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Shapes of implied volatility with positive mass at zero. arXiv:1310.1020 [qfin.PR
, 2013
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Fast NinomiyaVictoir calibration of the doublemeanreverting model. 2013. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2210420
 24 T. Bollerslev and
, 2011
"... We consider the three factor double mean reverting (DMR) model of Gatheral (2008), a model which can be successfully calibrated to both VIX options and SPX options simultaneously. One drawback of this model is that calibration may be slow because no closed form solution for European options exists. ..."
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We consider the three factor double mean reverting (DMR) model of Gatheral (2008), a model which can be successfully calibrated to both VIX options and SPX options simultaneously. One drawback of this model is that calibration may be slow because no closed form solution for European options exists. In this paper, we apply modified versions of the second order Monte Carlo scheme of Ninomiya and Victoir (2008) and compare these to the EulerMaruyama scheme with full truncation of Lord et al. (2010), demonstrating on the one hand that fast calibration of the DMR model is practical, and on the other that suitably modified NinomiyaVictoir schemes are applicable to the simulation of much more complicated timehomogeneous models than may have been thought previously. 1
Local Variance Gamma and Explicit Calibration to Option Prices
"... In some options markets (e.g. commodities), options are listed with only a single maturity for each underlying. In others, (e.g. equities, currencies), options are listed with multiple maturities. In this paper, we analyze a special class of pure jump Markov martingale models and provide an algorith ..."
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In some options markets (e.g. commodities), options are listed with only a single maturity for each underlying. In others, (e.g. equities, currencies), options are listed with multiple maturities. In this paper, we analyze a special class of pure jump Markov martingale models and provide an algorithm for calibrating such model to match the market prices of European options of multiple strikes and maturities. This algorithm matches option prices exactly and only requires solving several onedimensional rootsearch problems and applying elementary functions. We show how to construct a timehomogeneous process which meets a single smile, and a piecewise timehomogeneous process which can meet multiple smiles.
Mathieu Rosenbaum
, 2014
"... Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that logvolatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. This leads us to ad ..."
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Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that logvolatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault [16]. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H < 1/2. We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatility has been widely accepted as a stylized fact. Finally, we provide a quantitative market microstructurebased foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.
Bspline techniques for volatility modeling
, 2013
"... This paper is devoted to the application of Bsplines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitragefree implied volatility surface calibrated to sparse option data. We use an extension of ..."
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This paper is devoted to the application of Bsplines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitragefree implied volatility surface calibrated to sparse option data. We use an extension of classical Bsplines obtained by including basis functions with infinite support. We first come back to the application of shapeconstrained Bsplines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. An application is the Monte Carlo calibration of stochastic local volatility models by Markov projection. Then we present a new technique for the calibration of an implied volatility surface to sparse option data. We use a Bspline parameterization of the RadonNikodym derivative of the underlying’s riskneutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitragefree implied volatility surfaces. Finally, we propose a Galerkin method with Bspline finite elements to the solution of the differential equation satisfied by the Radon Nikodym derivative.