B-spline techniques for volatility modeling

This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first come back to the application of shape-constrained B-splines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. An application is the Monte Carlo calibration of stochastic local volatility models by Markov projection. Then we present a new technique for the calibration of an implied volatility surface to sparse option data. We use a B-spline parameterization of the Radon-Nikodym derivative of the underlying's risk-neutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch a Galerkin method with B-spline finite elements to the solution of the partial differential equation satisfied by the Radon-Nikodym derivative.Comment: 25 page

Moody's Correlated Binomial Default Distributions for Inhomogeneous Portfolios

This paper generalizes Moody's correlated binomial default distribution for homogeneous (exchangeable) credit portfolio, which is introduced by Witt, to the case of inhomogeneous portfolios. As inhomogeneous portfolios, we consider two cases. In the first case, we treat a portfolio whose assets have uniform default correlation and non-uniform default probabilities. We obtain the default probability distribution and study the effect of the inhomogeneity on it. The second case corresponds to a portfolio with inhomogeneous default correlation. Assets are categorized in several different sectors and the inter-sector and intra-sector correlations are not the same. We construct the joint default probabilities and obtain the default probability distribution. We show that as the number of assets in each sector decreases, inter-sector correlation becomes more important than intra-sector correlation. We study the maximum values of the inter-sector default correlation. Our generalization method can be applied to any correlated binomial default distribution model which has explicit relations to the conditional default probabilities or conditional default correlations, e.g. Credit Risk${}^{+}$, implied default distributions. We also compare some popular CDO pricing models from the viewpoint of the range of the implied tranche correlation.Comment: 29 pages, 17 figures and 1 tabl

Bayesian emulation for optimization in multi-step portfolio decisions

We discuss the Bayesian emulation approach to computational solution of multi-step portfolio studies in financial time series. "Bayesian emulation for decisions" involves mapping the technical structure of a decision analysis problem to that of Bayesian inference in a purely synthetic "emulating" statistical model. This provides access to standard posterior analytic, simulation and optimization methods that yield indirect solutions of the decision problem. We develop this in time series portfolio analysis using classes of economically and psychologically relevant multi-step ahead portfolio utility functions. Studies with multivariate currency, commodity and stock index time series illustrate the approach and show some of the practical utility and benefits of the Bayesian emulation methodology.Comment: 24 pages, 7 figures, 2 table

SDSS Standard Star Catalog for Stripe 82: the Dawn of Industrial 1% Optical Photometry

We describe a standard star catalog constructed using multiple SDSS photometric observations (at least four per band, with a median of ten) in the $ugriz$ system. The catalog includes 1.01 million non-variable unresolved objects from the equatorial stripe 82 ($|\delta_{J2000}|<$ 1.266$^\circ$) in the RA range 20h 34m to 4h 00m, and with the corresponding $r$ band (approximately Johnson V band) magnitudes in the range 14--22. The distributions of measurements for individual sources demonstrate that the photometric pipeline correctly estimates random photometric errors, which are below 0.01 mag for stars brighter than (19.5, 20.5, 20.5, 20, 18.5) in $ugriz$, respectively (about twice as good as for individual SDSS runs). Several independent tests of the internal consistency suggest that the spatial variation of photometric zeropoints is not larger than $\sim$0.01 mag (rms). In addition to being the largest available dataset with optical photometry internally consistent at the $\sim$1% level, this catalog provides practical definition of the SDSS photometric system. Using this catalog, we show that photometric zeropoints for SDSS observing runs can be calibrated within nominal uncertainty of 2% even for data obtained through 1 mag thick clouds, and demonstrate the existence of He and H white dwarf sequences using photometric data alone. Based on the properties of this catalog, we conclude that upcoming large-scale optical surveys such as the Large Synoptic Survey Telescope will be capable of delivering robust 1% photometry for billions of sources.Comment: 63 pages, 24 figures, submitted to AJ, version with correct figures and catalog available from http://www.astro.washington.edu/ivezic/sdss/catalogs/stripe82.htm