Processus stochastiques en finance (1ère partie)

Este documento é um texto didático destinado aos estudantes e pesquisadores em econometria e finanças. Baseia-se na experiência dos autores em cursos de pós-graduação nos dois lados do Atlântico: na ULB, Bruxelles e na FGV IEPGE, Rio. Não há a pretensão de rigor matemático, e nem a de cobrir todas as aplicações financeiras da teoria dos processos estocásticos. Esta primeira parte discute as martingales e o movimento browniano, os processos de difusão e a integral estocástica, o lema de Itô e o modelo de Black e Scholes.

A note on an application of Arrow's theorem: sufficient conditions for Lucas' inflation and welfare

Bellman's methods for dynamic optimization constitute the present mainstream in economics. However, some results associated with optimal controI can be particularly usefuI in certain problems. The purpose of this note is presenting such an example. The value function derived in Lucas' (2000) shopping-time economy in Infiation and Welfare need not be concave, leading this author to develop numerical analyses to determine if consumer utility is in fact maximized along the balanced path constructed from the first order conditions. We use Arrow's generalization of Mangasarian's results in optimal control theory and develop sufficient conditions for the problem. The analytical conclusions and the previous numerical results are compatible .

A general-equilibrium closed-form solution to the welfare costs of inflation (Forthcoming, Revista Brasileira de Economia)

This work presents closed-form solutions to Lucasís (2000) generalequilibrium expression for the welfare costs of ináation, as well as to the di§erence between the general-equlibrium measure and Baileyís (1956) partial-equilibrium measure. In Lucasís original work only numerical solutions are provided.

Welfare costs of inflation: the case for interest-bearing money and empirical estimates for Brazil

We provide in this paper a closed fonn for the Welfare Cost of Inflation which we prove to be closer than Bailey's expression to the correct solution of the corresponding non-separable differential equation. Next. we extend this approach to ao economy with interest-bearing money, once again presenting a better appoximation than the one given by Bailey's approach. Fmally, empirical estimates for Brazil are presented.

An intra-household approach to the welfare costs of inflation

The literature on the welfare costs of in‡ation universally assumes that the many-person household can be treated as a single economic agent. This paper explores what the heterogeneity of the agents in a household might imply for such welfare analyses. First, we show that allowing for a single-unity or for a multi-unity transacting technology impacts the money demand function and, therefore, the welfare costs of in‡ation. Second, we derive su¢cient conditions that make the welfare assessments which depart directly from the knowledge of the money demand function (as in Lucas (2000)) robust under this alternative setting. Third, we compare our general-equilibrium measure with Bailey’s (1956) partial-equilibrium one.

A note on growth and inflation

A simple exercise on growth and inflationary financing of public expenditures is presented in this note. In a parameterized overlapping generations mode1 where government expenses positivc1y affects the growth rate of human capital, steady state capital and output increase with inflation, reproducing the so called Tobin effect. For large inflation rates, however, government authorities cannot affect real variables and there are only nominal effects. It is also shown that the optimal policy implies some inflation but not growth maximization.

Two additions to Lucas's "inflation and welfare"

This work adds to Lucas (2000) by providing analytical solutions to two problems that are solved only numerically by the author. The first part uses a theorem in control theory (Arrow' s sufficiency theorem) to provide sufficiency conditions to characterize the optimum in a shopping-time problem where the value function need not be concave. In the original paper the optimality of the first-order condition is characterized only by means of a numerical analysis. The second part of the paper provides a closed-form solution to the general-equilibrium expression of the welfare costs of inflation when the money demand is double logarithmic. This closed-form solution allows for the precise calculation of the difference between the general-equilibrium and Bailey's partial-equilibrium estimates of the welfare losses due to inflation. Again, in Lucas's original paper, the solution to the general-equilibrium-case underlying nonlinear differential equation is done only numerically, and the posterior assertion that the general-equilibrium welfare figures cannot be distinguished from those derived using Bailey's formula rely only on numerical simulations as well.

Endogenous time-dependent rules and inflation inertia: preliminary version

When policy rules are changed, the effect of nominal rigidities should be modelled through endogenous pricing rules. We endogenize Taylor (1979) type pricing rule to examine the output effects of monetary disinflations. We derive optimal fixed-price time-dependent rules in inflationary steady states and during disinflations. We also develop a methodology to aggregate individual pricing rules which vary through disinflation. This allows us to reevaluate the output costs of monetary disinflation, including aspects as the role of the initial leveI of inflation and the importance of the degree of credibility of the policy change.

Inflation and debt indexation : the equivalence of two alternative schemes for the case of periodic payments

The presence of inflation has induced the financial institutions to implement procedures devised to protect the real values of theirs loans. Two of such procedurcs, the floaaing rale scheme and the monetary correction mechanism, tend to lead to very different streams of payments. However, whenever the floating rate scheme follows the rule of Strict adhercnce to lhe Fisher equation, lhe two procedures are financially equivalent.

An intra-household approach to the welfare costs of inflation (Revised Version, Forthcoming 2006, Estudos Econômicos)

The literature on the welfare costs of ináation universally assumes that the many-person household can be treated as a single economic agent. This paper explores what the heterogeneity of the agents in a household might imply for such welfare analyses. First, we show that allowing for a one-person or for a many-person transacting technology impacts the money demand function and, therefore, the welfare costs of ináation. Second, more importantly, we derive su¢ cient conditions under which welfare assessments which depart directly from the knowledge of the money demand function (as in Lucas (2000)) are robust (invariant) under the number of persons considered in the household. Third, we show that Baileyís (1956) partial-equilibrium measure of the welfare costs of ináation can be obtained as a Örst-order approximation of the general-equilibrium welfare measure derived in this paper using a many-person transacting technology.

Essays on asymmetric information problems without the single-crossing property: applications to corporate finance

The objective of this dissertation is to re-examine classical issues in corporate finance, applying a new analytical tool. The single-crossing property, also called Spence-irrlees condition, is not required in the models developed here. This property has been a standard assumption in adverse selection and signaling models developed so far. The classical papers by Guesnerie and Laffont (1984) and Riley (1979) assume it. In the simplest case, for a consumer with a privately known taste, the single-crossing property states that the marginal utility of a good is monotone with respect to the taste. This assumption has an important consequence to the result of the model: the relationship between the private parameter and the quantity of the good assigned to the agent is monotone. While single crossing is a reasonable property for the utility of an ordinary consumer, this property is frequently absent in the objective function of the agents for more elaborate models. The lack of a characterization for the non-single crossing context has hindered the exploration of models that generate objective functions without this property. The first work that characterizes the optimal contract without the single-crossing property is Araújo and Moreira (2001a) and, for the competitive case, Araújo and Moreira (2001b). The main implication is that a partial separation of types may be observed. Two sets of disconnected types of agents may choose the same contract, in adverse selection problems, or signal with the same levei of signal, in signaling models.

A few more moments to finance theory: introducing higher moments in investment theory and econometrics

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Inflation targeting in Brazil: a Keynesian approach

O Regime de Meta de Inflação se Tornou Dominante na Formulação de Políticas dos Bancos Centrais nos Últimos 15 Anos. a Teoria Subjacente, Particularmente a Regra de Taylor, Pode ser Vista como uma Competente Generalização Desse Comportamento. de um Ponto de Vista Keynesiano, Ele Será Aceitável se Encararmos a Taxa de Juros de Equilíbrio como Apenas uma Convenção Variável e se a Combinarmos ou com uma Taxa de Câmbio ou com uma Meta de Emprego. no Caso do Brasil, Porém, Além Dessa Ressalva Teórica e da Condição do Duplo Mandato, o Regime de Metas de Inflação Enfrenta um Problema de Incoerência. esta é uma Política que se Destinava a ser Utilizada na Administração da Política Monetária, não na Mudança do Regime de Política Monetária . a Política de Metas de Inflação foi Introduzida no Brasil em 1999 como um Substituto para a Âncora Cambial, que Havia Sido Usada Desastrosamente entre 1995 e 1998. Durante Muitos Anos, o País Havia Enfrentado uma Armadilha de Alta Taxa de Juros / Taxa de Câmbio Valorizada E, Portanto, Precisava Mudar seu Regime de Política Monetária Antes de Eventualmente Adotar o Regime de Meta de Inflação. Essa Mudança, que Começou com a Flutuação de Janeiro de 1999, Deveria ter Sido Completada com Reformas Específicas (Fim da Indexação dos Serviços Públicos e dos Próprios Juros Básicos). no Entanto, em Lugar de Desenvolver uma Estratégia para Reduzir a Taxa de Juros, o Governo Continuou a Definir a Inflação como o Principal Problema a ser Enfrentado e Adotou uma Política Formal de Metas de Inflação. a Conseqüência é que Desde 1999 Essa Política se Tornou o Obstáculo que a Economia Brasileira Enfrenta para Escapar da Armadilha da Taxa de Juros

Taylor principle and inflation stability in emerging market countriesw

The goal of this paper is to evaluate the validity of the Taylor principle for inflation control in 12 developing countries that use inflation targeting regimes: Brazil, Chile, Colombia, Hungary, Israel, Mexico, Peru, Philippines, Poland, South Africa, Thailand and Turkey. The test is based on a state-space model to determine when each country has followed the principle; then a threshold unit root test is used to verify if the stationarity of the deviation of the expected inflation from its target depends on compliance with the Taylor principle. The results show that such compliance leads to the stationarity of the deviation of the expected inflation from its target in all cases. Furthermore, in most cases, non-compliance with the Taylor principle leads to nonstationary deviation of the expected inflation.