scholarly journals Asymptotic results for exponential functionals of Lévy processes

2018 ◽  
Vol 128 (1) ◽  
pp. 108-131 ◽  
Author(s):  
Zenghu Li ◽  
Wei Xu
2011 ◽  
Vol 14 (07) ◽  
pp. 1045-1090 ◽  
Author(s):  
MITYA BOYARCHENKO ◽  
MARCO DE INNOCENTIS ◽  
SERGEI LEVENDORSKIĬ

We calculate the leading term of asymptotics of the prices of barrier options and first-touch digitals near the barrier for wide classes of Lévy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carr's randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Lévy processes commonly used in option pricing.


2019 ◽  
Vol 56 (4) ◽  
pp. 1086-1105
Author(s):  
Ekaterina T. Kolkovska ◽  
Ehyter M. Martín-González

AbstractWe study the distribution of the negative Wiener–Hopf factor for a class of two-sided jump Lévy processes whose positive jumps have a rational Laplace transform. The positive Wiener–Hopf factor for this class of processes was studied by Lewis and Mordecki (2008). Here we obtain a formula for the Laplace transform of the negative Wiener–Hopf factor, as well as an explicit expression for its probability density in terms of sums of convolutions of known functions. Under additional regularity conditions on the Lévy measure of the studied processes, we also provide asymptotic results as $u\to-\infty$ for the distribution function F(u) of the negative Wiener–Hopf factor. We illustrate our results in some particular examples.


2010 ◽  
Vol 13 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Ernst Eberlein ◽  
Dilip Madan

Author(s):  
UWE FRANZ

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.


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