9,028 research outputs found

    Continuous-time VIX dynamics: on the role of stochastic volatility of volatility

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    This paper examines the ability of several different continuous-time one- and two-factor jump-diffusion models to capture the dynamics of the VIX volatility index for the period between 1990 and 2010. For the one-factor models we study affine and non-affine specifications, possibly augmented with jumps. Jumps in one-factor models occur frequently, but add surprisingly little to the ability of the models to explain the dynamic of the VIX. We present a stochastic volatility of volatility model that can explain all the time-series characteristics of the VIX studied in this paper. Extensions demonstrate that sudden jumps in the VIX are more likely during tranquil periods and the days when jumps occur coincide with major political or economic events. Using several statistical and operational metrics we find that non-affine one-factor models outperform their affine counterparts and modeling the log of the index is superior to modeling the VIX level directly

    Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model

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    Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present a method to adapt formulas for both the path-integral propagators and the option prices themselves, so that jump processes are taken into account in conjunction with the usual drift and diffusion terms. In particular, we focus on stochastic volatility models, such as the exponential Vasicek model, and extend the pricing formulas and propagator of this model to incorporate jump diffusion with a given jump size distribution. This model is of importance to include non-Gaussian fluctuations beyond the Black-Scholes model, and moreover yields a lognormal distribution of the volatilities, in agreement with results from superstatistical analysis. The results obtained in the present formalism are checked with Monte Carlo simulations.Comment: 9 pages, 2 figures, 1 tabl

    Time-Changed Fast Mean-Reverting Stochastic Volatility Models

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    We introduce a class of randomly time-changed fast mean-reverting stochastic volatility models and, using spectral theory and singular perturbation techniques, we derive an approximation for the prices of European options in this setting. Three examples of random time-changes are provided and the implied volatility surfaces induced by these time-changes are examined as a function of the model parameters. Three key features of our framework are that we are able to incorporate jumps into the price process of the underlying asset, allow for the leverage effect, and accommodate multiple factors of volatility, which operate on different time-scales

    Term Structure of Volatility and Price Jumps in Agricultural Markets - Evidence from Option Data

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    Empirical evidence suggests that agricultural futures price movements have fat-tailed distributions and exhibit sudden and unexpected price jumps. There is also evidence that the volatility of futures prices contains a term structure depending on both calendar-time and time to maturity. This paper extends Bates (1991) jump-diffusion option pricing model by including both seasonal and maturity effects in volatility. An in-sample fit to market option prices on wheat futures shows that our model outperforms previous models considered in the literature. A numerical example illustrates the economic significance of our results for option valuation.Option pricing, Futures, Term structure of volatility, Jump-diffusion, Agricultural markets, Demand and Price Analysis,

    Robust Hedging of Variance Swaps: Discrete Sampling & Co-maturing European Options

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    In the practice of quantitative finance, model risk has raised significant concern and thus model-independent hedging is of particular interest to both academia and industry. In this thesis, we review two methods of constructing robust and model-independent hedging portfolios of variance swaps. One of them assumes a continuum of European options trade but does not require the underlying asset's price path to be continuous. However, the other assumes finite number of options quoted but requires the continuity of underlying asset's price path. We explore numerically the hedging performance as well as upper and lower bounds of several numerical examples by implementing these two methods. Finally, we try to combine these two methods and use an example to show an idea of a possible approach of doing this
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