Explain the concept of stochastic calculus and its application in pricing and risk management of financial instruments.

Stochastic calculus is a branch of mathematics that deals with the study of systems that evolve randomly over time. It's a powerful tool for modeling and analyzing random processes, particularly in financial markets. At its core, stochastic calculus utilizes concepts from probability theory and calculus to describe the behavior of processes that are affected by random fluctuations. These random fluctuations are often modeled using Brownian motion, a continuous-time process that resembles the random movement of a particle suspended in a fluid. One of the key concepts in stochastic calculus is the Ito integral, which allows us to calculate the integral of a function with respect to a stochastic process, like Brownian motion. This is crucial for modeling the price dynamics of financial instruments, whic....