a b s t r a c tMuch 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.

This paper describes a maximum likelihood method for estimating the parameters of Heston's model of stochastic volatility using data on an underlying market index and the prices of options written on that index. Parameters of the physical measure (associated with the index) and the parameters of the risk-neutral measure (associated with the options) are identified including the equity and volatility risk premia. The estimation is implemented using a particle filter. The computational load of this estimation method, which previously has been prohibitive, is managed by the effective use of parallel computing using Graphical Processing Units. A byproduct of this focus on easing the computational burden is the development of a simplification of the closedform approximation used to price European options in Heston's model. The efficacy of the filter is demonstrated under simulation and an empirical investigation of the fit of the model to the S&P 500 Index is undertaken. All the parameters of the model are reliably estimated and, in contrast to previous work, the volatility premium is well estimated and found to be significant.

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the halfrange cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

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