Hyperbolic Distributions in Finance

Statistical estimation of Lévy-type stochastic volatility models

“…Indeed, for more than a decade, several subclasses of Lévy processes have been used as alternatives to the Wiener process in financial models. Among the better known models are the variance Gamma model of [7], the CGMY model of [5], and the generalized hyperbolic motion of [3,12] (see also [4,11]). While preserving the simple statistical properties of the increments of a Wiener process (namely, independent and stationary increments), Lévy processes {Z t } t≥0 can exhibit flexible marginal distributions with heavy tails, high-kurtosis, and asymmetry.…”

“…suggesting the use of the statisticŝ 12) to recover τ (T ) by making a → 0. Still, ν(|x| ≥ a) is needed and to overcome this issue we impose a semiparametric assumption on the behavior of ν(|x| ≥ a) as a → 0.…”

A jump-diffusion Libor model and its robust calibration

“…For example we may freeze L j− at zero (see Glasserman and Merener (2003)), hence replace L j− with L j (0) in (9). As an alternative, if the r i are small enough and the magnitudes of δ j L j are small enough as well, one could drop in (9) the terms of order (δ j L j ) 2 and higher.…”

“…We underline that the structure of the dynamics (9), hence the feasibility of standard Monte Carlo simulation of every forward Libor in the terminal measure, is a consequence of our model design in Sections 3.1 and 3.2. In particular it is due to the special product structure of the generally high dimensional jump measure p and the linear structure of the log-Libor factor loadings (6).…”

“…In particular, these models are well-suited to cover jumps, heavy tails, and to match implied volatility surfaces observed in stock and interest rate markets. For modelling stock prices, pure jump Lévy processes were already proposed in Eberlein and Keller (1995), and Eberlein, Keller and Prause (1998). In Cont and Tankov (2003) regularized approaches for calibrating jump-diffusion stock price models were considered.…”