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Why do financial markets asymmetrically smile? A simple formula in the multi-factor Heston model
A simple approach to determining the Gaussian kernel that constitutes the backbone of the multi-factor Heston model is proposed based on a suitable expansion in powers of volatilities of volatilities. This analysis provides Black-Scholes-like formulas for pricing European vanilla options, allowing for accurate approximations of the option prices under the multi-factor Heston model up to volatilities of volatilities on the order of 50%. The analysis also leads to a simple formula for the implied volatility showing that changes in the convexity of the volatility smile are due only to price skewness, and an easy formula to reproduce volatility indices via the realized volatility. Interestingly, the variance of the Gaussian kernel is equal to the variance of the continuously compounded return in the case of the Heston model. The empirical analyses presented assess the potential of our approach to capture market distortions while adequately forecasting the dynamics of the VIX index
Multi-species mean-field spin-glasses. Rigorous results
We study a multi-species spin glass system where the density of each species
is kept fixed at increasing volumes. The model reduces to the
Sherrington-Kirkpatrick one for the single species case. The existence of the
thermodynamic limit is proved for all densities values under a convexity
condition on the interaction. The thermodynamic properties of the model are
investigated and the annealed, the replica symmetric and the replica symmetry
breaking bounds are proved using Guerra's scheme. The annealed approximation is
proved to be exact under a high temperature condition. We show that the replica
symmetric solution has negative entropy at low temperatures. We study the
properties of a suitably defined replica symmetry breaking solution and we
optimise it within a ziggurat ansatz. The generalized order parameter is
described by a Parisi-like partial differential equation.Comment: 17 pages, to appear in Annales Henri Poincar\`
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