Cost-efficiency in multivariate Lévy models

Abstract

In this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence and uniqueness results for the Esscher parameter and determine cost efficient strategies in particular in the case of price processes driven by multivariate NIG- and VG-processes. From a monotonicity characterization of efficient payoffs we obtain that basket options are generally inefficient in Lévy markets when pricing is based on the Esscher measure.We determine efficient versions of the basket options in real market data and show that the proposed cost efficient strategies are also feasible from a numerical viewpoint. As a result we find that a considerable efficiency loss may arise when using the inefficient payoffs.

Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing

“…The parameter h is called the Esscher parameter. For general exponential Lévy market models, Theorems 4.4-4.5 in [33] ( also see [51], their Theorem 2.6) state that h is unique, provided the driving Lévy process does not degenerate under P in the sense of Definition 24.16 of [52]. An application of this result yields that our market model (3.1) admits a unique h , provided rank(A) ≥ k, rank(M ) ≥ d and det Σ > 0.…”

“…Esche and Schweizer [18] show that the minimal entropy martingale measure for a multivariate Lévy process preserves the Lévy property because it corresponds to an Esscher transform. Rüschendorf and Wolf [51] provide explicit necessary conditions for the existence of Esscher parameters for multivariate Lévy processes and further show that the multivariate Esscher parameter is unique if it exists. Tankov [57] provides an introduction to the pricing theory in the context of exponential Lévy processes.…”

Optimal portfolio choice with benchmarks

“…This observation allows us to conclude that we can use the algorithm to deal with portfolio optimization problems for which explicit solutions are not available. Numerical examples were implemented in a Black-Scholes setting but more general market models can also be considered, such as Lévy markets with agents using Esscher pricing to value payoffs (Von Hammerstein et al (2014), Rüschendorf & Wolf (2015)).…”