Optimizing a Portfolio of Equities, Equity Futures and Equity European Options by Minimizing Value-at-Risk - a Simulated Annealing Framework

Value-at-Risk, defined as a given quantile of loss distribution of a portfolio, is minimized to obtain optimal allocation of a unit capital into different elements of the portfolio. The considered portfolio includes equities, equity futures, and European options on the equities, which made the return and thus loss non-linear, requiring numerical techniques for optimization. The loss distribution for a given composition of a portfolio was computed using monte-carlo simulation, with the options being valued in a Blackscholes framework, while the optimization of this composition was performed using simulated annealing, with each iteration involving generation of random vectors from a Dirichlet distribution with a mean at the current step and variance controlled by the shape factor of the underlying gamma random variables. The algorithm was compared with a brute force method for a three element portfolio and the performance was found to be good. Code was implemented in R