The fundamental works of Bachelier, Markovitz, Sharpe and Lintner, Black, Scholes and Merton, and others, which laid out the basis of the discipline that today we refer to as Mathematical Finance, rest in general on few main assumptions: returns are Gaussians, prices are unique/linear, markets are arbitrage-free, and investors are expected utility maximizers.It has long been recognized, on the other hand, that none of these assumptions hold true in practice, so the traditional theoretical results of Mathematical Finance are only approximately true at best, and their applicability is limited. Typical examples are the volatility smile, the leptokurtic feature of returns, trade- and volume-dependence of prices, existence of infinitesimally small arbitrage opportunities and constant violations of the expected utility theorem axioms. In this work, I continue the exploration, started a few decades ago, of the consequences of relaxing the assumptions of normality of returns, linearity/uniqueness of prices, and certainty equivalent based financial objectives. Specifically, in Chapter I, I develop a pure jump model for pricing credit index options, that is based on the double gamma dynamics for the default intensity. In Chapter II, I apply several supervised and unsupervised learning techniques to provide additional evidence of investors’ behaviors that contradicts Expected Utility Theory. In Chapter III, I show that spectral risk measures, a well known class of nonlinear expectation operators for pure jump semimartingales, admit an integral representation and, based on it, I define a new class of convex risk measures that are not sublinear.