These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distri…
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48 results for “Levy Process”
Develops information geometry for Lévy processes in finance.
problem Understanding the statistical properties of Lévy processes for financial modeling.
method Deriving -divergences from Lévy triplets, identifying Fisher information matrix and -connection.
result Identifies statistical implications and differential-geometric structures of Lévy processes.
Efficient methods for Lévy models using SINH-regular processes.
problem Efficient numerical methods for evaluating Lévy models.
method Defining SL-processes and sSL-processes, deriving properties of characteristic exponent, and showing all popular Lévy processes can be subordinated to Brownian motion.
result All crucial properties of characteristic exponent are consequences of a specific representation, and all popular Lévy processes are SL- or sSL-subordinated Brownian motion.
Study shows convergence rates for BSDEs approximated by compound Poisson processes.
problem Analyzing convergence rates of BSDEs driven by Lévy processes.
method Approximating Lévy processes by compound Poisson processes and studying BSDEs.
result Optimal convergence rates derived for BSDEs in -norm and Wasserstein distance.
In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Levy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for inst…
We introduce an algorithm for the pricing of finite expiry American options driven by Lévy processes. The idea is to tweak Carr's `Canadisation' method, cf. Carr [9] (see also Bouchard et al [5]), in such a way that the adjusted algorithm is viable for any Lévy process whose law at an independent, exponentially distrib…
This brief manuscript provides an introduction to Lévy processes and their applications in finance as the random process that drives asset models. Characteristic functions and random variable generators of popular Lévy processes are presented in R.
The paper defines and analyzes Poissonian occupation times for negative Lévy processes.
problem Analyzing the time spent below zero for Lévy processes with interruptions.
method Introduces Poissonian occupation times for spectrally negative Lévy processes.
result Extends results on continuous observation to interrupted observation.
Study of bandit problem with Poisson decision times and Lévy processes.
problem Continuous-time multi-armed bandit problem with Poisson decision times.
method Gittins index policy applied to spectrally one-sided Lévy processes.
result Gittins index converges to classical Lévy bandit index.
The pricing of options in exponential Levy models amounts to the computation of expectations of functionals of Levy processes. In many situations, Monte-Carlo methods are used. However, the simulation of a Levy process with infinite Levy measure generally requires either to truncate small jumps or to replace them by a …
New method calibrates multivariate Lévy processes using neural networks.
problem Calibrating multivariate Lévy processes with less smooth densities.
method Approximate Lévy density with parametrized functional form, estimate characteristic function using numerical integration with deep neural networks.
result Deep neural networks robustly capture sharp transitions in Lévy densities.
Optimizes liquidation strategies for assets with Levy process price dynamics.
problem Maximizing cash received from asset sale with price impact.
method Almgren-Chriss framework, constant absolute risk aversion, Levy process approximation.
result Explicit expression for optimal liquidation trajectories.
Method verifies if observed data fits Lévy-Driven Ornstein-Uhlenbeck process.
problem Verifying if observed data fits Lévy-Driven Ornstein-Uhlenbeck process.
method Estimating parameters and approximating the driving process to test CAR(1) Lévy-driven hypothesis.
result Demonstrates method's effectiveness through simulations and real data examples.
Develops Bilateral Gamma processes for financial market modeling.
problem Modeling financial market fluctuations with Lévy processes.
method Exploration of bilateral Gamma distributions and their Lévy processes.
result Validates Bilateral Gamma processes on real financial data.
The paper studies Lévy processes on compact manifolds, proving properties of their semigroups.
problem Analyzing Lévy processes on compact Riemannian manifolds.
method Proving properties of Feller semigroups and generators on spaces.
result The generator has a discrete spectrum of eigenvalues and the semigroup is trace-class when the process has a non-trivial Brownian part.
Universal approximation for rough paths and Lévy processes.
problem Approximating continuous functionals of càdlàg paths.
method Linear functionals of time-extended signatures.
result Universal approximation theorem for continuous functionals of càdlàg paths.
New method estimates volatility for Lévy processes with unbounded jumps efficiently.
problem Efficient estimation of volatility for Lévy processes with unbounded jumps.
method Developed a new estimator based on high-order expansions of truncated moments.
result Method outperforms existing alternatives in estimating volatility.
Modeling Bitcoin prices and media attention using jump-type processes.
problem Capturing the dynamics of Bitcoin prices and media attention.
method Lévy processes and semiparametric estimation.
result Effective modeling of Bitcoin prices and media attention using Lévy processes.
Study cliquet options in a jump-diffusion model with Lévy processes.
problem Pricing cliquet options in a complex financial model with jumps.
method Developed semi-analytic expressions using Lévy process distribution and Fourier transform.
result Inferred semi-analytic expressions for cliquet option prices and derived Greeks.
This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known Lévy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson proc…
This paper considers magnitude, asymptotics and duration of drawdowns for some Lévy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative Lévy processes using an approximation approach. For any spectrally negative Lévy process whose scale functions are well-behaved at …
Study on pricing American Exchange options using Lévy processes.
problem Pricing American Exchange options driven by Lévy processes.
method Represented American Exchange options as European options plus early exercise premium; studied properties of free boundary and provided an approximative formula.
result Developed an approximative formula for American Exchange options.
In this paper, we investigate Parisian ruin for a Lévy surplus process with an adaptive premium rate, namely a refracted Lévy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al.…
Levy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Levy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced w…
Parisian excursion below a fixed level from the last record maximum of Levy insurance risk processmath.PR
This paper presents some new results on Parisian ruin under Levy insurance risk process, where ruin occurs when the process has gone below a fixed level from the last record maximum, also known as the high-water mark or drawdown, for a fixed consecutive periods of time. The law of ruin-time and the position at ruin is …
The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative Lévy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We con…
Paper calculates perpetual American put option pricing with drawdown event in Lévy market.
problem Pricing perpetual American put options with a drawdown event in a Lévy market.
method Derives explicit price using geometric Lévy process with downward jumps, optimal stopping rule, and martingale arguments.
result Optimal stopping rule is the first time asset price falls below a specific value.
In this note we give, for a spectrally negative Levy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period r. The formula involves only the scale function of the spectrally ne…
The paper explores risk-minimization for exponential additive models, providing mathematical expressions and numerical examples.
problem Risk-minimization in incomplete markets for exponential additive models.
method Derive explicit mathematical expressions for local risk-minimization strategies in exponential additive models.
result Provide necessary conditions for deriving expressions and confirm integrability conditions for specific models.
In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specifie…
Study determines Lévy exponent from derivative prices.
problem Determine Lévy exponent in asset pricing models.
method Analyzes power-payoff derivatives to infer Lévy exponent structure.
result Lévy exponent can be determined from derivative prices.
Researchers calculate the price of a perpetual put option in Lévy models.
problem Calculating the price of a perpetual American put option in Lévy models.
method Derive the explicit price using geometric spectrally negative Lévy processes and optimal threshold.
result The optimal exercise time is the first epoch when the asset price drops below an optimal threshold.
Generalized Expected Discounted Penalty Function at General Drawdown for Lévy Risk Processesq-fin.PR
The paper studies drawdown times in Lévy risk processes, generalizing previous results.
problem Analyzing the time of drawdown in spectrally negative Lévy risk processes.
method Using the joint distribution of drawdown times, maximums, and other related quantities.
result Obtained semi-explicit expressions for the joint distribution in terms of scale functions and Lévy measure.
Scale-invariant Truncated Lévy Processcond-mat.stat-mech
We develop a scale-invariant truncated Lévy (STL) process to describe physical systems characterized by correlated stochastic variables. The STL process exhibits Lévy stability for the probability density, and hence shows scaling properties (as observed in empirical data); it has the advantage that all moments are fini…
Clarifies when certain stochastic PDEs have affine solutions.
problem Existence of affine realizations for semilinear SPDEs driven by Lévy processes.
method Analyzes conditions for affine solutions to SPDEs driven by Lévy processes.
result Conditions for the existence of affine realizations are established.
This paper gives examples of explicit arbitrage-free term structure models with Lévy jumps via state price density approach. By generalizing quadratic Gaussian models, it is found that the probability density function of a Lévy process is a "natural" scale for the process to be the state variable of a market.
Characterizes Lévy-driven Ornstein-Uhlenbeck processes linked to tempered stable distributions.
problem Understanding Lévy-driven Ornstein-Uhlenbeck processes and their properties.
method Characterizes the Lévy triplet and deduces transition laws for finite variation Ornstein-Uhlenbeck processes associated with tempered stable distributions.
result Provides algorithms for generating skeleton of Ornstein-Uhlenbeck processes related to exponentially-modulated tempered stable laws.
The paper tests hypotheses on two Lévy process-driven streams of observations.
problem Testing hypotheses on two Lévy process-driven streams of observations.
method Infinitesimal generators and super/sub-solutions are used to compute bounds and analyze the model.
result Bounds for infinitesimal generators are computed in terms of super/sub-solutions.
Study analyzes Lévy process structure on manifolds with conjugate points.
problem Microlocal analysis of Lévy processes on manifolds with conjugate points.
method Microlocal analysis, pseudodifferential operators, Fourier integral operators.
result Generator can be expressed as sum of pseudodifferential and Fourier integral operators.
New GLPs split Lévy bridges into non-overlapping subprocesses.
problem Creating multivariate stochastic processes with specific properties.
method Defining GLPs by splitting Lévy bridges and using time changes.
result GLPs have terminal values and increments with generalised multivariate Liouville distributions.
The paper evaluates functions of stable Lévy processes and their extrema efficiently.
problem Efficiently evaluating functions of stable Lévy processes and their extrema.
method Integral representations, conformal acceleration technique, simplified trapezoid rule.
result Efficient numerical procedures for cumulative probability distribution functions (cpdfs) are developed.
Investigates existence of affine models for Lévy-driven term structures.
problem Existence of affine realizations for term structure models with jumps.
method Analyzes term structure models driven by Lévy processes, focusing on restrictions on volatility.
result More severe restrictions on volatility compared to diffusion models.
Optimizes dividend control in a bankruptcy process using a special Levy process.
problem Optimizing dividend payouts in a bankruptcy process.
method Using a non-standard spectrally negative Levy process with endogenous regime switching.
result Optimal dividend control is of the barrier type and the optimal barrier can be identified.
The paper finds optimal threshold strategies for insurance companies with a positive terminal value at creeping ruin.
problem Optimizing dividend payments in an insurance company's surplus process with a positive terminal value at creeping ruin.
method Using fluctuation theory, the paper derives explicit formulas for the objective function and shows the optimality of threshold strategies.
result Threshold strategies are optimal for the dividend optimization problem under certain conditions.
Paper models non-maturing deposits using a Lévy-driven Ornstein-Uhlenbeck process.
problem Managing non-maturing deposits as a major funding source for banks.
method Develops a multivariate Lévy-driven Ornstein-Uhlenbeck process with three sources of randomness.
result Models rare but severe events in deposit volumes with positive probability.
New insights into tail behavior of heavy-tailed random vectors and processes.
problem Understanding tail behavior of aggregates of heavy-tailed random vectors.
method Analyzing multivariate regularly varying random vectors and Lévy processes.
result More than one large jump can determine tail behavior of aggregates.
Analytical tools for pricing power options in Lévy models.
problem Pricing power options with exotic features in exponential Lévy models.
method Analytical pricing formulas using Mellin space and residues in complex analysis.
result Pricing formulas converge fast and are efficient for power options.
Bayesian inference for Levy density with Gibbs posterior in discrete sampling.
problem Inference on Levy density for financial models with jumps.
method Gibbs posterior framework using a loss function for intractable likelihood.
result Gibbs posterior achieves nearly optimal rate of convergence under certain conditions.