Optimization with generalized deviation measures in risk management

Our work provides an overview of the so-called generalized deviation measures and generalized risk measures, and develops stochastic optimization approaches utilizing them. These measures are designed to quantify risk when implied distributions are known. We provide useful examples of deviation and risk measures, which can be efficiently applied in situations, when the classical measures either do not properly account for risk, or do not satisfy properties desired for efficient application in stochastic optimization. We discuss the importance of considering alternative risk and deviation measures in the classical models, such as the capital asset pricing model and quantile regression. We apply stochastic optimization and risk management techniques based on the conditional value-at-risk (CVaR) to solve a dynamic sensor scheduling problem with robustness constraints on a wireless connectivity network. We also develop an efficient application of the generalized Capital Asset Pricing Model based on mixed CVaR deviation to estimating risk preferences of investors using S&P500 stock index option prices.;In the first part we provide an overview of the main classes of generalized deviation measures and corresponding risk measures, and compare them to the classical risk and deviation measures, such as maximum risk, value-at-risk and standard deviation. In addition, we provide a relation between deviation measures and measures of error, which are used in regression models. In some applications, such as simulation, a distribution of the residual term has to be specified. We apply the entropy maximization principle to identify the appropriate distribution for the quantile regression (factor) model.;In the second part we consider several classes of problems that deal with optimizing the performance of dynamic sensor networks used for area surveillance, in particular, in the presence of uncertainty. The overall efficiency of a sensor network is addressed from the aspects of minimizing the overall information losses, as well as ensuring that all nodes in a network form a robust connectivity pattern at every time moment, which would enable the sensors to communicate and exchange information in uncertain and adverse environments. The considered problems are solved using mathematical programming techniques that incorporate CVaR, which allows one to minimize or bound the losses associated with potential risks. The issue of robust connectivity is addressed by imposing explicit restrictions on the shortest path length between all pairs of sensors and on the number of connections for each sensor (i.e., node degrees) in a network. Specific formulations of linear 0-1 optimization problems and the corresponding computational results are presented.;In the third part we apply the generalized Capital Asset Pricing Model based on mixed CVaR deviation to calibrate risk preferences of investors. We introduce the new generalized beta to capture tail performance of S&P500 returns. Calibration is done by extracting information about risk preferences from option prices on S&P500. Actual market option prices are matched with the estimated prices from the pricing equation based on the generalized beta. These results can be used for various purposes. In particular, the structure of the estimated deviation measure conveys information about the level of fear among investors. High level of fear reflects a tendency of market participants to hedge their investments and signals investors’ anticipation of poor market trend. This information can be used in risk management and for optimal capital allocation.