Distributed Gaussian Processes

To scale Gaussian processes (GPs) to large data sets we introduce the robust Bayesian Committee Machine (rBCM), a practical and scalable product-of-experts model for large-scale distributed GP regression. Unlike state-of-the-art sparse GP approximations, the rBCM is conceptually simple and does not rely on inducing or variational parameters. The key idea is to recursively distribute computations to independent computational units and, subsequently, recombine them to form an overall result. Efficient closed-form inference allows for straightforward parallelisation and distributed computations with a small memory footprint. The rBCM is independent of the computational graph and can be used on heterogeneous computing infrastructures, ranging from laptops to clusters. With sufficient computing resources our distributed GP model can handle arbitrarily large data sets.Comment: 10 pages, 5 figures. Appears in Proceedings of ICML 201

A Conversation with Eric Ghysels

Published in Econometric Theory, 2012, https://doi.org/10.1017/S026646661100017X</p

Gaussian Estimation of Continuous Time Models of the Short Term Interest Rate

This paper proposes a Gaussian estimator for nonlinear continuous time models of the short term interest rate. The approach is based on a stopping time argument that produces a normalizing transformation facilitating the use of a Gaussian likelihood. A Monte Carlo study shows that the finite sample performance of the proposed procedure offers an improvement over the discrete approximation method proposed by Nowman (1997). An empirical application to U.S. and British interest rates is given.Gaussian estimation, nonlinear diffusion, normalizing transformation

Simulation-based Estimation of Contingent-claims Prices

A new methodology is proposed to estimate theoretical prices of financial contingent-claims whose values are dependent on some other underlying financial assets. In the literature the preferred choice of estimator is usually maximum likelihood (ML). ML has strong asymptotic justification but is not necessarily the best method in finite samples. The present paper proposes instead a simulation-based method that improves the finite sample performance of the ML estimator while maintaining its good asymptotic properties. The methods are implemented and evaluated here in the Black-Scholes option pricing model and in the Vasicek bond pricing model, but have wider applicability. Monte Carlo studies show that the proposed procedures achieve bias reductions over ML estimation in pricing contingent claims. The bias reductions are sometimes accompanied by reductions in variance, leading to significant overall gains in mean squared estimation error. Empirical applications to US treasury bills highlight the differences between the bond prices implied by the simulation-based approach and those delivered by ML. Some consequences for the statistical testing of contingent-claim pricing models are discussed.Bias reduction, Bond pricing, Indirect inference, Option pricing, Simulation-based estimation

A Two-Stage Realized Volatility Approach to the Estimation for Diffusion Processes from Discrete Observations

This paper motivates and introduces a two-stage method for estimating diffusion processes based on discretely sampled observations. In the first stage we make use of the feasible central limit theory for realized volatility, as recently developed in Barndorff-Nielsen and Shephard (2002), to provide a regression model for estimating the parameters in the diffusion function. In the second stage the in-fill likelihood function is derived by means of the Girsanov theorem and then used to estimate the parameters in the drift function. Consistency and asymptotic distribution theory for these estimates are established in various contexts. The finite sample performance of the proposed method is compared with that of the approximate maximum likelihood method of Ait-Sahalia (2002).Maximum likelihood, Girsnov theorem, Discrete sampling, Continuous record, Realized volatility

Jackknifing Bond Option Prices

In continuous time specifications, the prices of interest rate derivative securities depend crucially on the mean reversion parameter of the associated interest rate diffusion equation. This parameter is well known to be subject to estimation bias when standard methods like maximum likelihood (ML) are used. The estimation bias can be substantial even in very large samples and it translates into a bias in pricing bond options and other derivative securities that is important in practical work. The present paper proposes a very general method of bias reduction for pricing bond options that is based on Quenouille's (1956) jackknife. We show how the method can be applied directly to the options price itself as well as the coefficients in continuous time models. The method is implemented and evaluated here in the Cox, Ingersoll and Ross (1985) model, although it has much wider applicability. A Monte Carlo study shows that the proposed procedure achieves substantial bias reductions in pricing bond options with only mild increases in variance that do not compromise the overall gains in mean squared error. Our findings indicate that bias correction in estimation of the drift can be more important in pricing bond options than correct specification of the diffusion. Thus, even if ML or approximate ML can be used to estimate more complicated models, it still appears to be of equal or greater importance to correct for the effects on pricing bond options of bias in the estimation of the drift. An empirical application to U.S. interest rates highlights the differences between bond and option prices implied by the jackknife procedure and those implied by the standard approach. These differences are large and suggest that bias reduction in pricing options is important in practical applications.Bias Reduction, Option Pricing, Bond Pricing, Term Structure of Interest Rate, Re-sampling, Estimation of Continuous Time Models