Fractional Ornstein-Uhlenbeck Process Research Articles

Variance and volatility swaps and options under the exponential fractional Ornstein–Uhlenbeck model

Considering the fair strike values of variance and volatility swaps, we use a stochastic volatility model in which the log volatility is given by a fractional Ornstein–Uhlenbeck process with two versions; a stationary version and a version with a deterministic initial value. Under these versions, the fair strike formulas are obtained in exact form for variance swaps and approximated fair strike formulas are derived for volatility swaps based on the fact that an aggregation of log-normal variables is well-approximated by shifted log-normal or log-normal distribution. In addition, we obtain two approximate pricing formulas for European options on the realized variance and volatility. The accuracy and robustness of the approximated fair strike formulas are examined via Monte-Carlo computations. We conduct calibration experiments to show that the Hurst exponent and the mean-reversion property of the fractional Ornstein–Uhlenbeck process are able to produce various shapes resembling the market term-structures of variance swaps when they are put together.

The El Niño Southern Oscillation Recharge Oscillator with the Stochastic Forcing of Long-Term Memory

The influence of the fast-varying variables that have a long-term memory on the El Niño Southern Oscillation (ENSO) is investigated by adding a fractional Ornstein–Uhlenbeck (FOU) process stochastic noise on the simple recharge oscillator (RO) model. The FOU process noise converges to zero very slowly with a negative power law. The corresponding non-zero ensemble mean during the integration period can exert a pronounced influence on the ensemble-mean dynamics of the RO model. The state-dependent noise, also called the multiplicative noise, can present its influence by reducing the relaxation coefficient and by introducing periodic external forcing. The decreasing relaxation coefficient can enhance the oscillation amplitude and shorten the oscillation period. The forced frequency is close to the natural frequency. The two mechanisms together can further amplify the amplitude and shorten the period, compared with the state-independent noise or additive noise, which only exhibits its influence by introducing non-periodic external forcing. These two mechanisms explicitly elucidate the influence of the stochastic forcing on the ensemble-mean dynamics of the RO model. It provides comprehensive knowledge to better understand the interaction between the fast-varying stochastic forcing and the slow-varying deterministic system and deserves further investigation.

The long time behavior of the fractional Ornstein-Uhlenbeck process with linear self-repelling drift

The long time behavior of the fractional Ornstein-Uhlenbeck process with linear self-repelling drift

Quantile estimation in fractional Levy Ornstein-Uhlenbeck processes

First we study estimation of the drift parameter in the fractional Ornstein-Uhlenbeck process whose marginal distribution is Student t-distribution. We obtain Spearman’s correlation based estimator, quantile estimator and Brownian excursion based estimator of the drift parameter. Then we study method of moments estimator and quantile estimator in fractional inverse Gaussian and fractional gamma Ornstein-Uhlenbeck processes.

Moderate Deviations for Parameter Estimation in the Fractional Ornstein–Uhlenbeck Processes with Periodic Mean

Moderate Deviations for Parameter Estimation in the Fractional Ornstein–Uhlenbeck Processes with Periodic Mean

Asymptotically efficient estimation of Ergodic rough fractional Ornstein-Uhlenbeck process under continuous observations

Asymptotically efficient estimation of Ergodic rough fractional Ornstein-Uhlenbeck process under continuous observations

Statistical inference in discretely observed fractional Ornstein–Uhlenbeck processes

In this paper, we develop a two-stage method for estimating all the unknown parameters in the fractional Ornstein–Uhlenbeck model from discrete observations. The estimation procedure is built upon the marriage of the power variation and the minimum contrast estimation. In the first stage, the Hurst coefficient is estimated by the increment Bernoulli statistic and the volatility parameter is estimated by power variations. The asymptotic theory of the proposed estimators in the diffusion term are established under an in-fill asymptotic scheme. In the second stage, two drift parameters are estimated based on the minimum contrast estimation. Their asymptotic theory are analyzed via use of a double asymptotic scheme. The results from Monte Carlo studies illustrate that the proposed estimators have reasonable finite sample properties. An empirical illustration based on realized volatility indicates that the realized volatility is rough.

Parameter estimation of the fractional Ornstein-Uhlenbeck process based on quadratic variation.

Modern experiments routinely produce extensive data of the diffusive dynamics of tracer particles in a large range of systems. Often, the measured diffusion turns out to deviate from the laws of Brownian motion, i.e., it is anomalous. Considerable effort has been put in conceiving methods to extract the exact parameters underlying the diffusive dynamics. Mostly, this has been done for unconfined motion of the tracer particle. Here, we consider the case when the particle is confined by an external harmonic potential, e.g., in an optical trap. The anomalous particle dynamics is described by the fractional Ornstein-Uhlenbeck process, for which we establish new estimators for the parameters. Specifically, by calculating the empirical quadratic variation of a single trajectory, we are able to recover the subordination process governing the particle motion and use it as a basis for the parameter estimation. The statistical properties of the estimators are evaluated from simulations.

APPROXIMATE PRICING OF DERIVATIVES UNDER FRACTIONAL STOCHASTIC VOLATILITY MODEL

AbstractThis paper examines the issue of derivative pricing within the framework of a fractional stochastic volatility model. We present a deterministic partial differential equation system to derive an approximate expression for the derivative price. The proposed approach allows for the stochastic volatility to be expressed as a composition of deterministic functions of time and a fractional Ornstein–Uhlenbeck process. We apply this method to the European option pricing under the fractional Stein–Stein volatility model, demonstrating its feasibility and reliability through numerical simulations. Our numerical simulations also illustrate the impact of the parameters in the fractional stochastic volatility model on the option price.

Nonlinear biases in the roughness of a Fractional Stochastic Regularity Model

A Multifractional Process with Random Exponent (MPRE) is used to model the dynamics of log-prices in a financial market. Under this assumption, we show that the Hurst-Hölder exponent of the MPRE follows the fractional Ornstein–Uhlenbeck process which in the Fractional Stochastic Volatility Model of Comte and Renault (1998) describes the dynamics of the log-volatility. We provide evidence that several biases of the estimation procedures can generate artificial rough volatility in surrogated as well as real financial data.

Kolmogorov bounds in the CLT of the LSE for Gaussian Ornstein Uhlenbeck processes

In this paper, we consider the Ornstein–Uhlenbeck (OU) process defined as solution to the equation [Formula: see text], [Formula: see text], where [Formula: see text] is a Gaussian process with stationary increments, whereas [Formula: see text] is unknown parameter to be estimated. We provide an upper bound in Kolmogorov distance for normal approximation of the least squares estimator [Formula: see text] of the drift parameter [Formula: see text] on the basis of the continuous observation [Formula: see text], as [Formula: see text]. Our approach is based on some novel estimates involving a combination of Malliavin calculus and Stein’s method for normal approximation. We apply our result to fractional OU processes of the first kind, and improved the upper bound of the Kolmogorov distance for the LSE [Formula: see text] provided by [Y. Chen, N. Kuang and Y. Li, Berry–Esseen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes, Stoch. Dyn. 20(4) (2020) 2050023; Y. Chen and Y. Li, Berry–Esseen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes with the hurst parameter [Formula: see text], Commun. Stat. Theory Methods 50(13) (2021) 2996–3013], respectively, in the cases [Formula: see text] and [Formula: see text]. We also apply our approach to fractional OU processes of the second kind.

On the Kolmogorov Distance for the Least Squares Estimator in the Fractional Ornstein-Uhlenbeck Process

The paper shows that the distribution of the normalized least squares estimator of the drift parameter in the fractional Ornstein-Uhlenbeck process observed over [0, T] converges to the standard normal distribution with an uniform optimal error bound of the order O(T −1/2) for 0.5 ≤ H ≤ 0.63 and of the order O(T4H-3) for 0.63 < H < 0.75 where H is the Hurst exponent of the fractional Brownian motion driving the Ornstein-Uhlenbeck process. For the normalized quasi-least squares estimator, the error bound is of the order O(T−1/4) for 0.5 ≤ H ≤ 0.69 and of the order O(T4H−3) for 0.69 < H < 0.75.

Hypotheses Testing in Nonergodic Fractional Ornstein-Uhlenbeck Models

We obtain explicit form of fine large deviation theorems for the log-likelihood ratio in testing models with fractional nonergodic Ornstein-Uhlenbeck processes with Hurst parameter more than half and get explicit rates of decrease of the error probabilities of Neyman-Pearson, Bayes and minimax tests.

Multi-mixed fractional Brownian motions and Ornstein–Uhlenbeck processes

The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.

A class of fractional Ornstein–Uhlenbeck processes mixed with a Gamma distribution

We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.

An Exponential Nonuniform Berry–Esseen Bound for the Fractional Ornstein–Uhlenbeck Process

An Exponential Nonuniform Berry–Esseen Bound for the Fractional Ornstein–Uhlenbeck Process

Power‐type derivatives for rough volatility with jumps

AbstractThis paper proposes a novel analytical pricing–hedging framework for volatility derivatives which simultaneously takes into account rough volatility and volatility jumps. Directly targeting the instantaneous variance of a risky asset, our model consists of a generalized fractional Ornstein–Uhlenbeck process driven by a Lévy subordinator and an independent sinusoidal‐composite Lévy process, and allows the characteristic function of average forward variance to be obtainable in semiclosed form, without having to invoke any geometric‐mean approximations. Pricing–hedging formulae are proposed for a general class of power‐type derivatives, in the spirit of numerical Fourier transform. A comparative empirical study is conducted on two independent recent data sets on Volatility Index options, before and during the COVID‐19 pandemic, to demonstrate that the proposed framework is highly amenable to efficient model calibration under various choices of kernels. The price dynamics of the underlying asset can be readily considered and the possibility of studying rough volatility of volatility is given as well.

Statistical analysis of the non-ergodic fractional Ornstein–Uhlenbeck process with periodic mean

Consider a periodic, mean-reverting Ornstein–Uhlenbeck process \(X=\{X_t,t\ge 0\}\) of the form \(d X_{t}=\left( L(t)+\alpha X_{t}\right) d t+ dB^H_{t}, \quad t \ge 0\), where \(L(t)=\sum _{i=1}^{p}\mu _i\phi _i (t)\) is a periodic parametric function, and \(\{B^H_t,t\ge 0\}\) is a fractional Brownian motion of Hurst parameter \(\frac{1}{2}\le H<1\). In the “ergodic” case \(\alpha <0\), the parametric estimation of \((\mu _1,\ldots ,\mu _p,\alpha )\) based on continuous-time observation of X has been considered in Dehling et al. (Stat Inference Stoch Process 13:175–192, 2010; Stat Inference Stoch Process 20:1–14, 2016) for \(H=\frac{1}{2}\), and \(\frac{1}{2}<H<1\), respectively. In this paper we consider the “non-ergodic” case \(\alpha >0\), and for all \(\frac{1}{2}\le H<1\). We analyze the strong consistency and the asymptotic distribution for the estimator of \((\mu _1,\ldots ,\mu _p,\alpha )\) when the whole trajectory of X is observed.

A Special Study of the Mixed Weighted Fractional Brownian Motion

In this work, we present the analysis of a mixed weighted fractional Brownian motion, defined by ηt:=Bt+ξt, where B is a Brownian motion and ξ is an independent weighted fractional Brownian motion. We also consider the parameter estimation problem for the drift parameter θ&gt;0 in the mixed weighted fractional Ornstein–Uhlenbeck model of the form X0=0;Xt=θXtdt+dηt. Moreover, a simulation is given of sample paths of the mixed weighted fractional Ornstein–Uhlenbeck process.

Explicit construction of joint multipoint statistics in complex systems

Complex systems often involve random fluctuations for which self-similar properties in space and time play an important role. Fractional Brownian motions, characterized by a single scaling exponent, the Hurst exponent H, provide a convenient tool to construct synthetic signals that capture the statistical properties of many processes in the physical sciences and beyond. However, in certain strongly interacting systems, e.g., turbulent flows, stock market indices, or cardiac interbeats, multiscale interactions lead to significant deviations from self-similarity and may therefore require a more elaborate description. In the context of turbulence, the Kolmogorov–Oboukhov model (K62) describes anomalous scaling, albeit explicit constructions of a turbulent signal by this model are not available yet. Here, we derive an explicit formula for the joint multipoint probability density function of a multifractal field. To this end, we consider a scale mixture of fractional Ornstein–Uhlenbeck processes and introduce a fluctuating length scale in the corresponding covariance function. In deriving the complete statistical properties of the field, we are able to systematically model synthetic multifractal phenomena. We conclude by giving a brief outlook on potential applications which range from specific tailoring or stochastic interpolation of wind fields to the modeling of financial data or non-Gaussian features in geophysical or geospatial settings.