Intermediate
Understanding Stochastic Calculus: Brownian Motion, Ito's Lemma, and the Path to Black-Scholes
A rigorous, first-principles build-up of the stochastic calculus that underpins modern quant finance, for quant analyst aspirants, prop trading applicants, and systematizing traders who want the actual math, not just the formula. Starts from random walks, builds the Wiener process and its defining properties, develops Ito's lemma from scratch, and uses it to derive the Black-Scholes PDE and formula step by step. Grounded in Nifty and Indian market examples throughout.
MODULES
4
DURATION
~3.5 hrs
TRACK
Quantitative Finance
Access Level
LEARNER
Everything included
Full Text Playbooks
Actionable Exercises
Mobile Reading Mode
Lifetime Updates
Curriculum Breakdown
Chapter 1: The Wiener Process: From Random Walks to Brownian Motion
5 Lessons▶
Random Walks Revisited: The Discrete Building Block of Randomness9 min read
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Defining Brownian Motion: The Four Properties of a Wiener Process10 min read
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Why Brownian Motion Has No Derivative: Continuous but Nowhere Smooth10 min read
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Quadratic Variation: The Property That Breaks Ordinary Calculus11 min read
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Case Study: Simulating a Brownian Motion Path for a Nifty Stock11 min read
Chapter 2: Ito Calculus: The Rules for Working With Randomness
5 Lessons▶
From Deterministic to Stochastic Differential Equations9 min read
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Ito's Lemma: The Extra Term That Ordinary Calculus Misses11 min read
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Deriving Ito's Lemma: A Taylor Series Walkthrough12 min read
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Applying Ito's Lemma to Geometric Brownian Motion11 min read
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Common Ito Calculus Mistakes and How to Spot Them9 min read
Chapter 3: The Path to Black-Scholes
5 Lessons▶
Modeling Stock Prices as Geometric Brownian Motion, Rigorously10 min read
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Building the Riskless Hedge: Delta Hedging and the Replicating Portfolio11 min read
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Deriving the Black-Scholes Partial Differential Equation12 min read
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Solving the PDE: How the Closed-Form Formula Falls Out12 min read
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Risk-Neutral Pricing Through the Stochastic Calculus Lens10 min read
Chapter 4: Beyond Black-Scholes: Extensions of the Toolkit
5 Lessons▶
Pricing a Put and Verifying Put-Call Parity With Ito's Tools9 min read
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Multivariate Ito's Lemma: Correlated Stocks and Basket Options11 min read
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Stochastic Volatility: Why Constant Sigma Breaks Down10 min read
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Jump-Diffusion: Modeling Market Gaps With a Poisson Process10 min read
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Capstone Case Study: Pricing a Real Nifty Option and Reading the Volatility Smile12 min read