Abstract not found

Implicit in the prices of traded financial assets are Arrow-Debreu prices or, with continuous states, the state-price density (SPD). We construct a nonparametric estimator for the SPD implicit in option prices and derive its asymptotic sampling theory. This estimator provides an arbitrage-free method of pricing new, complex, or illiquid securities while capturing those features of the data that are most relevant from an asset-pricing perspective, e.g., negative skewness and excess kurtosis for asset returns, volatility "smiles" for option prices. We perform Monte Carlo experiments and extract the SPD from actual S&P 500 option prices.

Draft for the Notices of the AMS Learning is key to developing systems tailored to a broad range of data analysis and information extraction tasks. We outline the mathematical foundations of learning theory and describe a key algorithm of it. 1

Often researchers find parametric models restrictive and sensitive to deviations from the parametric specifications; semi-nonparametric models are more flexible and robust, but lead to other complications such as introducing infinite dimensional parameter spaces that may not be compact. The method of sieves provides one way to tackle such complexities by optimizing an empirical criterion function over a sequence of approximating parameter spaces, called sieves, which are significantly less complex than the original parameter space. With different choices of criteria and sieves, the method of sieves is very flexible in estimating complicated econometric models. For example, it can simultaneously estimate the parametric and nonparametric components in semi-nonparametric models with or without constraints. It can easily incorporate prior information, often derived from economic theory, such as monotonicity, convexity, additivity, multiplicity, exclusion and non-negativity. This chapter describes estimation of semi-nonparametric econometric models via the method of sieves. We present some general results on the large sample properties of the sieve estimates, including consistency of the sieve extremum estimates, convergence rates of the sieve M-estimates, pointwise normality of series estimates of regression functions, root-n asymptotic normality and efficiency of sieve estimates of smooth functionals of infinite dimensional parameters. Examples are used to illustrate the general results.

We consider the problem of one-step-ahead prediction of a real-valued, stationary, strongly mixing random process fX i g i=01 . The best mean-square predictor of X0 is its conditional mean given the entire infinite past fX i g i=01 . Given a sequence of observations X1 X2 111 XN, we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension, for example, neural networks and Legendre polynomials. The proposed estimators select both the model memory and the model dimension, in a data-driven fashion, by minimizing certain complexity regularized least squares criteria. When the underlying predictor function has a finite memory, we establish that the proposed estimators are memory-universal: the proposed estimators, which do not know the true memory, deliver the same statistical performance (rates of integrated mean-squared error) as that delivered by estimators that know the true memory. Furthermore, when the underlying predictor function does not have a finite memory, we establish that the estimator based on Legendre polynomials is consistent.

For nancial time series, the generation of error bars on the point prediction is important in order to estimate the corresponding risk. The Bayesian evidence framework, already successfully applied to design of multilayer perceptrons, is applied in this paper to Least Squares Support Vector Machine (LS-SVM) regression in order to infer nonlinear models for predicting a time series and the related volatility.

In this paper, we address the complex problem of sequential Bayesian estimation and model selection/averaging. This problem does not usually admit any type of closed-form analytical solutions and, as a result, one has to resort to numerical methods. We propose here an original and powerful sequential simulation-based strategy to perform the necessary computations. This strategy is based on Monte Carlo particle methods and model selection/averaging using predictive distributions. It combines sequential importance sampling, Rao-Blackwellisation, a selection procedure and reversible jump MCMC moves. We demonstrate the eectiveness of the method by performing inference and learning on a hybrid model consisting of a dynamic linear model and a dynamic mixture of kernel basis functions. 1 Authorship based on alphabetical order 2 Corresponding author 1 Introduction Model selection is a fundamental data analysis task that arises in many elds of science and engineering. Research in this a...

We discuss a novel strategy for training neural networks using sequential Monte Carlo algorithms and propose a new hybrid gradient descent/sampling importance resampling algorithm (HySIR). In terms of both computational time and accuracy, the hybrid SIR is a clear improvement over conventional sequential Monte Carlo techniques. The new algorithm may be viewed as a global optimisation strategy, which allows us to learn the probability distributions of the network weights and outputs in a sequential framework. It is well suited to applications involving on-line, nonlinear and non-Gaussian signal processing. We show how the new algorithm outperforms extended Kalman filter training on several problems. In particular, we address the problem of pricing option contracts, traded in financial markets. In this context, we are able to estimate the one-step-ahead probability density functions of the options prices. i Contents 1 Introduction 1 2 State Space Neural Network Modelling 2 3 The Bayes...

Please do not quote without permission * Chidambaran is visiting at NYU, on leave from Tulane. Lee holds joint appointments at Tulane and HKUST. Trigueros is at Tulane. We are grateful for the comments from participants at seminars at Tulane

In this paper, we show that a hierarchical Bayesian modelling approach to sequential learning leads to many interesting attributes such as regularisation and automatic relevance determination. We identify three inference levels within this hierarchy, namely model selection, parameter estimation and noise estimation. In environments where data arrives sequentially, techniques such as cross-validation to achieve regularisation or model selection are not possible. The Bayesian approach, with extended Kalman filtering at the parameter estimation level, allows for regularisation within a minimum variance framework. A multi-layer perceptron is used to generate the extended Kalman filter nonlinear measurements mapping. We describe several algorithms at the noise estimation level, which allow us to implement adaptive regularisation and automatic relevance determination of model inputs and basis functions. An important contribution of this paper is to show the theoretical links between adaptive...