non-Gaussian Ornstein Uhlenbeck Research Articles

Generalized Wiener–Hermite integrals and rough non-Gaussian Ornstein–Uhlenbeck process

We discuss several properties of the generalized Hermite process, which is a non-Gaussian self-similar processes with self-similarity index belonging to the whole interval . We also define Wiener integral with respect to this process and as an application, we define and study its associated Ornstein–Uhlenbeck process.

Moment-matching approximations for stochastic sums in non-Gaussian Ornstein–Uhlenbeck models

In this paper, we recall actuarial and financial applications of sums of dependent random variables that follow a non-Gaussian mean-reverting process and contemplate distribution approximations. Our work complements previous related studies restricted to lognormal random variables; we revisit previous approximations and suggest new ones. We then derive moment-based distribution approximations for random sums attuned to Asian option pricing and computation of risk measures of random annuities. Various numerical experiments highlight the speed–accuracy benefits of the proposed methods.

Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process

Abstract Barndorff-Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility follows the Ornstein–Uhlenbeck process driven by a positive Levy process without the Gaussian component. The parameter estimation of these models is challenging because the likelihood function is not available in a closed-form expression. A large number of estimation techniques have been proposed, mainly based on Bayesian inference. The main aim of the paper is to present an application of iterated filtering for parameter estimation of such models. Iterated filtering is a method for maximum likelihood inference based on a series of filtering operations, which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. An application to S&P500 index data shows the model perform well and diagnostic plots for iterated filtering ensure convergence iterated filtering to maximum likelihood estimates. Empirical application is accompanied by a simulation study that confirms the validity of the approach in the case of Barndorff-Nielsen and Shephard’s stochastic volatility models.

PRICING AND HEDGING OF VIX OPTIONS FOR BARNDORFF-NIELSEN AND SHEPHARD MODELS

The VIX call options for the Barndorff-Nielsen and Shephard models will be discussed. Derivatives written on the VIX, which is the most popular volatility measurement, have been traded actively very much. In this paper, we give representations of the VIX call option price for the Barndorff-Nielsen and Shephard models: non-Gaussian Ornstein–Uhlenbeck type stochastic volatility models. Moreover, we provide representations of the locally risk-minimizing strategy constructed by a combination of the underlying riskless and risky assets. Remark that the representations obtained in this paper are efficient to develop a numerical method using the fast Fourier transform. Thus, numerical experiments will be implemented in the last section of this paper.

Adaptive Robust Efficient Methods for Periodic Signal Processing Observed with Colours Noises

In this paper, we consider the problem of robust adaptive efficient estimating a periodic signal observed in the transmission channel with the dependent noise defined by non-Gaussian Ornstein-Uhlenbeck processes with unknown correlation properties. Adaptive model selection procedures, based on the shrinkage weighted least squares estimates, are proposed. The comparison between shrinkage and least squares methods is studied and the advantages of the shrinkage methods are analyzed. Estimation properties for proposed statistical algorithms are studied on the basis of the robust mean square accuracy defined as the maximum mean square estimation error over all possible values of unknown noise parameters. Sharp oracle inequalities for the robust risks have been obtained. The robust efficiency of the model selection procedure has been established.

Using degradation-with-jump measures to estimate life characteristics of lithium-ion battery

Degradation-with-jump measures are time series data sets containing the information of both continuous and randomly jumping degradation evolution of a system. Traditional maximum likelihood estimation and Bayesian estimation are not convenient for such general jump processes without closed-form distributions. Based on general degradation models derived using Lévy driven non-Gaussian Ornstein-Uhlenbeck (OU) processes, we propose a systematic statistical method using linear programing estimators and empirical characteristic functions. The point estimates of reliability function and lifetime moments are obtained by deriving their explicit expressions. We also construct bootstrap procedures for the confidence intervals. Simulation studies for a stable process and a stable driven OU process are performed. In the case study, we use a general Lévy process to fit the Li-ion battery life data, and then estimate the reliability and lifetime moments of the battery. By integrally analyzing degradation data series embedded with jump measures, our work provides the efficient and precise estimation for life characteristics.

Application of Iterated Filtering for Parametric Estimation of Instantaneous Variance in the Case of Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Processes

The article presents a method for parametric estimation of instantaneous variance in the case of non-Gaussian Ornstein-Uhlenbeck stochastic volatility process by means of the iterated filtering and realized variance estimator. The method is applied to realized variance of S&P500 index data. Empirical application is accompanied with simulation study to examine performance of the estimation technique.

A non-Gaussian Ornstein–Uhlenbeck model for pricing wind power futures

ABSTRACTThe recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modelling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein–Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard in the context of modelling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a constant level of installed capacity. Also, based on 1 year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases. To further demonstrate the benefits of our proposed model, we address the pricing of European options written on wind power futures, which can be achieved through Fourier techniques.

Stochastic Volatility Models Based on OU-Gamma Time Change: Theory and Estimation

We consider stochastic volatility models that are defined by an Ornstein–Uhlenbeck (OU)-Gamma time change. These models are most suitable for modeling financial time series and follow the general framework of the popular non-Gaussian OU models of Barndorff-Nielsen and Shephard. One current problem of these otherwise attractive nontrivial models is, in general, the unavailability of a tractable likelihood-based statistical analysis for the returns of financial assets, which requires the ability to sample from a nontrivial joint distribution. We show that an OU process driven by an infinite activity Gamma process, which is an OU-Gamma process, exhibits unique features, which allows one to explicitly describe and exactly sample from relevant joint distributions. This is a consequence of the OU structure and the calculus of Gamma and Dirichlet processes. We develop a particle marginal Metropolis–Hastings algorithm for this type of continuous-time stochastic volatility models and check its performance using simulated data. For illustration we finally fit the model to S&P500 index data.

A Non-Gaussian Ornstein-Uhlenbeck Model for Pricing Wind Power Futures

The recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modeling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below, and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein-Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard (2001) in the context of modeling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a recent level of installed capacity. Also, based on one year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases.

Lévy-driven non-Gaussian Ornstein–Uhlenbeck processes for degradation-based reliability analysis

ABSTRACTWe use Lévy subordinators and non-Gaussian Ornstein–Uhlenbeck processes to model the evolution of degradation with random jumps. The superiority of our models stems from the flexibility of such processes in the modeling of stylized features of degradation data series such as jumps, linearity/nonlinearity, symmetry/asymmetry, and light/heavy tails. Based on corresponding Fokker–Planck equations, we derive explicit results for the reliability function and lifetime moments in terms of Laplace transforms, represented by Lévy measures. Numerical experiments are used to demonstrate that our general models perform well and are applicable for analyzing a large number of degradation phenomena. More important, they provide us with a new methodology to deal with multi-degradation processes under dynamicenvironments.

GENERALIZED BARNDORFF-NIELSEN AND SHEPHARD MODEL AND DISCRETELY MONITORED OPTION PRICING

This paper proposes a generalization of the Barndorff-Nielsen and Shephard model, in which the log return on an asset is governed by a Lévy process with stochastic volatility modeled by a non-Gaussian Ornstein–Uhlenbeck process. Under the generalized model, we derive a closed-form expression of the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

Mean-Variance Hedging Based on an Incomplete Market with External Risk Factors of Non-Gaussian OU Processes

We prove the global risk optimality of the hedging strategy of contingent claim, which is explicitly (or called semiexplicitly) constructed for an incomplete financial market with external risk factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) processes. Analytical and numerical examples are both presented to illustrate the effectiveness of our optimal strategy. Our study establishes the connection between our financial system and existing general semimartingale based discussions by justifying required conditions. More precisely, there are three steps involved. First, we firmly prove the no-arbitrage condition to be true for our financial market, which is used as an assumption in existing discussions. In doing so, we explicitly construct the square-integrable density process of the variance-optimal martingale measure (VOMM). Second, we derive a backward stochastic differential equation (BSDE) with jumps for the mean-value process of a given contingent claim. The unique existence of adapted strong solution to the BSDE is proved under suitable terminal conditions including both European call and put options as special cases. Third, by combining the solution of the BSDE and the VOMM, we reach the justification of the global risk optimality for our hedging strategy.

Robust model selection for a semimartingale continuous time regression from discrete data

The paper considers the problem of estimating a periodic function in a continuous time regression model observed under a general semimartingale noise with an unknown distribution in the case when continuous observation cannot be provided and only discrete time measurements are available. Two specific types of noises are studied in detail: a non-Gaussian Ornstein–Uhlenbeck process and a time-varying linear combination of a Brownian motion and compound Poisson process. We develop new analytical tools to treat the adaptive estimation problems from discrete data. A lower bound for the frequency sampling, needed for the efficiency of the procedure constructed by discrete observations, has been found. Sharp non-asymptotic oracle inequalities for the robust quadratic risk have been derived. New convergence rates for the efficient procedures have been obtained. An example of the regression with a martingale noise exhibits that the minimax robust convergence rate may be both higher or lower as compared with the minimax rate for the “white noise” model. The results of Monte-Carlo simulations are given.

Generalized Barndorff-Nielsen and Shephard Model and Discretely Monitored Option Pricing

This paper proposes a generalization of the Generalized Barndorff-Nielsen and Shephard model, in which the log return on an asset price is governed by a Levy process with stochastic volatility modeled by a non-Gaussian Ornstein-Uhlenbeck process. Under the generalized model, we derive a closed-form expression for the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

Ornstein–Uhlenbeck processes for geophysical data analysis

In this work a three parameter stochastic process, termed the Gamma-Ornstein–Uhlenbeck process, has been implemented to analyze geophysical data. Such non-Gaussian Ornstein–Uhlenbeck processes offer the possibility of capturing important distributional deviations from Gaussianity and make the model flexible of dependence structures. It is shown that the Gamma-Ornstein–Uhlenbeck process is a possible candidate for earthquake data modeling. Rigorous regression analysis is provided and based on that the first-passage times are computed for different sets of data. It is shown that this model may be used to estimate parameters related to some major events—namely major earthquakes.

Modelling the Variance Risk Premium of Equity Indices: The Role of Dependence and Contagion

Understanding variance risk is of key importance in mathematical finance since it affects risk management, asset allocation and derivative pricing. Variance risk is priced in financial markets by the so-called variance risk premium (VRP), which refers to the premium demanded for holding assets whose variance is exposed to stochastic shocks.This paper identifies a new modelling framework for equity indices and presents for the first time explicit analytical formulas for their VRP in a multivariate stochastic volatility setting, which includes multivariate non-Gaussian Ornstein-Uhlenbeck processes and Wishart processes. Moreover, we propose to incorporate contagion within the equity index via a multivariate Hawkes process and find that the resulting dynamics of the VRP represent a convincing alternative to the models studied in the literature up to date.We show that our new model can explain the key stylised facts of both equity indices and individual assets and their corresponding VRP, while popular (multivariate) stochastic volatility models fail.We finally prove the existence of a structure-preserving risk neutral measure for our model. In particular, we establish the class of equivalent probabilities that preserve the self-affecting structure of the Hawkes process.

A truncated estimation method with guaranteed accuracy

This paper presents a truncated estimation method of ratio type functionals by dependent sample of finite size. This method makes it possible to obtain estimators with guaranteed accuracy in the sense of the \(L_m\)-norm, \(m\ge 2\). As an illustration, the parametric and non-parametric estimation problems on a time interval of a fixed length are considered. In particular, parameters of linear (autoregressive) and non-linear discrete-time processes are estimated. Moreover, the parameter estimation problem of non-Gaussian Ornstein-Uhlenbeck process by discrete-time observations and the estimation problem of a multivariate logarithmic derivative of a noise density of an autoregressive process with guaranteed accuracy are solved. In addition to non-asymptotic properties, the limit behavior of presented estimators is investigated. It is shown that all the truncated estimators have asymptotic properties of basic estimators. In particular, the asymptotic efficiency in the mean square sense of the truncated estimator of the dynamic parameter of a stable autoregressive process is established.

Efficient robust nonparametric estimation in a semimartingale regression model

Dans cette article nous considérons le problème d’estimation robuste d’une fonction périodique dans un modèle de régression en temps continu avec un bruit dépendant décrit par une semi martingale carrée intégrable de distribution inconnue. Un exemple de ce bruit est un processus d’Ornstein–Uhlenbeck non gaussien avec sauts (voir (J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167–241), (Ann. Appl. Probab. 18 (2008) 879–908)). Nous proposons une procédure adaptative de sélection de modèle basée sur les estimateurs des moindres carrés pondérés. Sous des conditions générales sur les deux premiers moments de la distribution du bruit, des inégalités d’Oracle non asymptotiques pointues pour des risques quadratiques robustes sont obtenues et l’efficacité robuste est établie. Nous avons établi aussi que dans le cas du processus d’Ornstein–Uhlenbeck non Gaussian, la borne inférieure pour le risque quadratique robuste est donnée par la limite de l’intensité du bruit quand la fréquence tend vers l’infini. Nous donnons un exemple d’un modèle de régression avec un bruit martingale où la vitesse de convergence du risque quadratique devient plus lente si l’intensité du bruit tend vers l’infini.

Stochastic Volatility of Volatility and Variance Risk Premia

This paper introduces a new class of stochastic volatility models which allows for stochastic volatility of volatility (SVV): Volatility modulated non-Gaussian Ornstein-Uhlenbeck (VMOU) processes. Various probabilistic properties of (integrated) VMOU processes are presented. Further we study the effect of the SVV on the leverage effect and on the presence of long memory.One of the key results in the paper is that we can quantify the impact of the SVV on the (stochastic) dynamics of the variance risk premium (VRP). Moreover, provided the physical and the risk -- neutral probability measures are related through a structure -- preserving change of measure, we obtain an explicit formula for the VRP.