Mathematical Finance is the field of mathematics that studies financial markets. Topics in financial markets studied include market trading mechanisms, called market microstructure, corporate management decision making, called corporate finance, investment management, and derivative securities. In each of these areas, sophisticated mathematics is utilized for modeling purposes. The theory of stochastic processes, stochastic optimization, partial differential equations, and simulation methods are just some of the mathematical tools employed. For example, in the area of derivatives, stochastic calculus is used to price a call option on a common stock. A call option is a financial security that gives its owner the right to buy a common stock at a fixed price on or before a fixed future date. Using stochastic calculus, the price of a call option can be characterized as the expected value of a nonlinear and random payoff at a future date. Numerical methods, such as Monte Carlo simulation, are often used to compute these expected values.