ECONOMETRIAJose F. Nieves Mendez

Balbino Garcia

Chapter 6Mathematical and Numerical Optimization

The concept of optimization is fundamental to finance theory. The seminal work of Harry Markowitz demonstrated that financial decision-making for a rational agent is essentially a question of achieving an optical trade-off between risk and returns.

From an application perspective, mathematical programming allows the rationalization of many business or technological decisions. Nevertheless, in practice, the computational tractability of the resulting analytical models is a key issue. It does not make much sense to formulate models that we are not able to solve in a reasonable timeframe.

Linear Programming (LP) refers to the problem of minimizing a linear function subject to linear equality and inequality constraints. The standard form of a linear program is given by

min c’x x

s.t. Ax=b x>0

where c is an N-dimensional vector, A is a J x N matrix, and b is a J-dimensional vector.

Linear Programming

Minimizing a quadratic objective function subject to linear equality and inequality constrain is referred to as quadratic programming (QP). This problem is represented in standard form as

min (1/2 x’Qx+c’x) x

s.t. Ax=b x>0

where Q is an NxN matrix, c is an N-dimensional vector, A is a JxN matrix, and b is a J-dimensional vector.

Quadratic Programming

Convex Programming is a large class of optimization problems that contains subclasses such as semi definite programs (SPD), second-order cones programs (SOCP), geometrics programs (GP), least squares (LS), convex quadratic programming (QS), and linear programming (LP). A convex program in standard form is given by

min f(x) x

s.t. gi(x) <0, i=1,…,1

Ax = bWhere f and giare convex functions, A is a JxN matrix, and b is a

J-dimensional vector.

Convex Programming

By replacing the nonnegative constrain in the standard form of a linear program with so-called conic inclusion constraints, we obtain the conic optimization problem

min c’x x

s.t. Ax =b x € C

Where c is an N-dimensional vector, A is a JxN matrix, b is a J-dimensional vector, and C is a closed convex cone.

Conic Optimization

So far our discussion has focused on optimization problems where the variables are continuous. When they are only allowed to take on discrete values such as binary values (0,1) or integer values (…., -2,-1,0,-1,-2…) we refer to the resulting mathematical programming problem as a combinatorial, discrete, or integer programming (IP) problem.

Integer and Combinatorial Programming

Which are often referred to as the fundamental theorem of linear programming:

If the linear program is feasible and bounded, then there is at least one optical solution. Furthermore, at least one of the optimal solution corresponds to one of the vertices of the feasible set.

If the linear program is feasible, then there is a basic feasible solution.

If the linear program has solutions, then at least one of these solutions is a basic feasible solution.

Linear Programming

In this section we describe the idea behind interior-point methods for the solutions of the convex optimization problem in standard form:

min f(x) x

s.t. gi(x) <0, i=1,….,I

Ax=bWhere f and gi are convex functions, A is a JxN

matrix, and b is a J-dimensional vector.

Barrier and Interior-Point methods

In the barrier method, the idea is to convert the general problem with both equality and inequality constraints into a sequence of equality constrained approximations, which then can be solved by the Newton Method.

A Barrier Method

It is not difficult to show that the method derived above is equivalent to applying the Newton method directly to the modified KKT equations.

A Primal-Dual Interior-Point Method

Part IIManaging Uncertainty in Practice

Chapter 7Equity Price Models

We begin our discussion of equity price models by introducing some definitions and fixing some notations. A financial time series in discrete time is a sequence of financial variables such as assets prices or returns observed at discrete points in time, for example, the end of a trading day or the last trading day of the month. Most model that we will consider in this book assume that the spacing between points is fixed, for example models of daily returns assume that returns are observed between consecutive trading days. In order to recover fixed spacing between time points due to weekends, holidays or periods when trading is suspended, a sequence of trading days difference from the sequence of calendars days is typically introduced.

Definitions

A model of returns is a mathematical representation of returns. In finance theory, different types of models are considered. There are models that represents the time evolution of returns and models that represents relationship between the return of different asset at any given moment. The former is exemplified by a random walk model, the latter by conditions of no arbitrage. The distinction is important because models that represent the time evolution of assets can be used to make probabilistic forecasts starting from initial conditions.

Theoretical and Econometric Models

The random walk model is a basic model of stock prices based on the assumption of market efficiency. The basic idea is that returns can be represented as unforecastable fluctuations around some mean return.

Random Walks Models

The distribution of prices is a product of normal distributions; it is not a normal distribution itself. This is major drawback of the geometric random walk model in discrete time. To avoid this problem, let us consider the logarithm of prices. Recall from the definitions given above that the log returns are the differences of log prices.

Lognormal Model

General Equilibrium Theories (GET) are global mathematical models of an economy. They are based on two key principles:

Supply/demand equilibrium Agent optimality

General Equilibrium Theories

The Capital Asset Pricing Model is an equilibrium asset pricing model that hinges on mean-variance portfolio selection. The CAPM is an abstraction of the real-world capital markets based on the following assumptions:

Investor make investment decisions based on the expected return and variance of returns.

Investor are rational and risk-averse. Investor subscribe to the Markowitz method of portfolio

diversification. Investors all invest for the same period of time. Investor have the same expectations about the expected return and

variance of all assets. There is a risk-free assets and investors can borrow or lend any

amount at the risk-free rate. Capital markets are (perfectly) competitive and frictionless.

Capital Asset Pricing Model (CAPM)

The arbitrage principle is perhaps the most fundamental principle in modern financial theory. Essentially it states that it is not possible to earn a risk-free return without investment. The Arbitrage Pricing Theory is a particular formulation of relative pricing theory based on the principle of absence of arbitrage. The ATP places restrictions on the prices of a set of assets.

Arbitrage Pricing Theory (ATP)