Dependence Structure for Levy Processes and Its Application in Finance

Abstract

In this paper, we introduce DSPMD, discretely sampled process with pre-specified marginals and pre-specified dependence, and SRLMD, series representation for Levy process with pre-specified marginals and pre-specified dependence.

In the DSPMD for Levy processes, some regular copula can be extracted from the discrete samples of a joint process so as to correlate discrete samples on the pre-specified marginal processes. We prove that if the pre-specified marginals and pre-specified joint processes are some Levy processes, the DSPMD converges to some Levy process. Compared with Levy copula, proposed by Tankov, DSPMD offers easy access to statistical properties of the dependence structure through the copula on the random variable level, which is difficult in Levy copula. It also comes with a simulation algorithm that overcomes the first component bias effect of the series representation algorithm proposed by Tankov. As an application and example of DSPMD for Levy process, we examined the statistical explanatory power of VG copula implied by the multidimensional VG processes. Several baskets of equities and indices are considered. Some basket options are priced using risk neutral marginals and statistical dependence.

SRLMD is based on Rosinski's series representation and Sklar's Theorem for Levy copula. Starting with a series representation of a multi-dimensional Levy process, we transform each term in the series component-wise to new jumps satisfying pre-specified jump measure. The resulting series is the SRLMD, which is an exact Levy process, not an approximation. We give an example of alpha-stable Levy copula which has the advantage over what Tankov proposed in the follow aspects: First, it is naturally high dimensional. Second, the structure is so general that it allows from complete dependence to complete independence and can have any regular copula behavior built in. Thirdly, and most importantly, in simulation, the truncation error can be well controlled and simulation efficiency does not deteriorate in nearly independence case. For compound Poisson processes as pre-specified marginals, zero truncation error can be attained.