scholarly journals Asymptotic results for time-changed Lévy processes sampled at hitting times

2011 ◽  
Vol 121 (7) ◽  
pp. 1607-1632 ◽  
Author(s):  
Mathieu Rosenbaum ◽  
Peter Tankov
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Hitting Times ◽  
Asymptotic Results

PRICES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS IN LÉVY-DRIVEN MODELS, NEAR BARRIER

2011 ◽  
Vol 14 (07) ◽  
pp. 1045-1090 ◽  
Author(s):  
MITYA BOYARCHENKO ◽  
MARCO DE INNOCENTIS ◽  
SERGEI LEVENDORSKIĬ
Keyword(s):  
Financial Markets ◽  
Lévy Processes ◽  
Levy Processes ◽  
Gamma Model ◽  
Asymptotic Results ◽  
Robust Test ◽  
Leading Term ◽  
Digital Options ◽  
Variance Gamma Model ◽  
Normal Inverse Gaussian

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.

scholarly journals The distribution and asympotic behaviour of the negative Wiener–Hopf factor for Lévy processes with rational positive jumps

2019 ◽  
Vol 56 (4) ◽  
pp. 1086-1105
Author(s):  
Ekaterina T. Kolkovska ◽  
Ehyter M. Martín-González
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Probability Density ◽  
Lévy Processes ◽  
Levy Processes ◽  
Regularity Conditions ◽  
Asymptotic Results ◽  
Lévy Measure ◽  
The Laplace Transform ◽  
Additional Regularity

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.

scholarly journals Hitting Times of Points and Intervals for Symmetric Lévy Processes

2016 ◽  
Vol 46 (4) ◽  
pp. 739-777 ◽  
Author(s):  
Tomasz Grzywny ◽  
Michał Ryznar
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Hitting Times

On correlating Lévy processes

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