Constrained nonlinear programming for volatility estimation with GARCH models

This paper proposes a constrained nonlinear programming view of generalized autoregressive conditional heteroskedasticity (GARCH) volatility estimation models in financial econometrics. These models are usually presented to the reader as unconstrained optimization models with recursive terms in the literature, whereas they actually fall into the domain of nonconvex nonlinear programming. Our results demonstrate that constrained nonlinear programming is a worthwhile exercise for GARCH models, especially for the bivariate and trivariate cases, as they offer a significant improvement in the quality of the solution of the optimization problem over the diagonal VECH and the BEKK representations of the multivariate GARCH model

Multivariate GARCH estimation via a Bregman-proximal trust-region method

The estimation of multivariate GARCH time series models is a difficult task mainly due to the significant overparameterization exhibited by the problem and usually referred to as the "curse of dimensionality". For example, in the case of the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the dimensionality of the problem. Moreover, these parameters are subjected to convoluted nonlinear constraints necessary to ensure, for instance, the existence of stationary solutions and the positive semidefinite character of the conditional covariance matrices used in the model design. So far, this problem has been addressed in the literature only in low dimensional cases with strong parsimony constraints. In this paper we propose a general formulation of the estimation problem in any dimension and develop a Bregman-proximal trust-region method for its solution. The Bregman-proximal approach allows us to handle the constraints in a very efficient and natural way by staying in the primal space and the Trust-Region mechanism stabilizes and speeds up the scheme. Preliminary computational experiments are presented and confirm the very good performances of the proposed approach.Comment: 35 pages, 5 figure

Using CAViaR models with implied volatility for value-at-risk estimation

This paper proposes VaR estimation methods that are a synthesis of conditional autoregressive value at risk (CAViaR) time series models and implied volatility. The appeal of this proposal is that it merges information from the historical time series and the different information supplied by the market’s expectation of risk. Forecast combining methods, with weights estimated using quantile regression, are considered. We also investigate plugging implied volatility into the CAViaR models, a procedure that has not been considered in the VaR area so far. Results for daily index returns indicate that the newly proposed methods are comparable or superior to individual methods, such as the standard CAViaR models and quantiles constructed from implied volatility and the empirical distribution of standardised residual. We find that the implied volatility has more explanatory power as the focus moves further out into the left tail of the conditional distribution of S&P500 daily returns

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Support Vector Regression Based GARCH Model with Application to Forecasting Volatility of Financial Returns

In recent years, support vector regression (SVR), a novel neural network (NN) technique, has been successfully used for financial forecasting. This paper deals with the application of SVR in volatility forecasting. Based on a recurrent SVR, a GARCH method is proposed and is compared with a moving average (MA), a recurrent NN and a parametric GACH in terms of their ability to forecast financial markets volatility. The real data in this study uses British Pound-US Dollar (GBP) daily exchange rates from July 2, 2003 to June 30, 2005 and New York Stock Exchange (NYSE) daily composite index from July 3, 2003 to June 30, 2005. The experiment shows that, under both varying and fixed forecasting schemes, the SVR-based GARCH outperforms the MA, the recurrent NN and the parametric GARCH based on the criteria of mean absolute error (MAE) and directional accuracy (DA). No structured way being available to choose the free parameters of SVR, the sensitivity of performance is also examined to the free parameters.recurrent support vector regression, GARCH model, volatility forecasting

Applying MGARCH models in finance

In this paper we give literature review about application of multivariate GARCH (MGARCH) models in modern finance and economy. First, we will present basic concept of multivariate volatility (GARCH) modeling. MGARCH models specify equations for how the covariance moves over time and these models have been designed to model the conditional covariance matrix of multiple time series. Problems of portfolio Value-at-Risk (VaR) estimates, portfolio optimization, risk assessment, volatility transmitting, asset allocation, hedging in futures markets, pricing of assets and derivatives, CAPM betas require a multivariate framework, because all problems mentioned above require covariances as inputs. This implicates very wide application of MGARCH models. Additionally, in this paper we will also describe the leverage effect in multivariate GARCH models

A new approach to modelling nonlinear time series: introducing the ExpAR-ARCH and ExpAR-GARCH models and applications

The analysis of time series has long been the subject of interest in different fields. For decades time series were analysed with linear models. Nevertheless, an issue that has been raised is whether there exist other models that can explain and fit real data better than linear ones. In this paper, new nonlinear time series models are proposed (namely the ExpAR-ARCH and the ExpAR-GARCH), which are combinations of a nonlinear model in the conditional mean and a nonlinear model in the conditional variance and have the potential of explaining observed data in various fields. Simulated data of these models are presented, while different algorithms (the Nelder-Mead simplex direct search method, the Quasi-Newton line search algorithm, the Active-Set algorithm, the Sequential Quadratic Programming algorithm, the Interior Point algorithm and a Genetic Algorithm) are used and compared in order to check their estimation performance when it comes to these suggested nonlinear models. Moreover, an application to the Dow Jones data is considered, showing that the new models can explain real data better than the AR-ARCH and AR-GARCH models. © Paraskevi Katsiampa

Likelihood based inference for diffusion driven models

This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility models and counting process models. The diffusions can potentially be non-stationary. Although our methods are sampling based, making use of Markov chain Monte Carlo methods to sample the posterior distribution of the relevant unknowns, our general strategies and details are different from previous work along these lines. The methods we develop are simple to implement and simulation efficient. Importantly, unlike previous methods, the performance of our technique is not worsened, in fact it improves, as the degree of latent augmentation is increased to reduce the bias of the Euler approximation. In addition, our method is not subject to a degeneracy that afflicts previous techniques when the degree of latent augmentation is increased. We also discuss issues of model choice, model checking and filtering. The techniques and ideas are applied to both simulated and real data.Bayes estimation, Brownian bridge, Non-linear diffusion, Euler approximation, Markov chain Monte Carlo, Metropolis-Hastings algorithm, Missing data, Simulation, Stochastic differential equation.

Ambiguity Aversion and the Term Structure of Interest Rates

This paper studies the term structure implications of a simple structural economy in which the representative agent displays ambiguity aversion, modeled by Multiple Priors Recursive Utility. Bond excess returns reflect a premium for ambiguity, which is observationally distinct from the risk premium of affine yield curve models. The ambiguity premium can be large even in the simplest logutility model and is non zero also for stochastic factors that have a zero risk premium. A calibrated low-dimensional two-factor economy with ambiguity is able to reproduce the deviations from the expectations hypothesis documented in the literature, without modifying in a substantial way the nonlinear mean reversion dynamics of the short interest rate. In this economy, we do not find any apparent tradeoffs between fitting the first and second moments of the yield curve and the large equity premium.General Equilibrium, Term Structure of Interest Rates, Ambiguity Aversion, Expectations Hypothesis, Campbell-Shiller Regression