Pricing, hedging and investment are arguably the three fundamental problems in quantitative finance and asset pricing. Naturally, the problems are closely connected. The present thesis consists of three chapters. Each chapter contributes to the literature dealing with aspects of the fundamental problems. Special emphasis is put on volatility and jump risk. According to the literature on empirical asset pricing we investigate the connection between market volatility and the risk premium. This connection crucially impacts the design of (near-)optimal investment strategies. Following the classical robustness definition that mother nature plays against the investor, we study robust strategies in market environments exposed to volatility and jump risk. Another interesting aspect is the joint influence of volatility and jumps on inter-temporal hedging demands. Accounting for the hedging demand, an investor protects herself against changes in the investment opportunity set. Volatility and jumps also play an important role in the literature on option pricing. The thesis contributes to the literature by introducing a pricing and hedging method for high-dimensional basket and basket spread options. The method applies to a very general class of continuous-time pricing models comprising stochastic volatility, stochastic correlation, or jump diffusion models. Methodically, the thesis contributes to the literature on econometric modeling (Chapter 2), numerical solutions of partial differential equations (Chapter 3), and Fourier analysis (Chapter 4).