Multifractal analysis of Lévy fields

We study the pointwise regularity properties of the Lévy fields introduced by T. Mori; these fields are the most natural generalization of Lévy processes to the multivariate setting. We determine their spectrum of singularities, and we show that their H?lder singularity sets satisfy a large intersection property in the sense of K. Falconer.     

Stability of Merton's portfolio optimization problem for Lévy models

Merton's classical portfolio optimization problem for an investor, who can trade in a risk-free bond and a stock, can be extended to the case where the driving noise of the logreturns is a pure jump process instead of a Brownian motion. Benth et al. [4,5] solved the problem and found the optimal control implicitly given by an integral equation in the hyperbolic absolute risk aversion (HARA) utility case. There are several ways to approximate a Levy process with infinite activity by neglecting the small jumps or approximating them with a Brownian motion, as discussed in Asmussen and Rosinski [1]. In this setting, we study stability of the corresponding optimal investment problems. The optimal controls are solutions of integral equations, for which we study convergence. We are able to characterize the rate of convergence in terms of the variance of the small jumps. Additionally, we prove convergence of the corresponding wealth processes and indirect utilities (value functions).     

A Cramér-von Mises test for symmetry of the error distribution in asymptotically stationary stochastic models

This paper has to do with a Cramér-von Mises test for symmetry of the error distribution in a class of absolutely regular and non-necessarily stationary heteroscedastic models. The test statistic is based on the empirical characteristic function. Its convergence, as well as that of the residual-based empirical distribution function are established. From these results, the null cumulative distribution function of the test statistic is approximated. A simulation experiment shows that the test performs well on the examples tested.     

On geometric ergodicity of skewed—SVCHARME models

Markov Chain Monte Carlo is repeatedly used to analyze the properties of intractable distributions in a convenient way. In this paper we derive conditions for geometric ergodicity of a general class of nonparametric stochastic volatility models with skewness driven by the hidden Markov Chain with switching.     

Right inverses of Lévy processes and stationary stopped local times

If X is a symmetric Lévy process on the line, then there exists a non-decreasing, càdlàg process H such that X(H(x)) = x for all x≥ 0 if and only if X is recurrent and has a non-trivial Gaussian component. The minimal such H is a subordinator K. The law of K is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of X. When X is Brownian motion, K is just the usual ladder times process and this result extends the classical result of Lévy that the maximum process has the same law as the local time at 0. Write G _t for last point in the range of K prior to t. In a parallel with classical fluctuation theory, the process Z := (X _t −X _Gt ) _{t ≥0} is Markov with local time at 0 given by (X _Gt ) _{t ≥0}. The transition kernel and excursion measure of Z are identified. A similar programme is outlined for Lévy processes on the circle. This leads to the construction of a stopping time such that the stopped local times constitute a stationary process indexed by the circle. Received: 7 September 1999 / Revised version: 9 November 1999 / Published online: 8 August 2000     

A geometric interpretation of the transition density of a symmetric Lévy process

We study for a class of symmetric Lévy processes with state space R n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t>0 and （δt）t>0. The first family of metrics describes the diagonal term pt（0）; it is induced by the characteristic exponent ψ of the Lévy process by dt（x, y） = ^1/2tψ（x-y）. The second and new family of metrics δt relates to ^1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt（x） = pt（0）e- δ2t （x,0） where pt（0） corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt（x）.     

A note on convergence of option prices and their Greeks for Lévy models

We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.     

A note on the mean correcting martingale measure for geometric Lévy processes

A martingale measure is constructed by using a mean correcting transform for the geometric Lévy processes model. It is shown that this measure is the mean correcting martingale measure if and only if, in the Lévy process, there exists a continuous Gaussian part. Although this measure cannot be equivalent to a physical probability for a pure jump Lévy process, we show that a European call option price under this measure is still arbitrage free.     

Cramér-type moderate deviations for stationary sequences of bounded random variables

We derive Cramér-type moderate deviations for stationary sequences of bounded random variables. Our results imply the moderate deviation principles and a Berry–Esseen bound. Applications to quantile coupling inequalities, functions of ?-mixing sequences, and contracting Markov chains are discussed.     

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion are investigated. The direct problem consists in finding a solution of the corresponding boundary value problem for given data on the boundary of the domain of the independent variable. The peculiarity of the direct problem consists in the inhomogeneity and irregularity of mixed boundary data. Solvability and stability conditions are specified for the direct problem. The inverse boundary value problem consists in finding some traces of the solution of the corresponding boundary value problem for given standard and additional data on a certain part of the boundary of the domain of the independent variable. The peculiarity of the inverse problem consists in its ill-posedness. Regularizing methods and solution algorithms are developed for the inverse problem.     

Option pricing and hedging for optimized Lévy driven stochastic volatility models

This paper pays attention to Ornstein-Uhlenbeck (OU) based stochastic volatility models with marginal law given by Classical Tempered Stable (CTS) distribution and Normal Inverse Gaussian (NIG) distribution, which are subclasses of infinite activity Lévy processes and are compared to finite activity Barndorff-Nielsen and Shephard (BNS) model. They are applied to option pricing and hedging in capturing leptokurtic features in asset returns and clustering effect in volatility that are consistently observed phenomena in underlying asset dynamics. The analytical formula of option pricing can be obtained through use of characteristic functions and Fast Fourier Transform (FFT) technique. Additionally, we introduce two hybrid optimization techniques such as hybrid Particle Swarm optimization (PSO) algorithm and hybrid Differential Evolution (DE) algorithm into parameters calibration schemes to improve the calibration quality for newly constructed models. Finally, we conduct experiments on Chinese emerging option markets to examine the performance of proposed models exploiting hybrid optimization techniques.     

Statistical inference for misspecified ergodic Lévy driven stochastic differential equation models

We consider the estimation problem of misspecified ergodic Lévy driven stochastic differential equation models based on high-frequency samples. We utilize a widely applicable and tractable Gaussian quasi-likelihood approach which focuses on mean and variance structure. It is shown that the Gaussian quasi-likelihood estimators of the drift and scale parameters still satisfy polynomial type probability estimates and asymptotic normality at the same rate as the correctly specified case. In their derivation process, the theory of extended Poisson equation for time-homogeneous Feller Markov processes plays an important role. Our result confirms the reliability of the Gaussian quasi-likelihood approach for SDE models.     

On occupation times in the red of Lévy risk models

In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level 0) up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running maximum.     

Radner equilibrium in incomplete Lévy models

We construct continuous-time equilibrium models based on a finite number of exponential utility investors. The investors’ income rates as well as the stock’s dividend rate are governed by discontinuous Lévy processes. Our main result provides the equilibrium (i.e., bond and stock price dynamics) in closed-form. As an application, we show that the equilibrium Sharpe ratio can be increased and the equilibrium interest rate can be decreased (simultaneously) when the investors’ income streams cannot be traded.     

A simplified proof of the André-Oort conjecture for products of modular curves

In this paper we give a short proof of the André-Oort conjecture for products of modular curves under the Generalised Riemann Hypothesis using only simple Galois-theoretic and geometric arguments. We believe this method represents a strategy for proving the conjecture for a general Shimura variety under GRH without using ergodic theory. We also demonstrate a short proof of the Manin–Mumford conjecture for Abelian varieties using similar arguments.     

Approximate indifference pricing in exponential Lévy models

Financial markets based on Lévy processes are typically incomplete and option prices depend on risk attitudes of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the non-linear partial integro-differential equation associated to the indifference price. In this work, we develop closed-form approximations to exponential utility indifference prices in exponential Lévy models. To this end, we first establish a new non-asymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity. Next, we use this formula to derive a closed-form approximation of the indifference price by treating the Lévy model as a perturbation of the Black–Scholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of Lévy processes (?erný et al. 2013) to non-linear and non-smooth functionals. Our formula represents the indifference price as the linear combination of the Black–Scholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying Lévy process, and the derivatives of the Black–Scholes price. As a by-product, we obtain a simple approximation for the spread between the buyer’s and the seller’s indifference price. This formula allows to quantify, in a model-independent fashion, how sensitive a given product is to jump risk when jump size is small.     

Jump tail dependence in Lévy copula models

This paper investigates the dependence of extreme jumps in multivariate Lévy processes. We introduce a measure called jump tail dependence, defined as the probability of observing a large jump in one component of a process given a concurrent large jump in another component. We show that this measure is determined by the Lévy copula alone and that it is independent of marginal Lévy processes. We derive a consistent nonparametric estimator for jump tail dependence and establish its asymptotic distribution. Regarding the economic relevance of the measure, a simulation study illustrates that jump tail dependence has a substantial impact on financial portfolio distributions and optimal portfolio weights.     

Transition Density Estimates for a Class of Lévy and?Lévy-Type Processes

We show on- and off-diagonal upper estimates for the transition densities of symmetric Lévy and Lévy-type processes. To get the on-diagonal estimates, we prove a Nash-type inequality for the related Dirichlet form. For the off-diagonal estimates, we assume that the characteristic function of a Lévy(-type) process is analytic, which allows us to apply the complex analysis technique.