Finance is one of the fastest growing areas in mathematics. In some senses it is not a discipline in its own right, but rather an application area in which mathematicians with backgrounds in probability theory, statistics, optimal control, convex and functional analysis and partial differential equations can bring to bear experiences and results from their own fields to problems of real world interest.
In this survey we begin with the simplest possible financial model, and then give an account of the Black–Scholes option pricing formula, in which the key ideas are the replication of option pay–offs and pricing under the risk–neutral measure. Then we move on to discuss other important problems in finance, including the general theory for semi–martingale price processes, pricing in incomplete markets, interest–rate models and credit risk. The emphasis is on techniques and methodologies from stochastic processes.