Relative Robust Portfolio Optimization

Considering mean-variance portfolio problems with uncertain model parameters, we contrast the classical absolute robust optimization approach with the relative robust approach based on a maximum regret function. Although the latter problems are NP-hard in general, we show that tractable inner and outer approximations exist in several cases that are of central interest in asset management

Dynamic evolution for risk-neutral densities

Option price data is often used to infer risk-neutral densities for future prices of an underlying asset. Given the prices of a set of options on the same underlying asset with different strikes and maturities, we propose a nonparametric approach for estimating the evolution of the risk-neutral density in time. Our method uses bicubic splines in order to achieve the desired smoothness for the estimation and an optimization model to choose the spline functions that best fit the price data. Semidefinite programming is employed to guarantee the nonnegativity of the densities. We illustrate the process using synthetic option price data generated using log-normal and absolute diffusion processes as well as actual price data for options on the S&P500 index. We also used the risk-neutral densities that we computed to price exotic options and observed that this approach generates prices that closely approximate the market prices of these options.FCT POCI/MAT/59442/2004, PTDC/MAT/64838/2006

Portfolio Optimization Using SPEA2 with Resampling

Proceeding of: Intelligent Data Engineering and Automated Learning – IDEAL 2011: 12th International Conference, Norwich, UK, September 7-9, 2011The subject of financial portfolio optimization under real-world constraints is a difficult problem that can be tackled using multiobjective evolutionary algorithms. One of the most problematic issues is the dependence of the results on the estimates for a set of parameters, that is, the robustness of solutions. These estimates are often inaccurate and this may result on solutions that, in theory, offered an appropriate risk/return balance and, in practice, resulted being very poor. In this paper we suggest that using a resampling mechanism may filter out the most unstable. We test this idea on real data using SPEA2 as optimization algorithm and the results show that the use of resampling increases significantly the reliability of the resulting portfolios.The authors acknowledge financial support granted by the Spanish Ministry of Science under contract TIN2008-06491-C04-03 (MSTAR) and Comunidad de Madrid (CCG10- UC3M/TIC-5029).Publicad

A note on calculating the optimal risky portfolio

Given a number of risky assets and a riskless asset, the set of efficient portfolios in the mean-variance optimization sense are combinations of the riskless asset and a unique optimal risky portfolio. This note shows how a simple modification of Markowitz' method of critical lines can be used to determine the optimal risky portfolio in a faster, more reliable, and more memory-efficient way than the standard approaches.Mean-variance optimization, optimal risky portfolio

Continuous Trajectories for Primal-Dual Potential-Reduction Methods

This article considers continuous trajectories of the vector fields induced by two different primal-dual potential-reduction algorithms for solving linear programming problems. For both algorithms, it is shown that the associated continuous trajectories include the central path and the duality gap converges to zero along all these trajectories. For the algorithm of Kojima, Mizuno, and Yoshise, there is a a surprisingly simple characterization of the associated trajectories. Using this characterization, it is shown that all associated trajectories converge to the analytic center of the primal-dual optimal face. Depending on the value of the potential function parameter, this convergence may be tangential to the central path, tangential to the optimal face, or in between. Key words: linear programming, potential functions, potential-reduction methods, central path, continuous trajectories for linear programming. AMS subject classification: 90C05

Dynamic evolution for risk-neutral densities

Option price data is often used to infer risk-neutral densities for future prices of an underlying asset. Given the prices of a set of options on the same underlying asset with different strikes and maturities, we propose a nonparametric approach for estimating the evolution of the risk-neutral density in time. Our method uses bicubic splines in order to achieve the desired smoothness for the estimation and an optimization model to choose the spline functions that best fit the price data. Semidefinite programming is employed to guarantee the nonnegativity of the densities. We illustrate the process using synthetic option price data generated using log-normal and absolute diffusion processes as well as actual price data for options on the S&P500 index. We also used the risk-neutral densities that we computed to price exotic options and observed that this approach generates prices that closely approximate the market prices of these options.FCT POCI/MAT/59442/2004, PTDC/MAT/64838/2006

Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity

We present a new approach to estimate the risk-neutral probability density function (pdf) of the future prices of an underlying asset from the prices of options written on the asset. The estimation is carried out in the space of cubic spline functions, yielding appropriate smoothness. The resulting optimization problem, used to invert the data and determine the corresponding density function, is a convex quadratic or semidefinite programming problem, depending on the formulation. Both of these problems can be efficiently solved by numerical optimization software.http://www.sciencedirect.com/science/article/B6VCT-4NC4M8X-2/1/fed44dd574a24dbc1e9cd48bdb8185b