Stochastic Calculus¶
Under development
This optional advanced module is part of the course scaffold.
This module builds the continuous-time mathematics underlying derivatives pricing and much of theoretical quantitative finance: Brownian motion, the Itô integral and Itô's lemma, stochastic differential equations, and how these connect to option pricing. It is for learners with solid probability who are targeting derivatives, volatility, or research roles where continuous-time models are the working language; it is not required for the systematic-equities track of the course.
Topics¶
- Brownian motion: defining properties, non-differentiability, and quadratic variation
- The Itô integral: why ordinary calculus fails for stochastic integrands
- Itô's lemma and its use as the everyday computational tool of the field
- Canonical SDEs: geometric Brownian motion and the Ornstein–Uhlenbeck process, with their trading interpretations
- The connection to option pricing: hedging arguments, the Black–Scholes PDE, and risk-neutral valuation
- Simulating SDEs: Euler–Maruyama and Milstein schemes, discretization error
- Where continuous-time models earn their complexity and where discrete-time treatment suffices