Sebastian Ferrando

Probabilistic matching pursuit with Gabor dictionaries

Abstract We propose a probabilistic extension of the matching pursuit adaptive signal processing algorithm introduced by Mallat and others. In adaptive signal processing, signals are expanded in terms of a large linearly dependent “dictionary” of functions rather than in terms of an orthonormal basis. Matching pursuit is a simple greedy algorithm for generating an expansion of a given signal. In probabilistic matching pursuit multiple random expansions are obtained as estimates for a given signal. The new algorithm is illustrated in the context of signal denoising. Although most of the random expansions generated by probabilistic matching pursuit are poorer estimates for the signal than those obtained by matching pursuit, our final estimate, obtained as an expected value computed by means of an ergodic average, can improve the result obtained by MP in some denoising situations. One of the major underlying ideas is a novel notion of coherence between a signal and the dictionary. Several simulated examples are presented.

Evaluating exact VARMA likelihood and its gradient when data are incomplete

A detailed description of an algorithm for the evaluation and differentiation of the likelihood function for VARMA processes in the general case of missing values is presented. The method is based on combining the Cholesky decomposition method for complete data VARMA evaluation and the Sherman-Morrison-Woodbury formula. Potential saving for pure VAR processes is discussed and formulae for the estimation of missing values and shocks are provided. A theorem on the determinant of a low rank update is proved. Matlab implementation of the algorithm is in a companion article.

Three dimensional distribution of Brownian motion extrema

This paper describes the joint distributions of minima, maxima and endpoint values for a three dimensional (3D) Wiener process. In particular, the results provide the joint cumulative distributions for the maxima and/or minima of the components of the process. The method of images is used to derive explicit expressions for the densities; the analysis can only be carried out for special correlation structures and requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used in several applied fields such as financial mathematics to obtain analytical expressions for prices of options for the 3D geometric Brownian motion process.

Trajectory-Based Models, Arbitrage And Continuity

In a nonprobabilistic setting, we prove general trajectory-based models to have no free lunch with vanishing risk. The main ingredient is a local continuity requirement on the final portfolio value considered as a functional on the trajectory space. This is shown to be a natural assumption by establishing that a large class of practical trading strategies, defined by means of trajectory-based stopping times, give rise to locally continuous functionals. The theory is applied to two specific trajectory models of practical interest. The established results are then used to derive no free lunch results for nonsemimartingale stochastic models.

Cell recognition using wavelet templates

The paper describes an algorithm to count and classify cells of different geometrical shapes on a given image. The algorithm assumes that it is known a priori the type of geometries to be recognized and it allows for many different geometrical shapes to appear in the same image with different sizes, locations and orientations. The algorithm combines classical tools, mainly the two dimensional Fourier transform, with newly developed tools for edge enhancements as well as the main technical contribution of the present paper, which consists in the definition of an over-complete set of spanning functions. These functions are constructed from geometrical templates of size comparable to the image cells; moreover, the resulting functions are scaled and rotated to assure the recognition of all image cells. We then describe an algorithm that decomposes the image in its most likely elements. The combination of ingredients used by the algorithm provides a cell recognition tool that is very robust, provides high resolution to discern among competing candidate cells and delivers practical computational efficiency.

Discrete, Non Probabilistic Market Models. Arbitrage and Pricing Intervals

The paper develops general, discrete, non-probabilistic market models and minmax price bounds leading to price intervals for European options. The approach provides the trajectory based analogue of martingale-like properties as well as a generalization that allows a limited notion of arbitrage in the market while still providing coherent option prices. Several properties of the price bounds are obtained, in particular a connection with risk neutral pricing is established for trajectory markets associated to a continuous-time martingale model.

Optimal Investment Under Multi-Factor Stochastic Volatility

We consider a model for multivariate intertemporal portfolio choice in complete and incomplete markets with a multi-factor stochastic covariance matrix of asset returns. The optimal investment strategies are derived in closed form. We estimate the model parameters and illustrate the optimal investment based on two stock indices: S&P500 and DAX. It is also shown that the model satisfies several stylized facts well known in the literature. We analyse the welfare losses due to suboptimal investment strategies and we find that investors who invest myopically, ignore derivative assets, model volatility by one factor and ignore stochastic covariance between asset returns can incur significant welfare losses.

Trajectory based models. Evaluation of minmax pricing bounds

The paper studies sub and super-replication price bounds for contingent claims defined on general trajectory based market models. No prior probabilistic or topological assumptions are placed on the trajectory space, trading is assumed to take place at a finite number of occasions but not bounded in number nor necessarily equally spaced in time. For a given option, there exists an interval bounding the set of possible fair prices; such interval exists under more general conditions than the usual no-arbitrage requirement. The paper develops a backward recursive method to evaluate the option bounds; the global minmax optimization, defining the price interval, is reduced to a local minmax optimization via dynamic programming. Trajectory sets are introduced for which existing non-probabilistic markets models are nested as a particular case. Several examples are presented, the effect of the presence of arbitrage on the price bounds is illustrated.

Barrier options in three dimensions

The paper provides closed-form expressions for the price of several barrier type derivatives with a three-dimensional geometric Wiener process as underlying. These solutions are found for special correlation matrices and are given by linear combinations of three-dimensional Gaussian cumulative distributions. The method of images is used as a key technique to establish the solutions. Two cases are described extending the results to a wider set of correlation matrices, one case deals with random variances and the other case with random correlations.

LOCALIZED MONTE CARLO ALGORITHM TO COMPUTE PRICES OF PATH DEPENDENT OPTIONS ON TREES

A new simulation based algorithm to approximate prices of path dependent European options is introduced. The algorithm is defined for tree-like approximations to the underlying process and makes extensive use of structural properties of the discrete approximation. We indicate the advantages of the new algorithm in comparison to standard Monte Carlo algorithms. In particular, we prove a probabilistic error bound that compares the quality of both approximations. The algorithm is of general applicability and, for a large class of options, it has the same computational complexity as Monte Carlo.