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 Risk minimization in financial markets modeled by Itô-Lévy processes

2014 ◽  
Vol 26 (5-6) ◽  
pp. 939-979 ◽  
Author(s):  
Bernt Øksendal ◽  
Agnès Sulem
Keyword(s):  
Financial Markets ◽  
Lévy Processes ◽  
Levy Processes ◽  
Risk Minimization

scholarly journals Existence Conditions of Super-Replication Cost in a Multinomial Model

2017 ◽  
Vol 9 (4) ◽  
pp. 185
Author(s):  
Mei Xing
Keyword(s):  
Stock Price ◽  
Gaussian Model ◽  
Time Interval ◽  
Inverse Gaussian ◽  
Multinomial Model ◽  
Liquidity Premium ◽  
Gamma Model ◽  
Existence Conditions ◽  
Variance Gamma Model ◽  
Normal Inverse Gaussian

This paper gives a theorem for the continuous time super-replication cost of European options in an unbounded multinomial market. An approximation multinomial scheme is put forward on a finite time interval [0,1] corresponding to a pure jump Lévy model with unbounded jumps. Under the assumption that the expected underlying stock price at time 1 is bounded, the limit of the sequence of the super-replication cost in a multinomial model is proved to be greater than or equal to an optimal control problem. Furthermore, it is discussed that the existence conditions of a super-replication cost and a liquidity premium for the multinomial model. This paper concentrates on a multinomial tree with unbounded jumps, which can be seen as an extension of the work of (Xing, 2015). The super-replication cost and the liquidity premium under the variance gamma model and the normal inverse Gaussian model are calculated and illustrated.

scholarly journals LÉVY SIMPLE STRUCTURAL MODELS

2007 ◽  
Vol 10 (04) ◽  
pp. 593-606 ◽  
Author(s):  
MARTIN BAXTER
Keyword(s):  
Risk Management ◽  
Lévy Processes ◽  
Historical Data ◽  
Structural Models ◽  
Levy Processes ◽  
Gamma Model ◽  
Model Parameter ◽  
Credit Portfolio

This paper considers credit portfolio models based on Levy processes in general, and the gamma model in particular. It describes both single-name and multi-name situations using the gamma model, along with calibration fits and a comparison of various simple Levy models. There is also extensive historical data, including the May 2005 Auto crisis, which can be described in terms of the model. Parameter-based risk management using the gamma model is also discussed along with implementation details.

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