A Multivariate Jump-Driven Financial Asset Model

Abstract

We discuss a Lévy multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility. We use it to describe the behavior of a series of stocks or indexes and to study a multi-firm, value-based default model.Starting from an independent Brownian world, we introduce jumps and other deviations from normality, including non-Gaussian dependence. We use a stochastic time-change technique and provide the details for a Gamma change.The main feature of the model is the fact that -opposite to other, non jointly Gaussian settings -its risk neutral dependence can be calibrated from univariate derivative prices, providing a surprisingly good fit. JEL classification numbers: G12, G10.

Valuation of Collateralized Funds of Hedge Fund Obligations: A Basket Option Pricing Approach

“…Modelling dependence in this way allows to introduce two sources of co-movement among the NAV of different hedge funds. First, the use of a common stochastic clock introduces a new business time, in which all the market operates, it means all prices jump simultaneously [2,3,6,9]. Secondly, jump sizes are correlated [2,5,8].…”

“…To evaluate dynamically the collateral portfolio NAV it is necessary to model the joint risk neutral evolution of the underlying hedge funds and at the same time any CFO's structural features like coupon payments, over collateralization test, liquidity profile, equity distribution rules, management fees and so on have to be taken into account. In [9] the physical dependence among hedge fund log-returns is introduced through a Gamma stochastic time change of a Multivariate Brownian motion with drift, with independent components [2,6]. From a methodological stand point, the main limitation of that model lies in a low degree of flexibility: in particular, it is not able to replicate the correlations observed in the market.…”

A multivariate Lévy process model with linear correlation

“…This simplicity in implementation should be a great advantage, compared to existing multivariate Lévy process models. For example, the stochastic volatility formulation of [16] is based on a time-changed Brownian motion, and usually requires an Euler-type discretization of sample paths for simulation, while the Lévy copula model [26] resorts to the series representation of Lévy processes, which is well known to be computationally very expensive for most simulation use.…”

“…In the literature, there has recently been an increasing interest in the multivariate Lévy process modeling. For example, the Lévy copula models of Kallsen and Tankov [13] and Tankov [26] completely characterize the law of a multivariate Lévy 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 (fully or partially) common time-changing stochastic process for every component. A general construction of multi-factor Lévy models from Lévy models on rays is studied in Boyarchenko and Levendorskiǐ [4].…”

Computation of Copulas by Fourier Methods

“…Multivariate stochastic processes spring immediately to mind, for example, Lévy or affine processes (cf. e.g., [3,4,17] or [15]), while in mathematical finance models using time changes or linear mixture models have been developed; see, e.g., [5,10,12,13] or [11], to mention just a small part of the existing literature. In these approaches, however the copula is typically not known explicitly.…”