Financial Mathematics · 12 lectures

Black–Scholes from Scratch

A twelve-lecture path from coin flips to the Black–Scholes formula: probability, the finance you need, Taylor series, differential equations, Brownian motion, SDEs, Itô's lemma, martingales and risk-neutral pricing, then the binomial model, the PDE, the formula, the Greeks and where it all breaks.

1. Probability: The Language of UncertaintyBefore we can price anything, we need a precise way to talk about "what might happen". Sample spaces, random variables, expectation and the law that makes randomness predictable in bulk.
2. Normal and Lognormal: The Shape of ReturnsPrices can't go negative, but returns can. The normal distribution and its exponential cousin, the lognormal, and the moment-generating trick that produces the famous −σ²/2.
3. Finance Basics: Interest, Forwards, Options and No-ArbitrageThe minimum finance you need: discounting, forwards, what an option actually is, and the one principle that drives all of derivatives pricing.
4. Taylor Series: Local Approximations That Run FinanceDuration, convexity, delta, gamma and Itô's lemma are all Taylor expansions in disguise. Here is the tool, its error term, and why the second-order term is the one that matters.
5. Differential Equations: From Compounding to the Heat EquationHow quantities evolve: ordinary differential equations, how to solve them by hand and by computer, and a first look at the partial differential equation Black–Scholes turns into.
6. From Random Walks to Brownian MotionShrink the coin-flip walk in time and scale it by √n, and a continuous but nowhere-smooth random path appears. Its strange properties are why ordinary calculus breaks.
7. Stochastic Differential EquationsAdd a noise term to an ODE and you get an SDE. What dX = μ dt + σ dW actually means, how to simulate it, and the model Black–Scholes assumes for stock prices.
8. Itô's Lemma: The Chain Rule for RandomnessTaylor-expand a function of Brownian motion, apply (dW)² = dt, and an extra term appears. The single most important formula in quantitative finance, derived and used.
9. Martingales and the Risk-Neutral MeasureFair games, conditional expectation, and the astonishing result that we can price as if every investor were indifferent to risk, by changing the probability measure.
10. The Binomial Model: Replication with ArithmeticAll of the previous lecture's measure theory, done with one step, two states and high-school algebra. Then stack the steps into a tree and watch it converge to Black–Scholes.
11. The Black–Scholes PDE and FormulaDelta-hedge away the randomness, demand the riskless return, and a PDE appears. Solve it two ways, via the heat equation and via risk-neutral expectation, and arrive at the formula.
12. Greeks, Hedging, Monte Carlo, and Where Black–Scholes BreaksHow traders actually use the model: sensitivities, discrete hedging P&L, pricing by simulation, implied volatility, and the smile that tells you the assumptions are wrong.