Continuous-time Autoregressive Moving Average Processes Research Articles

Analytical formulas for option prices under time-changed CARMA process

We consider the option pricing problem when the underlying asset price is driven by a continuous time autoregressive moving average (CARMA) process, time changed a Lévy subordinator or/and an absolutely continuous time change process. We derive the analytical formulas for the option prices by employing the orthogonal polynomial expansion method. Our method is based on the observation that the CARMA process belongs to the class of polynomial diffusion and the time variable and underlying state variables enter the polynomial expansion separately. We demonstrate the accuracy of the method through a number of numerical experiments. We also investigate the price sensitivities with respect to the key parameters that govern the dynamics of the underlying state and time change variables.

Modelling Temperature Using CARMA Processes with Stochastic Speed of Mean Reversion for Temperature Insurance Pricing

In this paper, we present a continuous time autoregressive moving average (CARMA) model with stochastic speed of mean reversion. This model allows the mean reversion rates to behave stochastically and governed by an Ornstein-Uhlenbeck process. We provide closed-form solution to the CARMA with stochastic speed of mean reversion and formulate the price of temperature insurance using spot-forward relationship framework. We demonstrate the insurance pricing based on the cumulative average temperatures (CAT) index by simulating the temperature variations. We found that our proposed model may explain the temperature evolution well and the price of CAT-based index insurance looks reasonable.

PKS 2155-304: A Case Study of Blazar Variability Power Spectrum at the Highest Energies and on the Longest Timescales

We present the results of our Power Spectral Density (PSD) analysis for the BL Lac object PKS 2155-304, utilizing the nightly-binned long-term light curve from the decade-long monitoring, as well as the minute-binned intra-night light curve from the High Energy Stereoscopic Survey (H.E.S.S.; >200 GeV). The source is unique for exhibiting the shortest flux-doubling timescale at Very High Energy (VHE) among its class and thus provides a rare opportunity to study the particle acceleration on the smallest spatial scales in blazar jets. The light curves are modeled in terms of the Continuous-Time Auto-Regressive Moving Average (CARMA) process. The combined long-term and intra-night PSD extends up to ∼6 decades in the temporal frequency range; unprecedented at the TeV energies for a blazar source. Our systematic approach reveals that PKS 2155-304 shows, on average, a complex shape of variability power spectrum, with more variability power on longer timescales. The long-term variability is best modeled by the CARMA(2,1) process, while the intra-night variability is modeled by a CARMA(1,0) process. We note that the CARMA(1,0) process refers to an Ornstein–Uhlenbeck process where the power-law PSD slope (PSD varies as a function of variability frequency to the power of the negative slope) changes from two to zero, above a certain “characteristic/relaxation” timescale. Even though the derived power spectrum of the intra-night light curve did not reveal a flattening, we speculate such relaxation must occur on timescales longer than a few hours for the source.

On non-negative modeling with CARMA processes

Two stationary and non-negative processes that are based on continuous-time autoregressive moving average (CARMA) processes are discussed. First, we consider a generalization of Cox–Ingersoll–Ross (CIR) processes. Next, we consider CARMA processes driven by compound Poisson processes with exponential jumps which are generalizations of Ornstein–Uhlenbeck (OU) processes driven by the same noise. The way in which the two processes generalize CIR and OU processes and the relation between them will be discussed. Furthermore, the stationary distribution, the autocorrelation function, and pricing of zero-coupon bonds are considered.

Multivariate stochastic delay differential equations and CAR representations of CARMA processes

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a CAR(∞) representation. Furthermore, we show how the CAR(∞) representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations.

Continuous time ARMA processes: Discrete time representation and likelihood evaluation

This paper explores the representation and estimation of mixed continuous time ARMA (autoregressive moving average) systems of orders p, q. Taking the general case of mixed stock and flow variables, we discuss new state space and exact discrete time representations and demonstrate that the discrete time ARMA representations widely used in empirical work, based on differencing stock variables, are members of a class of observationally equivalent discrete time ARMA(p+1, p) representations, which includes a more natural ARMA(p, p) representation. We compare and contrast two approaches to likelihood evaluation and computation, namely one based on an exact discrete time representation and another utilising a state space representation and the Kalman–Bucy filter. We demonstrate the value of our approach in two applications: a univariate study of the yield curve at different frequencies; and, a multivariate study of the relationship between US GDP and oil prices, taking account of the mixed frequencies with which these data are available.

Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

We address the issue of sample path simulation of Levy-driven continuous-time autoregressive moving average (CARMA) processes. Approximate discrete-time simulation schemes are constructed along with quantifiable error analysis for stable, second-order and non-negative CARMA processes, based upon the so-called series representation of infinitely divisible laws and associated Levy processes. We prove that under suitable conditions, the simulation scheme can be improved in terms of second-order structure, finite dimensional laws as well as sample path properties. The simulation procedure is often quite simple and allows one to conduct super-sampling without running the algorithm once again. The computational complexity of the proposed scheme is not affected much by the sampling scheme, such as sampling frequency and irregular spacing. Numerical results are presented throughout to illustrate the effectiveness of the proposed simulation scheme.

Time series models for credit default swap premiums

We present statistical models for the continuous-time Dynamics of credit Default swap (CDS) premium within an intensity-based credit risk modeling Framework. Based on historical daily CDS Premiums for a large set of different corporate refence entities from several developed countries, we fit continuous-time autoregressive moving-average processes of an appropriate order driven by a Levy process. We recover the driving noise process, which only Shows a stochastic volatility effect for particular branches. On a distributional Level, the increments of the noise process are, as a rule, best modeled by a normal inverse Gaussian distribution.

FLEXIBLE AND SCALABLE METHODS FOR QUANTIFYING STOCHASTIC VARIABILITY IN THE ERA OF MASSIVE TIME-DOMAIN ASTRONOMICAL DATA SETS

We present the use of continuous-time autoregressive moving average (CARMA) models as a method for estimating the variability features of a light curve, and in particular its power spectral density (PSD). CARMA models fully account for irregular sampling and measurement errors, making them valuable for quantifying variability, forecasting and interpolating light curves, and for variability-based classification. We show that the PSD of a CARMA model can be expressed as a sum of Lorentzian functions, which makes them extremely flexible and able to model a broad range of PSDs. We present the likelihood function for light curves sampled from CARMA processes, placing them on a statistically rigorous foundation, and we present a Bayesian method to infer the probability distribution of the PSD given the measured lightcurve. Because calculation of the likelihood function scales linearly with the number of data points, CARMA modeling scales to current and future massive time-domain data sets. We conclude by applying our CARMA modeling approach to light curves for an X-ray binary, two AGN, a long-period variable star, and an RR-Lyrae star, in order to illustrate their use, applicability, and interpretation.

A Unified Formulation of Gaussian Versus Sparse Stochastic Processes—Part II: Discrete-Domain Theory

This paper is devoted to the characterization of an extended family of continuous-time autoregressive moving average (CARMA) processes that are solutions of stochastic differential equations driven by white Lévy innovations. These are completely specified by: 1) a set of poles and zeros that fixes their correlation structure and 2) a canonical infinitely divisible probability distribution that controls their degree of sparsity (with the Gaussian model corresponding to the least sparse scenario). The generalized CARMA processes are either stationary or nonstationary, depending on the location of the poles in the complex plane. The most basic nonstationary representatives (with a single pole at the origin) are the Lévy processes, which are the non-Gaussian counterparts of Brownian motion. We focus on the general analog-to-discrete conversion problem and introduce a novel spline-based formalism that greatly simplifies the derivation of the correlation properties and joint probability distributions of the discrete versions of these processes. We also rely on the concept of generalized increment process, which suppresses all long range dependencies, to specify an equivalent discrete-domain innovation model. A crucial ingredient is the existence of a minimally supported function associated with the whitening operator L; this B-spline, which is fundamental to our formulation, appears in most of our formulas, both at the level of the correlation and the characteristic function. We make use of these discrete-domain results to numerically generate illustrative examples of sparse signals that are consistent with the continuous-domain model.

Spectral estimates for high-frequency sampled continuous-time autoregressive moving average processes

In this article, we consider a continuous-time autoregressive moving average (CARMA) process driven by either a symmetric α-stable Levy process with α ∈ (0,2) or a symmetric Levy process with finite second moments. In the asymptotic framework of high-frequency data within a long time interval, we establish a consistent estimate for the normalized power transfer function by applying a smoothing filter to the periodogram of the CARMA process. We use this result to propose an estimator for the parameters of the CARMA process and exemplify the estimation procedure by a simulation study.

High‐frequency sampling and kernel estimation for continuous‐time moving average processes

Interest in continuous‐time processes has increased rapidly in recent years, largely because of high‐frequency data available in many applications. We develop a method for estimating the kernel function g of a second‐order stationary Lévy‐driven continuous‐time moving average (CMA) process Y based on observations of the discrete‐time process YΔ obtained by sampling Y at Δ, 2Δ, …, nΔ for small Δ. We approximate g by gΔ based on the Wold representation and prove its pointwise convergence to g as Δ → 0 for continuous‐time autoregressive moving average (CARMA) processes. Two non‐parametric estimators of gΔ, on the basis of the innovations algorithm and the Durbin–Levinson algorithm, are proposed to estimate g. For a Gaussian CARMA process, we give conditions on the sample size n and the grid spacing Δ(n) under which the innovations estimator is consistent and asymptotically normal as n → ∞. The estimators can be calculated from sampled observations of any CMA process, and simulations suggest that they perform well even outside the class of CARMA processes. We illustrate their performance for simulated data and apply them to the Brookhaven turbulent wind speed data. Finally, we extend results of Brockwell et al. (2012) for sampled CARMA processes to a much wider class of CMA processes.

Noise recovery for Lévy-driven CARMA processes and high-frequency behaviour of approximating Riemann sums

We consider high-frequency sampled continuous-time autoregressive moving average (CARMA) models driven by finite-variance zero-mean Levy processes. An $L^2$-consistent estimator for the increments of the driving Levy process without order selection in advance is proposed if the CARMA model is invertible. In the second part we analyse the high-frequency behaviour of approximating Riemann sum processes, which represent a natural way to simulate continuous-time moving average models on a discrete grid. We compare their autocovariance structure with the one of sampled CARMA processes and show that the rule of integration plays a crucial role. Moreover, new insight into the kernel estimation procedure of Brockwell et al. [11] is given.

Integration of CARMA processes and spot volatility modelling

Continuous‐time autoregressive moving average (CARMA) processes with a non‐negative kernel and driven by a non‐decreasing Lévy process constitute a useful and very general class of stationary, non‐negative continuous‐time processes which have been used, in particular for the modelling of stochastic volatility. In the celebrated stochastic volatility model of Barndorff‐Nielsen and Shephard (2001), the spot (or instantaneous) volatility at time t, V(t), is represented by a stationary Lévy‐driven Ornstein‐Uhlenbeck process. This has the shortcoming that its autocorrelation function is necessarily a decreasing exponential function, limiting its ability to generate integrated volatility sequences, , with autocorrelation functions resembling those of observed realized volatility sequences. (A realized volatility sequence is a sequence of estimated integrals of spot volatility over successive intervals of fixed length, typically 1 day.) If instead of the stationary Ornstein–Uhlenbeck process, we use a CARMA process to represent spot volatility, we can overcome the restriction to exponentially decaying autocorrelation function and obtain a more realistic model for the dependence observed in realized volatility. In this article, we show how to use realized volatility data to estimate parameters of a CARMA model for spot volatility and apply the analysis to a daily realized volatility sequence for the Deutsche Mark/ US dollar exchange rate.

On the limit behavior of the periodogram of high-frequency sampled stable CARMA processes

In this paper we consider a continuous-time autoregressive moving average (CARMA) process (Yt)t∈R driven by a symmetric α-stable Lévy process with α∈(0,2] sampled at a high-frequency time-grid {0,Δn,2Δn,…,nΔn}, where the observation grid gets finer and the last observation tends to infinity as n→∞. We investigate the normalized periodogram In,YΔn(ω)=|n−1/α∑k=1nYkΔne−iωk|2. Under suitable conditions on Δn we show the convergence of the finite-dimensional distribution of both Δn2−2/α[In,YΔn(ω1Δn),…,In,YΔn(ωmΔn)] for (ω1,…,ωm)∈(R∖{0})m and of self-normalized versions of it to functions of stable distributions. The limit distributions differ depending on whether ω1,…,ωm are linearly dependent or independent over Z. For the proofs we require methods from the geometry of numbers.

High‐frequency sampling of a continuous‐time ARMA process

Continuous‐time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non‐uniformly spaced data and as a tool for dealing with high‐frequency data of the form ,n = 0, 1, 2,…, where Δ is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process , when Δ is small and the underlying continuous‐time process is a specified CARMA process.

Estimation for Non-Negative Lévy-Driven CARMA Processes

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a useful and very general class of stationary, nonnegative continuous-time processes that have been used, in particular, for the modeling of stochastic volatility. Brockwell, Davis, and Yang (2007) derived efficient estimates of the parameters of a nonnegative Lévy-driven CAR(1) process and showed how the realization of the underlying Lévy process can be estimated from closely-spaced observations of the process itself. In this article we show how the ideas of that article can be generalized to higher order CARMA processes with nonnegative kernel, the key idea being the decomposition of the CARMA process into a sum of dependent Ornstein–Uhlenbeck processes.

Multivariate CARMA processes

A multivariate Lévy-driven continuous time autoregressive moving average (CARMA) model of order ( p , q ), q < p , is introduced. It extends the well-known univariate CARMA and multivariate discrete time ARMA models. We give an explicit construction using a state space representation and a spectral representation of the driving Lévy process. Furthermore, various probabilistic properties of the state space model and the multivariate CARMA process itself are discussed in detail.

Representations of continuous-time ARMA processes

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.

Representations of continuous-time ARMA processes

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.