Research Seminar in Probability and Statistics II - #11

21 maio 2018, 22:40 • Manuel Cabral Morais

Estimation of the drift of a 2n-dimension OU process

Ana Prior (Instituto Superior de Engenharia de Lisboa, ISEL, Portugal)

June 12 (Tue.), 11:00 - Room P3.10 (Math. Building, IST, Av. Rovisco Pais, 1049-001 Lisboa)

A 2n-dimension Ornstein-Uhlenbeck (OU) process for which the diffusion matrix is singular is considered. This process is used as a model for the dynamic behavior of vibrating engineering structures such as bridges, buildings, dams, among others. We study the problem of estimating the vibration frequencies of the structure or, equivalently, the parameters of the stochastic differential equation (SDE) that governs the OU process.

Firstly, it is considered the case where the OU process is perturbed by an independent wiener process. The maximum likelihood estimator of the drift matrix is obtained and the properties of the estimator are established. The local asymptotic normality of the estimator is analyzed in detail. Since general regularity conditions do not hold in this case (the diffusion matrix is singular), theoretical results from the classic literature on the subject do not immediately apply and an alternative approach based on the Laplace transform is used.

Secondly, it is considered the case where the OU process is perturbed by two independent fractional brownian motions. Models involving fractional noises have not been widely used in engineering. However, many problems in engineering involve processes exhibiting long memory. For this reason, the estimation of the parameters of multidimensional state space linear models, described by SDEs and disturbed by fractional Brownian motion, has a potential application in different areas of engineering. We analyze the problem of estimating the drift parameters of a 2- dimension linear stochastic differential equation perturbed by two independent fractional Brownian motions with the same Hurst parameter belonging to (1/2,1). The maximum likelihood estimator of the drift parameters is obtained after a transformation of the original model and making use of the so called fundamental martingale.

In both cases, a simulation study is presented in the context of a real world situation that illustrates the asymptotic behavior of the maximum likelihood estimator of the drift matrix.

https://math.tecnico.ulisboa.pt/seminars/pe/index?action=planned

Research Seminar in Probability and Statistics II - #10

21 maio 2018, 21:31 • Manuel Cabral Morais

Multiple-valued symbolic data clustering: heuristic and model-based approaches

José G. Dias (Instituto Universitário de Lisboa (ISCTE-IUL), BRU-IUL, Lisboa, Portugal)

May 29 (Tue.), 11:00 - Room P3.10 (Math. Building, IST, Av. Rovisco Pais, 1049-001 Lisboa)

Symbolic data analysis (SDA) has been developed as an extension of the data analysis to handle more complex data structures. In this general framework the pair observation/variable is characterized by more than one value: from two (e.g., interval-value data defined by minimum and maximum values) to multiple-valued variables (e.g., frequencies or proportions). 

This research discusses the clustering of multiple-valued symbolic data. First, we discuss an extension of heuristic clustering based on the symmetric Kullback-Leibler distance combined with a complete-linkage rule within the hierarchical clustering framework. Then, we propose a new model-based clustering framework. These new family of models based on the Dirichlet distribution includes mixture of regression/expert models. Results are illustrated with synthetic and demographic (population pyramids) data.

https://math.tecnico.ulisboa.pt/seminars/pe/index?action=planned

Research Seminar in Probability and Statistics II - #9

29 abril 2018, 13:36 • Manuel Cabral Morais

Market Risk Measurement - Theory and Practice

Mirela Predescu (BNP Paribas - Risk Analytics and Modelling Team, London)

May 22 (Tue.), 11:00 - Room P3.10 (Math. Building, IST, Av. Rovisco Pais, 1049-001 Lisboa)

Topics that will be covered in this talk

- Value-at-Risk (VaR)

- Expected Shortfall (ES)

- VaR/ES Measurement

- Historical Simulation

- Model Building Approach

- Monte Carlo Simulation Approach

- VaR Backtesting

https://math.tecnico.ulisboa.pt/seminars/pe/index?action=planned

Research Seminar in Probability and Statistics II - #8

11 abril 2018, 08:16 • Manuel Cabral Morais

Evaluation of volatility models for forecasting Value-at-Risk and Expected Shortfall in the Portuguese Stock Market

Nuno Sobreira (Department of Mathematics, Lisbon School of Economics & Management, Universidade de Lisboa)

May 8 (Tue.), 11:00 - Room P3.10 (Math. Building, IST, Av. Rovisco Pais, 1049-001 Lisboa)

The objective of this paper is to run a forecasting competition of different parametric volatility time series models to estimate Value-at-Risk (VaR) and Expected Shortfall (ES) within the Portuguese Stock Market. This work is also intended to bring new insights about the methods used throughout this exercise. Finally, we want to relate the timing of the exceptions (extreme losses surpassing the VaR) with events at the firm level and with national/international economic conditions.

For these purposes, a number of models from the General Autoregressive Conditional Heteroscedasticity (GARCH) class are used with different distribution functions for the innovations, in particular, Normal, Student-t and Generalized Error Distribution (GED) and corresponding skewed versions. The GARCH models are also used in conjunction with the Generalized Pareto Distribution through the use of extreme value theory.

The performance of these different models to forecast 1% and 5% VaR and ES for 1-day, 5-days and 10-days horizons are analyzed for a set of companies traded in the EURONEXT Lisbon stock exchange. The results obtained for the VaRs and ESs are evaluated with backtesting procedures based on a number of statistical tests and compared with the use of different loss functions.

The final results are analyzed in several dimensions. Preliminary analysis show that the use of extreme value theory generally leads to better results, especially for low values of alpha. This is more evident in the case of the statistical backtests dealing with ES. Moreover, skewed distributions generally do not seem to perform better than their centered counterparts.

http://math.tecnico.ulisboa.pt/seminars/pe/

Research Seminar in Probability and Statistics II - #7

10 abril 2018, 14:22 • Manuel Cabral Morais

Dynamic Capital Structure Choice and Investment Timing

Peter M. Kort (Department of Econometrics & Operations Research and CentER, Tilburg University; Department of Economics, University of Antwerp)

Apr. 24 (Tue.), 11:00 - Room P3.10 (Math. Building, IST, Av. Rovisco Pais, 1049-001 Lisboa)

The paper considers the problem of an investor that has the option to acquire a firm. Initially this firm is run as to maximize shareholder value, where the shareholders are risk averse. To do so it has to decide each time on investment and dividend levels. The firm's capital stock can be financed by equity and debt, where less solvable firms pay a higher interest rate on debt. Revenue is stochastic.

We find that the firm is run such that capital stock and dividends develop in a fixed proportion to the equity. In particular, it turns out that more dividends are paid if the economic environment is more uncertain. We also derive an explicit expression for the threshold value of the equity above which it is optimal for the investor to acquire the firm. This threshold increases in the level of uncertainty reflecting the value of waiting that uncertainty generates.

Joint work with Engelbert J. Dockner (deceased) and Richard F. Hartl.

http://math.tecnico.ulisboa.pt/seminars/pe/