Definition
Stochastic calculus extends ordinary calculus to handle random (stochastic) processes like stock price movements. Itô's lemma is the fundamental theorem, enabling derivation of the Black-Scholes formula and other pricing models. Geometric Brownian motion (GBM) is the standard model for stock price evolution, though real markets exhibit fat tails and volatility clustering not captured by GBM.
functions Formula
dS = μS·dt + σS·dW (Geometric Brownian Motion)
lightbulb Example
A stock follows GBM with 10% drift and 20% volatility. Itô's lemma transforms this into the log-normal distribution, providing the probability distribution of future prices and enabling Black-Scholes option pricing.
verified_user Key Points
- Mathematical foundation of continuous-time finance
- Itô's lemma is the core theorem
- Geometric Brownian Motion models stock prices
- Real markets deviate from GBM assumptions