A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Levy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.

Stochastic interest rate volatility modeling with a continuous-time GARCH(1, 1) model

In this paper we introduce a fractionally integrated exponential continuous time GARCH(p, d, q) process. It is defined in such a way that it is a continuous time extension of the discrete time FIEGARCH(p, d, q) process. We investigate stationarity and moment properties of the new model. It is also shown that the long memory effect introduced in the log-volatility propagates to the volatility process.

We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the density of which, at a fixed instant in time, we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the volatility density. An expansion of the bias and a bound on the variance are derived.

If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting “MRCA age” process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright–Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on ℝ++×ℝ++ with intensity measures of the form λ⊗μ where λ is Lebesgue measure, and μ (the “family lifetime measure”) is an arbitrary, absolutely continuous measure satisfying μ((0, ∞))=∞ and μ((x, ∞))<∞ for all x>0. Special cases of this construction describe the time evolution of the MRCA age in (1+β)-stable continuous state branching processes conditioned on nonextinction—a particular case of which, β=1, is Feller’s continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist. Moreover, we introduce a new family of Markov processes that stands in a relation with respect to the general (1+β)-stable continuous state branching process and its conditioned version that is similar to the one between the family of Bessel-squared diffusions and the unconditioned and conditioned Feller continuous state branching process.

In this work, an attempt is made to apply the Local Lagged Adapted Generalized Method of Moments (LLGMM) to estimate state and parameters in stochastic differential dynamic models. The development of LLGMM is motivated by parameter and state estimation problems in continuous-time nonlinear and non-stationary stochastic dynamic model validation problems in biological, chemical, engineering, energy commodity markets, financial, medical, military, physical sciences and social sciences. The byproducts of this innovative approach (LLGMM) are the balance between model specification and model prescription of continuous-time dynamic process and the development of discrete-time interconnected dynamic model of local sample mean and variance statistic process (DTIDMLSMVSP). Moreover, LLGMM is a dynamic non-parametric method. The DTIDMLSMVSP is an alternative approach to the GARCH(1,1) model, and it provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. Furthermore, applications of LLGMM to energy commodities price, U.S. Treasury Bill interest rate and the U.S.–U.K. foreign exchange rate data strongly exhibit its unique role, scope and performance, in particular, in forecasting and confidence-interval problems in applied statistics.

The aim of this study is to extend the scope and applicability of the level-crossing method to discrete-time stochastic processes and generalize it to enable us to study multiple discrete-time stochastic processes. In previous versions of the level-crossing method, problems with it correspond to the fact that this method had been developed for analyzing a continuous-time process or at most a multiple continuous-time process in an individual manner. However, since all empirical processes are discrete in time, the already-established level-crossing method may not prove adequate for studying empirical processes. Beyond this, due to the fact that most empirical processes are coupled; their individual study could lead to vague results. To achieve the objectives of this study, we first find an analytical expression for the average frequency of crossing a level in a discrete-time process, giving the measure of the time experienced for two consecutive crossings named as the "waiting time." We then introduce the generalized level-crossing method by which the consideration of coupling between the components of a multiple process becomes possible. Finally, we provide an analytic solution when the components of a multiple stochastic process are independent Gaussian white noises. The comparison of the results obtained for coupled and uncoupled processes measures the strength and efficiency of the coupling, justifying our model and analysis. The advantage of the proposed method is its sensitivity to the slightest coupling and shortest correlation length.

We consider variance-optimal hedging in general continuous-time affine stochastic volatility models. The optimal hedge and the associated hedging error are determined semiexplicitly in the case that the stock price follows a martingale. The integral representation of the solution opens the door to efficient numerical computation. The setup includes models with jumps in the stock price and in the activity process. It also allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

In this paper, we analyze the robustness and sensitivity of various continuous-time rough Volterra stochastic volatility models in relation to the process of market calibration. Model robustness is examined from two perspectives: the sensitivity of option price estimates and the sensitivity of parameter estimates to changes in the option data structure. The following sensitivity analysis consists of statistical tests to determine whether a given studied model is sensitive to changes in the option data structure based on the distribution of parameter estimates. Empirical study is performed on a data set consisting of Apple Inc. equity options traded on four different days in April and May 2015. In particular, the results for RFSV, rBergomi and alpha RFSV models are provided and compared to the results for Heston, Bates, and AFSVJD models.

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour asn →∞ of the process conditioned on non-absorption until timen.By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.

In this paper, the authors propose a new outlook to model Tokyo stock exchange market’s volatility (Nikkei225) in the concept of continuous-time GARCH(1,1) driven by Meixner Levy Process for the first time. GARCH(1,1) model is estimated as an appropriate discrete-time model for the timeseries, then Meixner Levy process driven continuous-time model is developed to analyze the volatility characteristics of standardized log returns of Nikkei225. Pseudo Maximum Likelihood (PML), is employed as the parameter estimation process for the Meixner-COGARCH(1,1) model. The data covers the period from 2005 to 2013. The empirical results show that continuous-time GARCH(1,1) driven by a Meixner Levy process successfully captures volatility clustering and heavy-tail behaviour of Nikkei225.

We have seen that there is strong empirical and theoretical evidence supporting the conclusion that the volatility of log price changes of a financial asset is a time-dependent stochastic process. In this chapter we discuss an approach for describing stochastic processes characterized by a time-dependent variance (volatility), the ARCH processes introduced by Engle in 1982 [50]. ARCH models have been applied to several different areas of economics. Examples include (i) means and variances of inflation in the UK, (ii) stock returns, (iii) interest rates, and (iv) foreign exchange rates. ARCH models are widely studied in economics and finance and the literature is huge. They can also be very attractive for describing physical systems. ARCH models are simple models able to describe a stochastic process which is locally nonstationary but asymptotically stationary. This implies that the parameters controlling the conditional probability density function f t ( x ) at time t are fluctuating. However, such a ‘local’ time dependence does not prevent the stochastic process from having a well defined asymptotic pdf P ( x ). ARCH processes are empirically motivated discrete-time stochastic models for which the variance at time t depends, conditionally, on some past values of the square value of the random signal itself. ARCH processes define classes of stochastic models because each specific model is characterized by a given number of control parameters and by a specific form of the pdf, called the conditional pdf, of the process generating the random variable at time t . In this chapter we present some widely used ARCH processes. We focus our attention on the shape of the asymptotic probability density function and on the scaling properties observed.

In the previous chapter, we studied a discrete-time stochastic process {X n, n ≥ 0} on finite state space with Markov property at times n = 0, 1, 2 ···. Now we would like to study a continuous-time stochastic process {X(t), t ≥ 0} on a finite state space with Markov property at each time t ≥ 0. We shall call such a process continuous-time Markov Chain (CTMC). We shall see in the next chapter that a finite-state CTMC spends an exponentially distributed amount of time in a given state before jumping out of it. Thus exponential distributions play an important role in CTMCs. In addition, the Poisson distribution and Poisson processes also form the foundation of many CTMC models. Hence we study these topics in this chapter.KeywordsPoisson ProcessComputational ProblemCompound Poisson ProcessExponential Random VariablePoisson Random VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We propose a novel flexible bivariate conditional Poisson (BCP) INteger‐valued Generalized AutoRegressive Conditional Heteroscedastic (INGARCH) model for correlated count time series data. Our proposed BCP‐INGARCH model is mathematically tractable and has as the main advantage over existing bivariate INGARCH models its ability to capture a broad range (both negative and positive) of contemporaneous cross‐correlation, which is a non‐trivial advancement. Properties of stationarity and ergodicity for the BCP‐INGARCH process are developed. Estimation of the parameters is performed through conditional maximum likelihood (CML), and the finite‐sample behavior of the estimators is investigated through simulation studies. Asymptotic properties of the CML estimators are derived. Hypothesis testing methods for the presence of contemporaneous correlation between the time series are presented and evaluated. A Granger causality test is also addressed. We apply our methodology to monthly counts of hepatitis cases in two nearby Brazilian cities, which are highly cross‐correlated. The data analysis demonstrates the importance of considering a bivariate model allowing for a wide range of contemporaneous correlation in real‐life applications.

This thesis studies the design of optimal investment strategies. A strategy is considered optimal when it minimizes the variance of terminal portfolio wealth for a given level of expected terminal portfolio wealth, or equivalently, maximizes an investor's utility. We study this issue in two particular situations: when asset returns follow a continuous-time path-independent process, and when they follow a discrete-time path-dependent process. Continuous-time path-independent return models are popular but controversial in the literature. We formulate the criteria for portfolio rules to be considered as optimal in this framework. We construct a portfolio consisting of a risky and a risk-free asset where the return on the risky asset follows a Gaussian diffusion process with non-constant drift and diffusion. Portfolio rules satisfying the specified criteria are shown to be compatible with the objective of utility maximization. Discrete-time path-dependent return models are more realistic, given the fact that almost all historical return data are measured in discrete time and exhibit serial correlation. Hence, we develop a novel methodology to assess the market efficiency and predictability of Australian index returns on equities, debts and cash, which we show are path-dependent empirically. We propose a one-step Multivariate Semi-parametric Maximum Likelihood Estimation (one-step MSMLE) technique to estimate a Vector-autoregressive Multivariate Generalized Autoregressive Conditional Heteroskedasticity (VAR-MGARCH) model of Australian asset returns. The estimation is done using a rolling historical window approach so as to highlight and capture path-dependency in asset returns as well as allow for parameter changes. Serial correlation is found in both the return and the volatility levels of the Australian assets that we consider. Having shown this, we then extend a class of reactive portfolio controls to the case when returns follow a VAR-MGARCH process. The portfolio controls are formulated by solving the Lagrangian which minimizes the variance in next period wealth for a given targeted next period wealth. We quantitatively demonstrate that this class of reactive portfolio controls are efficient, even under the existence of market impacts.