18,981 research outputs found
Nonlinear Parabolic Equations arising in Mathematical Finance
This survey paper is focused on qualitative and numerical analyses of fully
nonlinear partial differential equations of parabolic type arising in financial
mathematics. The main purpose is to review various non-linear extensions of the
classical Black-Scholes theory for pricing financial instruments, as well as
models of stochastic dynamic portfolio optimization leading to the
Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both
problems can be represented by solutions to nonlinear parabolic equations.
Qualitative analysis will be focused on issues concerning the existence and
uniqueness of solutions. In the numerical part we discuss a stable
finite-volume and finite difference schemes for solving fully nonlinear
parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387
Portfolio selection problems in practice: a comparison between linear and quadratic optimization models
Several portfolio selection models take into account practical limitations on
the number of assets to include and on their weights in the portfolio. We
present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset
Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional
Value-at-Risk (LACVaR) models, where the assets are limited with the
introduction of quantity and cardinality constraints. We propose a completely
new approach for solving the LAM model, based on reformulation as a Standard
Quadratic Program and on some recent theoretical results. With this approach we
obtain optimal solutions both for some well-known financial data sets used by
several other authors, and for some unsolved large size portfolio problems. We
also test our method on five new data sets involving real-world capital market
indices from major stock markets. Our computational experience shows that,
rather unexpectedly, it is easier to solve the quadratic LAM model with our
algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of
the best commercial codes for mixed integer linear programming (MILP) problems.
Finally, on the new data sets we have also compared, using out-of-sample
analysis, the performance of the portfolios obtained by the Limited Asset
models with the performance provided by the unconstrained models and with that
of the official capital market indices
Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem
In this paper we propose and analyze a method based on the Riccati
transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation
arising from the stochastic dynamic optimal allocation problem. We show how the
fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a
quasi-linear parabolic equation whose diffusion function is obtained as the
value function of certain parametric convex optimization problem. Although the
diffusion function need not be sufficiently smooth, we are able to prove
existence, uniqueness and derive useful bounds of classical H\"older smooth
solutions. We furthermore construct a fully implicit iterative numerical scheme
based on finite volume approximation of the governing equation. A numerical
solution is compared to a semi-explicit traveling wave solution by means of the
convergence ratio of the method. We compute optimal strategies for a portfolio
investment problem motivated by the German DAX 30 Index as an example of
application of the method
Using Column Generation to Solve Extensions to the Markowitz Model
We introduce a solution scheme for portfolio optimization problems with
cardinality constraints. Typical portfolio optimization problems are extensions
of the classical Markowitz mean-variance portfolio optimization model. We solve
such type of problems using a method similar to column generation. In this
scheme, the original problem is restricted to a subset of the assets resulting
in a master convex quadratic problem. Then the dual information of the master
problem is used in a sub-problem to propose more assets to consider. We also
consider other extensions to the Markowitz model to diversify the portfolio
selection within the given intervals for active weights.Comment: 16 pages, 3 figures, 2 tables, 1 pseudocod
Mean-risk models using two risk measures: A multi-objective approach
This paper proposes a model for portfolio optimisation, in which distributions are characterised and compared on the basis of three statistics: the expected value, the variance and the CVaR at a specified confidence level. The problem is multi-objective and transformed into a single objective problem in which variance is minimised while constraints are imposed on the expected value and CVaR. In the case of discrete random variables, the problem is a quadratic program. The mean-variance (mean-CVaR) efficient solutions that are not dominated with respect to CVaR (variance) are particular efficient solutions of the proposed model. In addition, the model has efficient solutions that are discarded by both mean-variance and mean-CVaR models, although they may improve the return distribution. The model is tested on real data drawn from the FTSE 100 index. An analysis of the return distribution of the chosen portfolios is presented
Theory and Applications of Robust Optimization
In this paper we survey the primary research, both theoretical and applied,
in the area of Robust Optimization (RO). Our focus is on the computational
attractiveness of RO approaches, as well as the modeling power and broad
applicability of the methodology. In addition to surveying prominent
theoretical results of RO, we also present some recent results linking RO to
adaptable models for multi-stage decision-making problems. Finally, we
highlight applications of RO across a wide spectrum of domains, including
finance, statistics, learning, and various areas of engineering.Comment: 50 page
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