Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 10

Stochastic Integrals

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model variables that behave randomly. The best-known stochastic process to which stochastic calculus is applied is the Wiener process, which is used for modeling Brownian motion and other diffusion processes subject to random fluctuations. Since the 1970s, the Wiener process has been widely applied in mathematical finance to model the evolution in time of stock prices and interest rates. While an ordinary integral is either a number or a function, a stochastic integral is a random variable or a stochastic process. Stochastic integrals allow financial modelers to differentiate the time-varying behavior of asset returns from the stochastic behavior of asset returns. Using stochastic integrals:

• One can define random movements and state-dependent nature of asset prices more rigorously.
• One can convert the physical measure which describes the probability that an underlying instrument (such as a stock price or interest rate) will take a particular value or values to the risk-neutral measure which is a useful tool for pricing derivatives on the underlying asset.
• One can convert a financial asset into a martingale, which is useful for defining expected future prices of underlying assets that determine the value of derivative ...