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This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a time-varying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for short-dated options.

Bayesian methods have been efficient in estimating parameters of stochastic volatility models for analyzing financial time series. Recent advances made it possible to fit stochastic volatility models of increasing complexity, including covariates, leverage effects, jump components and heavy-tailed distributions. However, a formal model comparison via Bayes factors remains difficult. The main objective of this paper is to demonstrate that model selection is more easily performed using the deviance information criterion (DIC). It combines a Bayesian measure-of-fit with a measure of model complexity. We illustrate the performance of DIC in discriminating between various different stochastic volatility models using simulated data and daily returns data on the S&P100 index.

In this paper a stochastic volatility model is presented that directly prescribes the stochastic development of the implied Black-Scholes volatilities of a set of given standard options. Thus the model is able to capture the stochastic movements of a full term structure of implied volatilities. The conditions are derived that have to be satisfied to ensure absence of arbitrage in the model and its numerical implementation is discussed.

i Abstract: The volatility smile changed drastically around the crash of 1987 and new option pricing models have been proposed in order to accommodate that change. Deterministic volatility models allow for more °exible volatility surfaces but refrain from introducing additional risk-factors. Thus, options are still redundant securities. Alternatively, stochastic models introduce additional risk-factors and options are then needed for spanning of the pricing kernel. We develop a statistical test based on this di®erence in spanning. Using daily S&P500 index options data from 1986-1995, our tests suggest that both in- and out-of-the-money options are needed for spanning. The ¯ndings are inconsistent with deterministic volatility models but are consistent with stochastic models which incorporate additional priced risk-factors such as stochastic volatility, interest rates, or jumps. What is a good model to price equity derivatives and to manage risk? Starting from Black and Scholes (1973), a common approach in the derivative pricing literature has been to model the underlying asset as a geometric Brownian motion with constant volatility. Early tests of options on stocks such as Rubinstein (1985) more or less supported the empirical implications of a geometric

This paper provides a simple unified framework for assessing the empirical linkages between returns and realized and implied volatilities. First, we show that whereas the volatility feedback effect as measured by the sign of the correlation between contemporaneous return and realized volatility depends importantly on the underlying structural model parameters, the correlation between return and implied volatility is unambiguously positive for all reasonable parameter configurations. Second, the lagged return-volatility asymmetry, or the leverage effect, is always stronger for implied than realized volatility. Third, implied volatilities generally provide downward biased forecasts of subsequent realized volatilities. Our results help explain previous findings reported in the extant empirical literature, and is further corroborated by new estimation results for a sample of monthly returns and implied and realized volatilities for the aggregate S&P market index. JEL Classification: G12, C51, C22.

We compare the predictive ability of Stochastic Volatility (SV) models to that of volatility forecasts implied by option prices. An SV model is proposed with implied volatility as an explanatory variable in the variance equation which allows the use of statistical testing; we refer to this model as the SVX model. Next we obtain a Stochastic Implied Volatility (SIV) model by restricting the volatility persistence parameter in the SVX model to equal zero. All SV models are estimated by exact maximum likelihood using Monte Carlo importance sampling methods. The performance of the models is evaluated both within-sample and out-of-sample for daily returns on the Standard & Poor’s 100 index. Our in-sample results confirm the information content of implied volatility measures as the SVX and SIV models produce more effective estimates of the underlying volatility process than the standard SV model based solely on historical returns. The out-of-sample volatility forecasts are evaluated against daily squared returns and intraday volatility measures for forecasting horizons ranging from 1 to 20 days. For both the squared daily returns and the cumulative intraday squared 10-minute returns we find that the SIV model outperforms both the SV and the SVX model on several evaluation criteria but that the SV model produces volatility forecasts with the smallest bias. All models underestimate the volatility process on average which in our opinion is closely related to the fact that the average level of volatility in the estimation samples is lower than in the evaluation sample.

Abstract: In 2003, the Chicago Board Options Exchange (CBOE) made two key enhancements to the volatility index (VIX) methodology based on S&P options. The new VIX methodology seems to be based on a complicated formula to calculate expected volatility. In this paper, with the use of Thailand’s SET50 Index Options data, we modify the VIX formula to a very simple relationship, which has a higher negative correlation between the VIX for Thailand (TVIX) and SET50 Index Options. We show that TVIX provides more accurate forecasts of option prices than the simple expected volatility (SEV) index, but the SEV index outperforms TVIX in forecasting expected volatility. Therefore, the SEV index would seem to be a superior tool as a hedging diversification tool because of the high negative correlation with the volatility index.

Abstract: The authors study time-variation in the co-movements between daily stock and Treasury bond returns over 1986 to 2000. Their innovation is to examine whether variation in stock-bond return dynamics can be linked to non-return-based measures of stock market uncertainty, specifically the implied volatility (IV) from equity index options and detrended stock turnover (DTVR). The authors investigate two empirical questions suggested by recent literature on stock market uncertainty and cross-market hedging. First, from a forward-looking perspective, they find that the levels of IV and DTVR are both negatively associated with the future correlation between stock and bond returns. The probability of a negative correlation between daily stock and bond returns over the next month is several times greater following relatively high values of IV and DTVR. Second, from a contemporaneous perspective, the authors find that bond returns tend to be relatively high (low) during days when IV increases (decreases) and during days when stock turnover is unexpectedly high (low). Their findings suggest that stock market uncertainty has cross-market pricing influences that play an important role in understanding joint stock-bond price formation. Further, our results imply that stock-bond diversification benefits increase with stock market uncertainty.

We show how contracts paying the realized covariance between two currencies can be constructed by combining static positions in a continuum of options with continuous trading in underlying futures or forward contracts. The construction is general in that the volatilities and correlations are arbitrary. We thank Mark Broadie, Tony Corso, Jose Lopez, and Nedia Miller for comments. Any errors are our own. I Introduction Volatility swaps have recently emerged on several over-the-counter markets (see [5] for example). These contracts pay the difference between the realized volatility over a specified time interval and a constant 1 agreed upon at the outset of the contract. The motivation for contracts whose payoffs are tied to volatility has been discussed by several authors. For example, Gastineau[13] and Galai[11] propose the development of option indices which can be used as the underlying for derivative contracts. Brenner and Galai[2] propose the development of realized volatility in...