Abstract
In this paper, we develop a multivariate risk-neutral Lévy process model and discuss its applicability in the context of the volatility smile of multiple assets. Our formulation is based upon a linear combination of independent univariate Lévy processes and can easily be calibrated to a set of one-dimensional marginal distributions and a given linear correlation matrix. We derive conditions for our formulation and the associated calibration procedure to be well-defined and provide some examples associated with particular Lévy processes permitting a closed-form characteristic function. Numerical results of the option premiums on three currencies are presented to illustrate the effectiveness of our formulation with different linear correlation structures.
Highlights
Pure-jump Levy processes and their marginal infinite divisibility have attracted considerable attention amongst practitioners and academics for the primal reason that the flexibility of their distributions, such as heavy tails and asymmetry, suits very well to various practical objects in, for example, finance, telecommunications, turbulence, to mention just a few
The Levy copula models of Kallsen and Tankov [13] and Tankov [26] completely characterize the law of a multivariate Levy process, and are applied in pricing basket options, while Luciano and Schoutens [16] and Moosbrucker [19] propose to produce some dependence among components in the variance gamma process framework by setting a common time-changing stochastic process for every component
We propose a multivariate risk-neutral Levy process model, which is based on a linear combination of independent univariate Levy processes, and which can be calibrated to a set of one-dimensional marginal distributions and a given linear correlation matrix
Summary
There is still no market consensus on how to model both volatility smiles and an intended correlation structure. We propose a multivariate risk-neutral Levy process model, which is based on a linear combination of independent univariate Levy processes, and which can be calibrated to a set of one-dimensional marginal distributions and a given linear correlation matrix. Applying Levy processes permitting a parametric form of the characteristic function, we can derive the conditions for both our formulation and the Carr-Madan method [8] to be well defined (Proposition 3.2). Third and perhaps most importantly, compared to existing models, our formulation provides a considerably easier way to simulate the resultant multivariate Levy process, as a linear combination of independent univariate Levy processes.
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