Marginal consistent dependence modelling using weak subordination for Brownian motions

Abstract

We present an approach for modelling dependencies in exponential Lévy market models with arbitrary margins originated from time changed Brownian motions. Using weak subordination of Buchmann et al. [Bernoulli, 2017], we face a new layer of dependencies, superior to traditional approaches based on pathwise subordination, since weakly subordinated processes are not required to have independent components considering multivariate stochastic time changes. We apply a subordinator being able to incorporate any joint or idiosyncratic information arrivals. We emphasize multivariate variance gamma and normal inverse Gaussian processes and state explicit formulae for the Lévy characteristics. Using maximum likelihood, we estimate multivariate variance gamma models on various market data and show that these models are highly preferable to traditional approaches. Consistent values of basket-options under given marginal pricing models are achieved using the Esscher transform, generating a non-flat implied correlation surface.

Keywords:

• Lévy processes
• Dependence modelling
• Weak multivariate subordination
• Maximum likelihood estimation
• Variance gamma
• Normal inverse Gaussian

AMS Subject Classifications:

• C51
• G12

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Let and as in definition 3.1 of Luciano et al. (2016) and , where is the identity matrix. To define a 2d-dimensional subordinator analogous to (6(6) ), let . Then, for some , , where a factor-based subordinated Brownian motion of Luciano et al. (2016).

2 Note that for the algorithm makes great demands on the working memory, which usually cannot be achieved using a personal computer.

3 Note that the Weak Reassembled VG model converges to the Multivariate VG model for and .

4 In case the upperbound is attained, such that , the Factor-Based VG model preferable.

5 In fact, 36 is the smallest square number greater than stated as reasonable number of cells.

6 We use the definition of the implied correlation stated by Da Fonseca et al. (2007): For marginal volatilities fixed to the implied levels, the implied correlation is given by inverting the Black-Scholes pricing formula for best-of call options derived by Stulz (1982).

Additional information

Funding

This work was supported by Australian Research Council [grant number DP160104737]; Deutsche Forschungsgemeinschaft [grant number 282079427].