Nonparametric volatility density estimation

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

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.

Stochastic volatility (SV) models take a very important role in financial market, while there still exist some difficulties in estimating parameters in SV models. In this paper, a unified parameter estimation algorithm is proposed to estimate continuous-time SV model. Parameters in equity prices and volatilities stochastic processes are estimated separately after orthogonalization of Brownian motion in SV models. A closed-form of Maximum Likelihood Estimation (MLE) and moment estimation results for four parameters in SV models are deducted. Five typical SV models are simulated to reveal the characteristics in empirical volatilities data. Parameter estimation results are presented to prove the convergence of estimation algorithm. We also compare our algorithm with Least Square (LS) estimations and numerical solutions of all the parameters in with MLE method, respectively. LS can only work in a specific form of SV models and the convergence speed is limited. Numerical solutions of the likelihood function with four parameters are unstable and can not converge to true values of all the parameters. Thus our algorithm works more efficiently than other methods.

Market Prices of Orthogonal Risk and Risk Aversion in Complete Stochastic Volatility Models: Theoretical and Empirical

Why Do Absolute Returns Predict Volatility So Well?

Estimation of continuous-time stochastic volatility models with jumps using high-frequency data

Attainability of European Path-Independent Claims in Incomplete Markets

Kalman Filtering Techniques

In this paper we investigate the performance of a linear wavelet-type deconvolution estimator for weakly dependent data. We show that the rates of convergence which are optimal in the case of i.i.d. data are also (almost) attained for strongly mixing observations, provided the mixing coefficients decay fast enough. The results are applied to a discretely observed continuous-time stochastic volatility model.

Hyperbolic Normal Stochastic Volatility Model

A signal is defined as continuous within a time range if its amplitude is specified for all t in that range, except possibly at a finite number of discontinuities. If a continuous time variable is quantized, that is, if t is represented by a set of distinct values denoted by tk, where k takes on positive integer values exclusively, the signal is defined only at discrete-time instants, known as sampling instants. Signals which are defined only at discrete instants of time are called discrete-time, or just discrete signals and can be represented by a sequence or collection of numbers.

Harmonic Balance Method for non-linear periodic flows is presented in this paper. As- sumption of temporal periodicity can be used to convert harmonic transient equations into a set of coupled steady state equations. Solution of coupled equations yields flow fields at discrete instants of time throughout a representative harmonic period. The method uses second-order accurate, polyhedral Finite Volume Method. It is developed for a general transport equation and generalised to coupled non-linear equation sets, including incom- pressible Navier-Stokes equations and is implemented in OpenFOAM. Verification of the model is carried out by comparing the results with transient simulations. Preliminary re- sults for weakly and strongly harmonic incompressible flow around oscillating airfoil are presented.

The presented effort is the extract of a composite work. Primarily to model a dc servo-system within the available framework through observed data. The observed data of the hardware are classified as input and output samples measured at discrete instants of time and collected as an array of data for finite duration. Several input-output data sets are collected and processed to find out the model of the dc servo motor through System Identification Tool Box of MATLAB, a Graphical User Interface (GUI) based approach. The fitness of identified model is checked every time new data set is applied to choose the best fitted model. Secondarily to demonstrate in steps the ways of design and implementation of Internal Model Principle (IMP) & Finite Dimensional Repetitive Control (FDRC) based Proportional Integral Derivative (PID) controller for the plant. A similar test has been made with the system using conventional PID control. Finally, a valid comparison is carried out to show advantage, disadvantage or limitation of applied control schemes.

The problem of H∞ filtering design is studied for non-linear sampled-data systems using the Takagi–Sugeno (T–S) fuzzy model approach. The sampled-data filtering is to estimate the states of a continuous-time system using only sampled measurements at discrete instants of time. Traditionally, the sufficient conditions for the existence of such an H∞ filter are characterised in terms of the solution of a differential Hamilton–Jacobi inequality with jumps, which is equivalent to solving the partial differential inequality with jumps. In general, there is no analytical solution for this non-linear partial differential inequality with jumps. First, in this study, the T–S fuzzy model is proposed to represent a class of non-linear sampled-data systems. Next, by using the T–S fuzzy model, the H∞ fuzzy filtering design problem for non-linear sampled-data system is characterised in terms of a linear matrix inequality (LMI) problem. Hence, the H∞ fuzzy filter of non-linear sampled-data systems can be given via solving LMIs instead of solving a differential Hamilton–Jacobi inequality with jumps. To illustrate the results, a numerical example is included.