received: December 2003, Mathematics Subject Classification (2000): 91B70, 60F05, 60H30JEL Classification: C13, G13Jean-Pierre Fouque: Work partially supported by NSF grant DMS-0071744.Ronnie Sircar: Work supported by NSF grant DMS-0090067. We are grateful to Peter Thurston for research assistance.We thank a referee for his/her comments which improved the paper.     

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.     

Inferring future volatility from the information in implied volatility in Eurodollar options: a new approach

We study the information content of implied volatility fromseveral volatility specifications of the Heath-Jarrow-Morton(1992) (HJM) models relative to popular historical volatilitymodels in the Eurodollar options market. The implied volatilityfrom the HJM models explains much of the variation of realizedinterest rate volatility over both daily and monthly horizons.The implied volatility dominates the GARCH terms, the Glostenet al. (1993) type asymmetric volatility terms, and the interestrate level. However, it cannot explain that the impact of interestrate shocks on the volatility is lower when interest rates arelow than when they are high.     

The informational content of implied volatility

Implied volatility is widely believed to be informationallysuperior to historical volatility, because it is the 'market's'forecast of future volatility. But for S&P 1 00 index options,the most actively traded contract in the United States, we findimplied volatility to be a poor forecast of subsequent realizedvolatility. In aggregate and across subsamples separated bymaturity and strike price, implied volatility has virtuallyno correlation with future volatility, and it does not incorporatethe information contained in recent observed volatility.     

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.     

A semiparametric factor model for implied volatility surface dynamics

We propose a semiparametric factor model, which approximatesthe implied volatility surface (IVS) in a finite dimensionalfunction space. Unlike standard principal component approachestypically used to reduce complexity, our approach is tailoredto the degenerated design of IVS data. In particular, we onlyfit in the local neighborhood of the design points by exploitingthe expiry effect present in option data. Using DAX index optiondata, we estimate the nonparametric components and a low-dimensionaltime series of latent factors. The modeling approach is completedby studying vector autoregressive models fitted to the latentfactors.     

The exact Taylor formula of the implied volatility

In a model driven by a multidimensional local diffusion, we study the behavior of the implied volatility \({\sigma}\) and its derivatives with respect to log-strike \(k\) and maturity \(T\) near expiry and at the money. We recover explicit limits of the derivatives \({\partial_{T}^{q}} \partial_{k}^{m} \sigma\) for \((T,x-k)\) approaching the origin within the parabolic region \(|x-k|\leq\lambda\sqrt{T}\), with \(x\) denoting the spot log-price of the underlying asset and where \(\lambda\) is a positive and arbitrarily large constant. Such limits yield the exact Taylor formula for the implied volatility within the parabola \(|x-k|\leq\lambda\sqrt{T}\). In order to include important models of interest in mathematical finance, e.g. Heston, CEV, SABR, the analysis is carried out under the weak assumption that the infinitesimal generator of the diffusion is only locally elliptic.     

Volatility linkages of the equity,bond and money markets: an implied volatility approach

This study proposes an alternative approach for examining volatility linkages between Standard & Poor's 500, Eurodollar futures and 30 year Treasury Bond futures markets using implied volatility from the three markets. Simple correlation analysis between implied volatilities in the three markets is used to assess market correlations. Spurious correlation effects are considered and controlled for. I find that correlations between implied volatilities in the equity, money and bond markets are positive, strong and robust. Furthermore, I replicate the approach of Fleming, Kirby and Ostdiek (1998) to check the substitutability of the implied volatility approach and find that the results are nearly identical; I conclude that my approach is simple, robust and preferable in practice. I also argue that the results from this paper provide supportive evidence on the information content of implied volatilities in the equity, bond and money markets.     

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.     

On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be a diffusion or a Markov process, as the examples in Sect. 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus. E. Alòs’ research is supported by grants MEC FEDER MTM 2006 06427 and SEJ2006-13537. J.A. León’s research is partially supported by the CONACyT grant 45684-F. J. Vives’ research is supported by grant MEC FEDER MTM 2006 06427.     

The implied volatility term structure of stock index options

This paper tests the expectations hypothesis of the term structure of implied volatility for several national stock market indexes. The tests indicate that the slope of at-the-money implied volatility over different maturities has predictive ability for future short-dated implied volatility, although not to the extent predicted by the expectations hypothesis. The low forecast power may be due to failure to control for a risk premium in the prices of the options. Evidence is presented that a time-varying risk premium proportional to the level of market volatility is consistent with the results.     

Predictable dynamics in implied volatility surfaces from OTC currency options

Recent empirical studies report predictable dynamics in the volatility surfaces that are implied by observed index option prices, such as those prescribed by general equilibrium models. Using an extensive data set from the over-the-counter options market, we document similar predictability in the factors that capture the daily variation of surfaces implied by options on 25 different foreign exchange rates. We proceed to demonstrate that simple vector autoregressive specifications for the factors can help produce accurate out-of-sample forecasts of the systematic component of the surface at short horizons. Profitable delta-hedged positions can be set up based on these forecasts; however, profits disappear when typical transaction costs are taken into account and when trading rules on wide segments of the surface are sought.     

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.     

The effect of mergers on implied volatility of equity options

This paper uses a sample of 25 large mergers from 1996 to 2004 to study the effect of mergers on the implied volatilities of equity options. The results indicate a statistically significant increase in volatility beyond the amount predicted if the transaction were effectively nothing more than a portfolio combination of the target and acquirer. The disparity suggests that, at least for the first 18 months after the transaction becomes effective, market participants expect mergers to increase risk. Integration risk and uncertainty about the extent to which efficiency gains and greater market power are realized are possible explanations for the discrepancy.     

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.     

Determining the implied volatility for American options using the QAM

This paper describes an efficient numerical procedure which may be used to determine implied volatilities for American options using the quadratic approximation method. Simulation results are presented. The procedure usually converges in five or six iterations with extreme accuracy under a wide variety of option market conditions. A comparison of American implied volatilities with European model implied volatilities indicates that significant differences may arise. This suggests that reliance on European model volatilities estimates may lead to significant pricing errors.     

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.     

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.