The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles

TL;DR

This paper introduces a multivariate diffusion model that accurately prices multi-asset derivatives and aligns single-asset and index volatility smiles within a consistent, tractable framework without requiring complex Fourier techniques.

Contribution

The paper presents a novel multivariate mixture dynamics model that ensures arbitrage-free pricing and consistent volatility smiles for single assets and baskets, with explicit formulas and dependence analysis.

Findings

Model is arbitrage-free and consistent with market smiles.

Provides explicit formulas for basket options.

Demonstrates improved dependence modeling and tractability.

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.