Pedro Mota

In this paper we construct a new set of basis functions for the numerical solution of nonhomogeneous heat conduction problems with Dirichlet boundary conditions and null initial data. These functions can be seen as Newtonian potentials of plane waves for the heat equation and satisfy a null initial condition. Density results for adapted waves will be established and several numerical simulations will be presented in order to discuss the accuracy and feasibility of the proposed method. An application of the method for heat problems with non null initial temperature will also be discussed.

Motivated by the need to describe regime switching in stock prices, we introduce and study a stochastic process in continuous time with two regimes and one threshold driving the change in regimes. When the difference between the regimes is simply given by different sets of real-valued parameters for the drift and diffusion coefficients, we show that there are consistent estimators for the threshold as long as we know how to classify a given observation of the process as belonging to one of the two regimes.

The list of approximations to the Normal cumulative distribution function is long and, eventually, not fully known due to the large number of published articles in the last decades. In this paper we will present new improvements in some well known approximations, without increasing the complexity of the formulas.

Motivated by the need to describe bear-bull market regime switching in stock prices, we introduce and study a stochastic process in continuous time with two regimes, threshold and delay, given by a stochastic differential equation. When the difference between the regimes is simply given by a different set of real valued parameters for the drift and diffusion coefficients, with changes between regimes depending only on these parameters, we show that if the delay is known there are consistent estimators for the threshold as long we know how to classify a given observation of the process as belonging to one of the two regimes. When the drift and diffusion coefficients are of geometric Brownian motion type we obtain a model with parameters that can be estimated in a satisfactory way, a model that allows differentiating regimes in some of the NYSE 21 stocks analyzed and also, that gives very satisfactory results when compared to the usual Black–Scholes model for pricing call options.

The geometric Brownian motion (GBM) is very popular in modeling the dynamics of stock prices. However, the constant volatility assumption is questionable and many models with nonconstant volatility have been developed. In the papers [7,12] the authors introduce a regime switching process where in each regime the process is driven by GBM and the change in regime is defined by the crossing of a threshold. In this paper we used Akaike's and Bayesian information criteria to show that the GBM with regimes provides a better fit than the GBM. We also perform a forecasting comparison of the models for two selected companies.

The normality of the log-returns for the price of the stocks is one of the most important assumptions in mathematical finance. Usually is assumed that the price dynamics of the stocks are driven by geometric Brownian motion and, in that case, the log-return of the prices are independent and normally distributed. For instance, for the Black-Scholes model and for the Black-Scholes pricing formula [4] this is one of the main assumptions. In this paper we will investigate if this assumption is verified in the real world, that is, for a large number of company stock prices we will test the normality assumption for the log-return of their prices. We will apply the KolmogorovSmirnov [10, 5], the Shapiro-Wilks [17, 16] and the Anderson-Darling [1, 2] tests for normality to a wide number of company prices from companies quoted in the Nasdaq composite index.

Regime switching diffusion processes with one or two thresholds and regime switching occurring by a change in the diffusion drift and/or volatility functions parameters of a stochastic differential equation, whose solution defines a continuous time diffusion process, were defined in previous works; the change in regime occurring whenever the trajectory of the process crosses a threshold, possibly with some delay. In this paper we generalise the previous
results by allowing the underlying diffusion process to change from one family of diffusions in one regime to an entirely different one in the other regime; these families of diffusions are characterised by specific functional forms for drift and volatility coefficients depending on parameters. We propose an estimation procedure for all the parameters, namely the thresholds, the delay and, for both regimes, diffusion’s parameters and we apply the introduced estimation procedure to both simulated and real data.

When (Xt)t≥0 is an ergodic process, the density function of Xt converges to some invariant density as t →∞. We will compute and study some asymptotic properties of pseudo moments estimators obtained from this invariant density, for a specific class of ergodic processes. In this class of processes we can find the Cox-Ingersoll & Ross or Dixit & Pindyck processes, among others. A comparative study of the proposed estimators with the usual estimators obtained from discrete approximations of the likelihood function will be carried out.

This paper inspects how risk is affected by news sign and size within - depreciation, appreciation, stability - distinct exchange rate trends, and by volatility model choice, taking on various asymmetric Generalized Autoregressive Conditional Heteroskedasticity models to daily Mozambique New Metical against South Africa Rand, MZN/ZAR, exchange rate over January 2010 - December 2014. Our results show that risk measurement
and asymmetry of shocks to volatility depend on exchange rate trend, being that estimating the full sample conceals the actual behavior, and model choice, specifically the degree of nonlinearity and persistence. In particular, we find that when positive/negative news type matches the sign of the exchange rate trend, risk increases by more. Interestingly, this means that in times of appreciation the good news turns out to be bad,
likely because they raise the fear of overvaluation, under monitoring in natural resource producers and exporting countries. The findings contribute to the growing concern on nonlinear economic policy design, exchange rate targeting and surely international trade and investment decisions, where an incorrect assessment exchange rate risk and asymmetry may lead to mispricing of assets, namely options, and eventual underestimation of
measures, as Value at Risk, relevant for Basel agreement.

The normality of the log-return of stock prices is often assumed by the market players in order to use some useful results, as for instance, the Black-Scholes formula for pricing European options. However, several studies regarding different indexes have shown that the normality assumption of the returns usually fails.
In this paper we analyse the normality assumption for intra-day and inter-day log-returns, comparing opening prices and/or closing prices for a large number of companies quoted in the Nasdaq Composite index. We use the Pearson's Chi-Square, Kolmogorov-Smirnov, Anderson-Darling, Shapiro-Wilks and Jarque-Bera goodness-of-fit tests to study the normality assumption.
We find that the failure rate in the normality assumption for the log-return of stock prices is not the same for intra-day and inter-day prices, is somewhat test dependent and strongly dependent on some extreme price observations.
To the best of our knowledge, this is the first study on the normality assumption for the log-return of stock prices dealing simultaneously with a large number of companies and normality tests, and at the same time considering various scenarios of intra-day, inter-day prices and data trimming.