Tracking error with minimum guarantee constraints

In recent years the popularity of indexing has greatly increased in financial markets and many different families of products have been introduced. Often these products also have a minimum guarantee in the form of a minimum rate of return at specified dates or a minimum level of wealth at the end of the horizon. Period of declining stock market returns together with low interest rate levels on Treasury bonds make it more difficult to meet these liabilities. We formulate a dynamic asset allocation problem which takes into account the conflicting objectives of a minimum guaranteed return and of an upside capture of the risky asset returns. To combine these goals we formulate a double tracking error problem using asymmetric tracking error measures in the multistage stochastic programming framework.Minimum guarantee, benchmark, tracking error, dynamic asset allocation, scenario

Tracking Error: a multistage portfolio model

We study multistage tracking error problems. Different tracking error measures, commonly used in static models, are discussed as well as some problems which arise when we move from static to dynamic models. We are interested in dynamically replicating a benchmark using only a small subset of assets, considering transaction costs due to rebalancing and introducing a liquidity component in the portfolio. We formulate and solve a multistage tracking error model in a stochastic programming framework. We numerically test our model by dynamically replicating the MSCI Euro index. We consider an increasing number of scenarios and assets and show the superior performance of the dynamically optimized tracking portfolio over static strategies.

Time and nodal decomposition with implicit non-anticipativity constraints in dynamic portfolio optimization

We propose a decomposition method for the solution of a dynamic portfolio optimization problem which fits the formulation of a multistage stochastic programming problem. The method allows to obtain time and nodal decomposition of the problem in its arborescent formulation applying a discrete version of Pontryagin Maximum Principle. The solution of the decomposed problems is coordinated through a fixed- point weighted iterative scheme. The introduction of an optimization step in the choice of the weights at each iteration allows to solve the original problem in a very efficient way.Stochastic programming, Discrete time optimal control problem, Iterative scheme, Portfolio optimization

Combining stochastic programming and optimal control to solve multistage stochastic optimization problems

In this contribution we propose an approach to solve a multistage stochastic programming problem which allows us to obtain a time and nodal decomposition of the original problem. This double decomposition is achieved applying a discrete time optimal control formulation to the original stochastic programming problem in arborescent form. Combining the arborescent formulation of the problem with the point of view of the optimal control theory naturally gives as a first result the time decomposability of the optimality conditions, which can be organized according to the terminology and structure of a discrete time optimal control problem into the systems of equation for the state and adjoint variables dynamics and the optimality conditions for the generalized Hamiltonian. Moreover these conditions, due to the arborescent formulation of the stochastic programming problem, further decompose with respect to the nodes in the event tree. The optimal solution is obtained by solving small decomposed subproblems and using a mean valued fixed-point iterative scheme to combine them. To enhance the convergence we suggest an optimization step where the weights are chosen in an optimal way at each iteration.Stochastic programming, discrete time control problem, decomposition methods, iterative scheme

MATHEMATICAL METHODS IN ECONOMICS AND FINANCE

Mathematical Methods in Economics and Finance is a journal published by the Ca' Foscari Department of Economics since 2012, and formerly published by the Department of Applied Mathematics of the same University from 2006 to 2011. This journal replaces the former Rendiconti, a series in Italian issued annually from 1969 to 2005. The main features of the journal are: 1. Publication of original and unpublished papers that present theoretical results, methodological contributions, and applications in the areas of actuarial mathematics, financial mathematics, management science, mathematical economics, quantitative finance, and operational research. 2. Peer review process based on double-blind refereeing by at least two anonymous referees. 3. Inclusion in the MathSciNet list of journals. 4. Published papers online are free to access and download

Combining stochastic programming and optimal control to decompose multistage stochastic optimization problems

The paper suggests a possible cooperation between stochastic programming and optimal control for the solution of multistage stochastic optimization problems. We propose a decomposition approach for a class of multistage stochastic programming problems in arborescent form (i.e. formulated with implicit non-anticipativity constraints on a scenario tree). The objective function of the problem can be either linear or nonlinear, while we require that the constraints are linear and involve only variables from two adjacent periods (current and lag 1). The approach is built on the following steps. First, reformulate the stochastic programming problem into an optimal control one. Second, apply a discrete version of Pontryagin maximum principle to obtain optimality conditions. Third, discuss and rearrange these conditions to obtain a decomposition that acts both at a time stage level and at a nodal level. To obtain the solution of the original problem we aggregate the solutions of subproblems through an enhanced mean valued fixed point iterative scheme.The paper suggests a possible cooperation between stochastic programming and optimal control for the solution of multistage stochastic optimization problems. We propose a decomposition approach for a class of multistage stochastic programming problems in arborescent form (i.e. formulated with implicit non-anticipativity constraints on a scenario tree). The objective function of the problem can be either linear or nonlinear, while we require that the constraints are linear and involve only variables from two adjacent periods (current and lag 1). The approach is built on the following steps. First, reformulate the stochastic programming problem into an optimal control one. Second, apply a discrete version of Pontryagin maximum principle to obtain optimality conditions. Third, discuss and rearrange these conditions to obtain a decomposition that acts both at a time stage level and at a nodal level. To obtain the solution of the original problem we aggregate the solutions of subproblems through an enhanced mean valued fixed point iterative schem

MINIMIZING A FUZZY FUNCTION

The present paper considers the unconstrained optimization of a continuous fuzzyn function with respect to fuzzy numbers of the L-R type. We extend some methods of solution of non-fuzzy parametric mathematical programming to the fuzzy case