The Multivariate Mixture Dynamics Model: Shifted Dynamics and Correlation Skew

24 Pages Posted: 16 Dec 2015 Last revised: 9 Nov 2018

See all articles by Damiano Brigo

Damiano Brigo

Imperial College London - Department of Mathematics

Camilla Pisani

Aarhus University - Department of Business and Economics

Francesco Rapisarda

Bloomberg L.P.

Date Written: October 31, 2018

Abstract

The Multi Variate Mixture Dynamics model is a tractable, dynamical, arbitrage-free multivariate model characterized by transparency on the dependence structure, since closed form formulae for terminal correlations, average correlations and copula function are available. It also allows for complete decorrelation between assets and instantaneous variances.

Each single asset is modeled according to a lognormal mixture dynamics model, and this univariate version is widely used in the industry due to its flexibility and accuracy. The same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows for consistency of single asset and index/portfolio smile.

In this paper, we generalize the MVMD model by introducing shifted dynamics and we propose a definition of implied correlation under this model. We investigate whether the model is able to consistently reproduce the implied volatility of FX cross rates once the single components are calibrated to univariate shifted lognormal mixture dynamics models. We consider in particular the case of the Chinese renminbi FX rate, showing that the shifted MVMD model correctly recovers the CNY/EUR smile given the EUR/USD smile and the USD/CNY smile, thus highlighting that the model can also work as an arbitrage free volatility smile extrapolation tool for cross currencies that may not be liquid or fully observable.

We compare the performance of the shifted MVMD model in terms of implied correlation with those of the shifted Simply Correlated Mixture Dynamics model where the dynamics of the single assets are connected naively by introducing correlation among their Brownian motions.

Finally, we introduce a model with uncertain volatilities and correlation. The Markovian projection of this model is a generalization of the shifted MVMD model.

Keywords: MVMD model, Mixture of densities, Multivariate local volatility, Correlation Skew, Random Correlation, Calibration, Cross exchange rates, FX smile, Index volatility smile, renminbi-USD smile, renminbi-EUR smile, CNY-USD smile, CNY-EUR smile, SCMD model

JEL Classification: G13

Suggested Citation

Brigo, Damiano and Pisani, Camilla and Rapisarda, Francesco, The Multivariate Mixture Dynamics Model: Shifted Dynamics and Correlation Skew (October 31, 2018). Available at SSRN: https://ssrn.com/abstract=2703722 or http://dx.doi.org/10.2139/ssrn.2703722

Damiano Brigo

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

Camilla Pisani (Contact Author)

Aarhus University - Department of Business and Economics ( email )

Fuglesangs Alle 4
Aarhus, 8210
Denmark

Francesco Rapisarda

Bloomberg L.P. ( email )

39 Finsbury Square
London, EC2A 1HD
United Kingdom

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