The Dynamics of Implied Volatilities: A Common Principal Components Approach

It is common practice to identify the number and sources of shocks that move, e.g., ATM implied volatilities by principal components analysis. This approach, however, is likely to result in a loss of information, since the surface structure of implied volatilities is neglected. In this paper we analyze the implied volatility surface along maturity slices with a common principal components analysis (CPC), known from morphometrics. In CPC analysis, the space spanned by the eigenvectors is identical across groups, whereas variances associated with the common principal components vary. Our analysis shows that implied volatility surface dynamics can be traced back to a common eigenstructure in maturity slices. This empirical result is used to set up a factor model for implied volatility surface dynamics. This revised version was published online in June 2006 with corrections to the Cover Date.     

A closed-form GARCH option valuation model

This paper develops a closed-form option valuation formula fora spot asset whose variance follows a GARCH(p, q) process thatcan be correlated with the returns of the spot asset. It providesthe first readily computed option formula for a random volatilitymodel that can be estimated and implemented solely on the basisof observables. The single lag version of this model containsHeston's (1993) stochastic volatility model as a continuous-timelimit. Empirical analysis on S&P500 index options showsthat the out-of-sample valuation errors from the single lagversion of the GARCH model are substantially lower than thead hoc Black-Scholes model of Dumas, Fleming and Whaley (1998)that uses a separate implied volatility for each option to fitto the smirk/smile in implied volatilities. The GARCH modelremains superior even though the parameters of the GARCH modelare held constant and volatility is filtered from the historyof asset prices while the ad hoc Black-Scholes model is updatedevery period. The improvement is largely due to the abilityof the GARCH model to simultaneously capture the correlationof volatility, with spot returns and the path dependence involatility.     

Does implied volatility provide any information beyond that captured in model-based volatility forecasts?

This paper contributes to our understanding of the informational content of implied volatility. Here we examine whether the S&P 500 implied volatility index (VIX) contains any information relevant to future volatility beyond that available from model based volatility forecasts. It is argued that this approach differs from the traditional forecast encompassing approach used in earlier studies. The findings indicate that the VIX index does not contain any such additional information relevant for forecasting volatility.     

“Life Expectancy in the Future: A Summary of a Discussion among Experts”, Robert B. Friedland,October 1998

Abstract

In this paper we present an econometric model of implied volatilities of S&;P500 index options. First, we model the dynamics the CBOE VIX index as a proxy for the general level of implied volatilities. We then describe a parametric model of the implied volatility surface for options with a term of up to two years. We show that almost all of the variation in the implied volatility surface can be explained by the VIX index and one or two other uncorrelated factors. Finally, we present a model of the dynamics of these factors.     

Asymptotic Expansion Formula of Option Price Under Multifactor Heston Model

The stochastic volatility model of Heston (Rev Financ Stud 6(2):327–343, 1993) has found difficulty in describing some of the important features of implied volatility dynamics, leading to a quest for multifactor extensions as well as the incorporation of time-dependent model parameters. In this paper, an asymptotic expansion approach to the multifactor Heston model with time-dependent parameters is developed. The results of Benhamou et al. (SIAM J Financ Math 1(1):289–325, 2010) are extended and it is shown that the extension to the multifactor model involves an extra expansion term that captures the interaction between variance factors. The expansion formula under constant parameters can be explicitly computed and the incorporation of time-dependent parameters is straightforward under the framework. As illustration, a two-factor model is calibrated to data of index options and variance swaps and it is found that it is possible to distinguish a short-term and long-term variance factor from the implied volatility surface and variance swap rates. Moreover, the two-factor model is able to reproduce the shapes of the implied volatility surface during various market scenarios.     

Asymmetric information about volatility: How does it affect implied volatility,option prices and market liquidity?

This paper develops a model of asymmetric information in which an investor has information regarding the future volatility of the price process of an asset and trades an option on the asset. The model relates the level and curvature of the smile in implied volatilities as well as mispricing by the Black-Scholes model to net options order flows (to the market maker). It is found that an increase in net options order flows (to the market maker) increases the level of implied volatilities and results in greater mispricing by the Black-Scholes model, besides impacting the curvature of the smile. The liquidity of the option market is found to be decreasing in the amount of uncertainty about future volatility that is consistent with existing evidence. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asset pricing and energy consumption risk

This paper proposes energy consumption in the US as a new measure for the consumption capital asset pricing model. We find that (i) industrial energy growth produces reasonable values for the relative risk aversion coefficient and the implied risk-free rate; (ii) compared to alternative consumption measures, industrial energy performs well in explaining the cross-sectional variation in stock returns with the lowest implied risk aversion and pricing errors; (iii) the industrial energy consumption risk model performs equally well as the Fama–French three-factor model in the cross-sectional asset pricing tests; and (iv) total energy consumption risk is priced in the presence of the Fama–French factor risks.     

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.     

Mean-reversion properties of implied volatilities

In this paper, we present a new stylized fact for options whose underlying asset is a stock index. Extracting implied volatility time series from call and put options on the Deutscher Aktien index (DAX) and financial times stock exchange index (FTSE), we show that the persistence of these volatilities depends on the moneyness of the options used for its computation. Using a functional autoregressive model, we show that this effect is statistically significant. Surprisingly, we show that the diffusion-based stochastic volatility models are not consistent with this stylized fact. Finally, we argue that adding jumps to a diffusion-based volatility model help recovering this volatility pattern. This suggests that the persistence of implied volatilities can be related to the tails of the underlying volatility process: this corroborates the intuition that the liquidity of the options across moneynesses introduces an additional risk factor to the one usually considered.     

Risk aversion and the yield of corporate debt

This paper develops a model to estimate the implied default probability of corporate bonds. The model explicitly considers the risk averse behavior of investors to provide a more precise framework for estimating the implied default probability. A Kalman filter method is used to estimate time-varying risk premium associated with the investor's risk aversion. The results of nonlinear regressions indicate that previous risk-neutrality models consistently overestimate the implied default rates of corporate bonds. The results also suggest that investors may have been adequately compensated for investment in risky bonds.     

Options markets, self-fulfilling prophecies, and implied volatilities

This paper answers the following often asked question in option pricing theory: if the underlying asset's price does not satisfy a lognormal distribution, can market prices satisfy the Black-Scholes formula just because market participants believe it should? In complete markets, if the underlying asset's objective distribution is not lognormal, then the answer is no. But, in an incomplete market, if the underlying asset's objective distribution is not lognormal and all traders believe it is, then the answer is yes! The Black-Scholes formula can be a self-fulfilling prophecy. The proof of this second assertion consists of generating an economy where self-confirming beliefs sustain the Black-Scholes formula as an equilibrium. An asymmetric information model is provided, where the underlying asset's price has stochastic volatility and drift. This model is distinct from the existing pricing models in the literature, and it provides new empirical implications concerning Black-Scholes implied volatilities and the bid/ask spread. Similar to stochastic volatility models, this model is consistent with the implied volatility “smile” pattern in strike prices. In addition, it is consistent with implied volatilities being biased predictors of future volatilities.     

Active Portfolio Management,Implied Expected Returns,and Analyst Optimism

This paper investigates whether implied expected returns based on the approach of CLAUS/THOMAS (2001) can be implemented in active portfolio management. This approach uses analysts' forecasts to derive return expectations by equating the present value of expected cash-flows to the current market price. It is found that active investment strategies which maximize implied expected returns significantly outperform a passive index investment. A significant part of this outperformance can be explained by the difference between the implied expected return and the return expectation justified by the CAPM. The empirical results suggest that a substantial part of this difference can be attributed to an optimism bias in analysts' forecasts.     

Model-driven statistical arbitrage on LETF option markets

In this paper, we study the statistical properties of the moneyness scaling transformation, which adjusts the moneyness coordinate of the implied volatility smile in an attempt to remove the discrepancy between the IV smiles for levered and unlevered ETF options. We construct bootstrap uniform confidence bands which indicate that the implied volatility smiles are statistically different after moneyness scaling has been performed. An empirical application shows that there are trading opportunities possible on the LETF market. A statistical arbitrage type strategy based on a dynamic semiparametric factor model is presented. This strategy presents a statistical decision algorithm which generates trade recommendations based on comparison of model and observed LETF implied volatility surface. It is shown to generate positive returns with a high probability. Extensive econometric analysis of the LETF implied volatility process is performed including out-of-sample forecasting based on a semiparametric factor model and a uniform confidence bands' study. These provide new insights into the latent dynamics of the implied volatility surface. We also incorporate Heston stochastic volatility into the moneyness scaling method for better tractability of the model.     

The information content of implied volatility, skewness and kurtosis: empirical evidence from long-term CAC 40 options

Implied standard deviation is widely believed to be the best available forecast of the volatility of returns over the remaining contract life (Jorion, 1995 ). In this paper, we take this result two steps further to the higher moments of the distribution (skewness and kurtosis) based on a Gram–Charlier series expansion of the normal distribution (Corrado and Su, 1996 ) using long-term CAC 40 option prices contract, named PXL. First, we found that implied first moments contain a substantial amount of information for future moments of CAC 40 returns although this amount decreases with respect to the moment's order. Secondly, we found that the different shapes of the volatility smile are consistent with different distribution of the underlying returns. Based on these results, we also observed that including other implied moments significantly improves the out-of-sample pricing performance of the Black–Scholes, (1973) model.     

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.     

Impact of US and UK macroeconomic news announcements on the return distribution implied by FTSE-100 index options

This study investigates the effect of scheduled US and UK macroeconomic news announcements on the return distribution implied by FTSE-100 option prices. The results provide new evidence for the whole implied return distribution being systematically affected by certain macroeconomic news announcements. After controlling the unexpected content of the news announcement for quality (good vs. bad news) it is found that good (bad) news causes implied volatility to decrease (increase), option-implied return distribution becomes less (more) left-skewed and kurtosis increases (decreases). The results are consistent with the behavioral model created by Barberis et al. [Barberis, N., Shleifer, A., and Vishny, R. (1998). A model of investor sentiment. Journal of Financial Economics, 49, 307-343.], in which good (bad) news is expected to be followed by good (bad) news.     

Testing the martingale restriction for option implied densities

This paper contributes to the forecasting literature by presenting a new evaluation method for density and probability estimates. This procedure is particularly well suited for analyzing time series of forecasts implied from option prices, although the results are very general and can be applied also outside this framework. A small scale simulation study documents that valid and accurate inference can be drawn for option implied densities with the proposed method. The new testing procedure is demonstrated in an empirical application on density estimates implied from $/£ currency options.     

Common asset pricing factors in volatilities and returns in futures markets

Factor-based asset pricing models have been used to explain the common predictable variation in excess asset returns. This paper combines means with volatilities of returns in several futures markets to explain their common predictable variation. Using a latent variables methodology, tests do not reject a single factor model with a common time-varying factor loading. The single common factor accounts for up to 53% of the predictable variation in the volatilities and up to 14% of the predictable variation in the means. S&P500 futures volatility predicted by the factor model is highly correlated with volatility implied in S&P500 futures options. But both the factor and implied volatilities are significant in predicting future volatility. In derivatives pricing, both implied volatility from options and factors extracted from asset pricing models should be employed.     

Forecasting foreign exchange volatility: Why is implied volatility biased and inefficient? And does it matter?

Research has consistently found that implied volatility is a conditionally biased predictor of realized volatility across asset markets. This paper evaluates explanations for this bias in the market for options on foreign exchange futures. Several recently proposed solutions – including a model of priced volatility risk – fail to explain a significant portion of the conditional bias found in implied volatility. Further, while implied volatility fails to subsume econometric forecasts in encompassing regressions, these forecasts do not significantly improve delta-hedging performance. Thus this paper argues that statistical metrics are inappropriate measures of the information content of implied volatility. Implied volatility appears much more useful when measured by a more relevant, economic metric.     

General approximation schemes for option prices in stochastic volatility models

In this paper we develop a general method for deriving closed-form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using purely probabilistic methods. A flexible way of approximating the equivalent implied volatility from the basic price expansion is also introduced. As an application of our method we derive closed-form approximations for call prices and implied volatilities in the Heston [Rev. Financial Stud., 1993, 6, 327–343] model. The accuracy of these approximations is studied and compared with numerically obtained values.