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

This paper characterizes all continuous price processes that are consistent with current option prices. This extends Derman and Kani (1994), Dupire (1994, 1997), and Rubinstein (1994), who only consider processes with deterministic volatility. Our characterization implies a volatility forecast that does not require a specific model, only current option prices. We show how arbitrary volatility processes can be adjusted to fit current option prices exactly, just as interest rate processes can be adjusted to fit bond prices exactly. The procedure works with many volatility models, is fast to calibrate, and can price exotic options efficiently using familiar lattice techniques.

Option Prices, Implied Price Processes, and Stochastic Volatility

http://www.cs.hunter.cuny.edu/~felisav/StochasticProcesses/Course_Material_files/BBG_jedc96.pdf

Monte Carlo methods for security pricing

The risk of default affects virtually every financial contract. Therefore the pricing of default risk has received much attention; both from traders, who have a strong interest in pricing transactions accurately, and from financial economists, who have much to learn from the way such risks are priced in markets. The standard theoretical paradigm for modeling credit risks is the contingent claims approach pioneered by Black and Scholes. Much of the literature follows Merton (1974) by explicitly linking the risk of a firm’s default to the variability in the firm’s asset value. Although this line of research has proven very useful in addressing the qualitatively important aspects of

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Estimating the Price of Default Risk

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

This paper proposes a simple modification to the standard Monte Carlo simulation procedure for computing the prices of derivative securities. The modification imposes the martingale property on the simulated sample paths of the underlying asset price. This procedure is referred to as the empirical martingale simulation (EMS). The EMS ensures that the price estimated by simulation satisfies rational option pricing bounds. The EMS also yields a substantial error reduction for the price estimate. The EMS can be easily coupled with the standard variance reduction methods to obtain greater computational efficiency. Simulation studies are conducted for European and Asian call options using both the Black and Scholes and GARCH option pricing frameworks. The results indicate that the EMS yields substantial variance reduction particularly for in- and at-the-money options. Cette etude propose une modification simple aux procedures traditionnelles de calcul de prix des produits derives par simulation de Monte Carlo. La modification impose la propriete de martingale aux trajectoires simulees de la variable d'etat sous-jacente. L'utilisation de cette procedure assure que l'estime de prix respecte les bornes rationnelles d'option tout en diminuant de facon substantielle la variance des estimes de prix. La procedure peut aisement etre jumelee aux methodes traditionnelles de reduction de variance afin d'obtenir une plus grande efficacite. Une etude de simulation est presentee pour des options d'achat Europeennes et Asiatiques. Les resultats indiquent que la methode obtient des reductions substantielle de la variance des estimes de prix et ce, particulierement pour les options in et at the money.

Empirical Martingale Simulation for Asset Prices

This paper proposes a simple modification to the standard Monte Carlo simulation procedure for computing the prices of derivative securities. The modification imposes the martingale property on the simulated sample paths of the underlying asset price. This procedure is referred to as the empirical martingale simulation (EMS). The EMS ensures that the price estimated by simulation satisfies the rational option pricing bounds. The EMS yields a substantial error reduction for the price estimate and can be easily coupled with the standard variance reduction methods. Simulation studies are conducted for European and Asian call options using both the Black and Scholes and GARCH option pricing frameworks. The results indicate that the EMS yields substantial variance reduction particularly for in- and at-the-money or longer-maturity options. The option price estimate based on the EMS is found to exhibit a minor small-sample bias only in few occasions. An analysis of the trade-off between computing time and price accuracy reveals that the EMS dominates the conventional simulation methods.

American option pricing under GARCH by a Markov chain approximation

Moody’s KMV method is a popular commercial implementation of the structural credit risk model pioneered by Merton (1974). It is an algorithm for estimating the unobserved asset value and the unknown parameters required for implementing such a model. This estimation method has found its way to the recent academic literature, but it has not yet been formally analyzed to assess its statistical properties. This paper fllls this gap and shows that, in the context of Merton’s model, the KMV estimates are identical to maximum likelihood estimates (MLE) developed in Duan (1994). Unlike the MLE method, however, the KMV algorithm is silent about the distributional properties of the estimates and thus ill-suited for statistical inference. The KMV algorithm also cannot generate estimates for capital-structure speciflc parameters. In contrast, the MLE approach is ∞exible and can be readily applied to difierent structural credit risk models.

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Jean-Guy Simonato

Papers

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

Empirical Martingale Simulation for Asset Prices

Empirical Martingale Simulation for Asset Prices

American option pricing under GARCH by a Markov chain approximation

On the Equivalence of the KMV and Maximum Likelihood Methods for Structural Credit Risk Models