Modeling price movements is central to quantitative finance. In Derivatives, we saw how options depend on underlying asset prices that evolve randomly over time, we need tools to analyze these stochastic processes.
Modeling a random process / random movement first appears in physics, where particles move in unpredictable ways due to collisions with other particles. This phenomenon is called Brownian motion or Wiener process. In finance, we could consider stock prices as a similar mechanism, each transaction is a “collision” (buy or sell force) that causes the price to change randomly. This leads us to model stock prices as stochastic processes.
Topics in stochastic calculus include: