Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

https://faculty.washington.edu/ezivot/econ589/EVT_Mcneil_Frey_2000.pdf

Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

Publisher Summary The class of stochastic volatility (SV) models has its roots in both, mathematical finance and financial econometrics. In fact, several variations of SV models originated from research looking at very different issues. Volatility plays a central role in the pricing of derivative securities. The Black-Scholes model for the pricing of an European option is by far the most widely used formula even when the underlying assumptions are known to be violated. The Black-Scholes model predicts a flat term structure of volatilities. In reality, the term structure of at-the-money implied volatilities is typically upward sloping when short term volatilities are low and the reverse when they are high. The Black-Scholes model is taken as a reference point from which several notions of volatility are presented. Several stylized facts regarding volatility and option prices are also presented. Both sections set the scene for a formal framework defining stochastic volatility. The chapter introduces the statistical models of stochastic volatility.

5 Stochastic volatility

We compare 330 ARCH-type models in terms of their ability to describe the conditional variance. The models are compared out-of-sample using DM–$ exchange rate data and IBM return data, where the latter is based on a new data set of realized variance. We find no evidence that a GARCH(1,1) is outperformed by more sophisticated models in our analysis of exchange rates, whereas the GARCH(1,1) is clearly inferior to models that can accommodate a leverage effect in our analysis of IBM returns. The models are compared with the test for superior predictive ability (SPA) and the reality check for data snooping (RC). Our empirical results show that the RC lacks power to an extent that makes it unable to distinguish ‘good’ and ‘bad’ models in our analysis. Copyright © 2005 John Wiley & Sons, Ltd.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/jae.800

A forecast comparison of volatility models: does anything beat a GARCH(1,1)?

This article develops an option pricing model and its corresponding delta formula in the context of the generalized autoregressive conditional heteroskedastic (GARCH) asset return process. the development utilizes the locally risk-neutral valuation relationship (LRNVR). the LRNVR is shown to hold under certain combinations of preference and distribution assumptions. the GARCH option pricing model is capable of reflecting the changes in the conditional volatility of the underlying asset in a parsimonious manner. Numerical analyses suggest that the GARCH model may be able to explain some well-documented systematic biases associated with the Black-Scholes model.

The garch option pricing model

This article develops a general methodology that uses the observed prices of a derivative contract to compute maximum likelihood parameter estimates for an unobserved asset value process. the use of this estimation methodology is demonstrated in two applications: Vasicek's term structure model and deposit insurance pricing. This methodology can also be useful in the empirical analysis of complex financial contracts involving embedded options.

Maximum likelihood estimation using price data of the derivative contract

Augmented GARCH (p,q) process and its diffusion limit

This paper proposes a unified state-space formulation for parameter estimation of exponential-affine term structure models. The proposed method uses an approximate linear Kalman filter which only requires specifying the conditional mean and variance of the system in an approximate sense. The method allows for measurement errors in the observed yields to maturity, and can simultaneously deal with many yields on bonds with different maturities. An empirical analysis of two special cases of this general class of model is carried out: the Gaussian case (Vasicek 1977) and the non-Gaussian case (Cox Ingersoll and Ross 1985 and Chen and Scott 1992). Our test results indicate a strong rejection of these two cases. A Monte Carlo study indicates that the procedure is reliable for moderate sample sizes.

Estimating and testing exponential-affine term structure models by Kalman filter

A forward intensity model for the prediction of corporate defaults over different future periods is proposed. Maximum pseudo-likelihood analysis is then conducted on a large sample of the US industrial and financial firms spanning the period 1991-2011 on a monthly basis. Several commonly used factors and firm-specific attributes are shown to be useful for prediction at both short and long horizons. Our implementation also factors in momentum in some variables and documents their importance in default prediction. The prediction is very accurate for shorter horizons. The accuracy deteriorates somewhat when the horizon is increased to two or three years, but its performance still remains reasonable. The forward intensity model is also amenable to aggregation, which allows for an analysis of default behavior at the portfolio and/or economy level.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2061925_code1075989.pdf?abstractid=1791222&mirid=4

Jin-Chuan Duan

Papers

The garch option pricing model

Maximum likelihood estimation using price data of the derivative contract

Augmented GARCH (p,q) process and its diffusion limit

Estimating and testing exponential-affine term structure models by Kalman filter

Multiperiod Corporate Default Prediction - A Forward Intensity Approach