The Arbitrage-Free Multivariate Mixture Dynamics Model: Consistent Single-Assets and Index Volatility Smiles

47 Pages Posted: 1 Mar 2013 Last revised: 24 Sep 2014

See all articles by Damiano Brigo

Damiano Brigo

Imperial College London - Department of Mathematics

Francesco Rapisarda

Bloomberg L.P.

Abir Sridi

Université Paris I Panthéon-Sorbonne

Date Written: September 24, 2014

Abstract

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.

Keywords: Mixture of densities, Volatility smile, Lognormal density, Multivariate local volatility, Complete Market, Option on a weighted Arithmetic average of a basket, Spread option, Option on a weighted geometric average of a basket, Markovian projection, Copula function

JEL Classification: G13

Suggested Citation

Brigo, Damiano and Rapisarda, Francesco and Sridi, Abir, The Arbitrage-Free Multivariate Mixture Dynamics Model: Consistent Single-Assets and Index Volatility Smiles (September 24, 2014). Available at SSRN: https://ssrn.com/abstract=2226053 or http://dx.doi.org/10.2139/ssrn.2226053

Damiano Brigo (Contact Author)

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://www.imperial.ac.uk/people/damiano.brigo

Francesco Rapisarda

Bloomberg L.P. ( email )

39 Finsbury Square
London, EC2A 1HD
United Kingdom

Abir Sridi

Université Paris I Panthéon-Sorbonne ( email )

17, rue de la Sorbonne
Paris, IL 75005
France

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