Optimization methods in financial engineering

Our study developed novel approaches to solving and analyzing challenging problems of financial engineering including options pricing, market forecasting, and portfolio optimization. We also make connections of the portfolio theory with general deviation measures to classical portfolio and asset pricing theories.;We consider a problem faced by traders whose performance is evaluated using the VWAP benchmark. Efficient trading market orders include predicting future volume distributions. Several forecasting algorithms based on CVaR-regression were developed for this purpose.;Next, we consider assumption-free algorithm for pricing European Options in incomplete markets. A non-self-financing option replication strategy was modelled on a discrete grid in the space of time and the stock price. The algorithm was populated by historical sample paths adjusted to current volatility. Hedging error over the lifetime of the option was minimized subject to constraints on the hedging strategy. The output of the algorithm consists of the option price and the hedging strategy defined by the grid variables.;Another considered problem was optimization of the Omega function. Hedge funds often use the Omega function to rank portfolios. We show that maximizing Omega function of a portfolio under positively homogeneous constraints can be reduced to linear programming.;Finally, we look at the portfolio theory with general deviation measures from the perspective of the classical asset pricing theory. We derive pricing form of generalized CAPM relations and stochastic discount factors corresponding to deviation measures. We suggest methods for calibrating deviation measures using market data and discuss the possibility of restoring risk preferences from market data in the framework of the general portfolio theory.