We introduce a multivariate diffusion model that is able to price derivative securities featuring multiple underlying assets. Each underlying shows a volatility smile and is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows to reconcile single name and index/basket volatility smiles in a consistent framework. Rather than simply correlating one-dimensional local volatility models for each asset, our approach could be dubbed a multidimensional local volatility approach with state dependent diffusion matrix. The model is quite tractable, leading to a complete market and not requiring Fourier techniques  , contrary to multivariate stochastic volatility models such as Wishart. We provide a semi-analytic formula for the price of European optionson a basket/index of securities . A comparison with the standard approach consisting in using Monte Carlo simulation that samples simply-correlated suitably discretized one-dimensional  paths is made . Our results show that our approach is promising in terms of basket option pricing . We also introduce a multivariate uncertain volatility model of which our multivariate local volatilities model is a multivariate markovian projectionand analyze the dependence structure induced by our multivariate dynamics in detail . A few numerical examples on simple contracts conclude the paper.
We introduce a multivariate diffusion model that is able to price derivative securities featuring multiple underlying assets. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows to reconcile single name and index/basket volatility smiles in a consistent framework. Our approach could be dubbed a multidimensional local volatility approach with vector-state dependent diffusion matrix. The model is quite tractable, leading to a complete market and not requiring Fourier techniques for calibration and dependence measures , contrary to multivariate stochastic volatility models such as Wishart. We prove existence and uniqueness of solutions for the model stochastic differential equations, provide formulas for a number of basket options, and analyze the dependence structure of the model in detail by deriving a number of results on covariances, its copula function and rank correlation measures and volatilities-assets correlations . A comparison with sampling simply-correlated suitably discretized one-dimensional mixture dynamical paths is made , both in terms of option pricing and of dependence, and first order expansion relationships between the two models' local covariances are derived . We also show existence of a multivariate uncertain volatility model of which our multivariate local volatilities model is a Markovian projection, highlighting that the projected model is smoother and avoids a number of drawbacks of the uncertain volatility version. We also show a consistency result where the Markovian projection of a geometric basket in the multivariate model is a univariate mixture dynamics model . A few numerical examples on basket and spread options pricing conclude the paper.
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underlying shows a volatility smile and
asset volatility smile
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Rather than simply correlating one-dimensional local volatility models for each asset, our
Our
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state
vector-state
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for calibration and dependence measures
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provide a semi-analytic formula for the price of European optionson a basket/index of securities
prove existence and uniqueness of solutions for the model stochastic differential equations, provide formulas for a number of basket options, and analyze the dependence structure of the model in detail by deriving a number of results on covariances, its copula function and rank correlation measures and volatilities-assets correlations
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the standard approach consisting in using Monte Carlo simulation that samples
sampling
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mixture dynamical
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. Our results show that our approach is promising
, both
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basket option pricing
option pricing and of dependence, and first order expansion relationships between the two models' local covariances are derived
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introduce
show existence of
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multivariate markovian projectionand analyze the dependence structure induced by our multivariate dynamics in detail
Markovian projection, highlighting that the projected model is smoother and avoids a number of drawbacks of the uncertain volatility version. We also show a consistency result where the Markovian projection of a geometric basket in the multivariate model is a univariate mixture dynamics model
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simple contracts
basket and spread options pricing