This chapter introduces mathematical finance. It begins by using classical analysis to derive the fundamental partial differential equation that determines the value of a European derivative. Provided with a terminal pay-off, the solution to this equation gives the current value of an option. Next, the modern martingale/probabilistic approach is used to obtain the same result. Additionally, the modern approach is used to value swaps, European options, caps/floors, and swaptions. The theoretical foundation that allows a function estimated under the real-world probability measure to be used directly in pricing methods, i.e. under the risk-neutral measure, is then discussed. This means that a prepayment model that is estimated on the basis of historical observations for prepayments can be used directly in various pricing methods. How to infer the risk-neutral distribution of an underlying instrument from traded call and put values is then shown. This distribution is then used to derive a formula for valuing generic European option payoffs known as the replication formula. The presentation is given from a practical perspective, and the technical aspects are left out.