scholarly journals Asymptotic results for heavy-tailed Lévy processes and their exponential functionals

Bernoulli ◽  
2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Wei Xu
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Asymptotic Results ◽  
Heavy Tailed

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 Application of Lévy processes in modelling (geodetic) time series with mixed spectra

2021 ◽  
Vol 28 (1) ◽  
pp. 121-134
Author(s):  
Jean-Philippe Montillet ◽  
Xiaoxing He ◽  
Kegen Yu ◽  
Changliang Xiong
Keyword(s):  
Time Series ◽  
Lévy Processes ◽  
Stable Process ◽  
Functional Model ◽  
Noise Spectrum ◽  
Levy Processes ◽  
Infinite Variance ◽  
Residual Time ◽  
First Case ◽  
Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Lévy processes, namely Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. The fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity properties. The stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, the model implies potential anxiety in the functional model selection, where missing geophysical information can generate such residual time series.

scholarly journals Application of Levy Processes in Modelling (Geodetic) Time Series With Mixed Spectra

2019 ◽  
Author(s):  
Jean-Philippe Montillet ◽  
Xiaoxing He ◽  
Kegen Yu
Keyword(s):  
Time Series ◽  
Lévy Processes ◽  
Stable Process ◽  
Functional Model ◽  
Noise Spectrum ◽  
Levy Processes ◽  
Residual Time ◽  
Heavy Tailed Distributions ◽  
First Case ◽  
Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the extraction of geophysical signals. The noise spectrum of these time series is generally modeled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series, after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three random variables (r.v.), with the last r.v. belonging to the family of Levy processes. This stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Levy processes: Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. Fractional Levy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity property. Stable process is characterized by a large variance, which can be satisfied in the case of heavy-tailed distributions. The application to geodetic time series implies potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.

scholarly journals Application of Lévy Processes in Modelling (Geodetic) Time Series With Mixed Spectra

2020 ◽  
Author(s):  
Jean-Philippe Montillet ◽  
Xiaoxing He ◽  
Kegen Yu ◽  
Changliang Xiong
Keyword(s):  
Time Series ◽  
Lévy Processes ◽  
Stable Process ◽  
Functional Model ◽  
Noise Spectrum ◽  
Levy Processes ◽  
Infinite Variance ◽  
Residual Time ◽  
First Case ◽  
Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series, after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series,we identify three classes of Lévy processes: Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. Fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity property. Stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, it implies potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.

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