LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Five pieces of work carry this lecture's map of modern quantitative finance. Louis Bachelier's 1900 dissertation put randomness into prices. William Sharpe's 1964 model separated market risk from everything else. Black-Scholes priced options by replication. Bruno Dupire's 1994 local-volatility paper fitted the smile. Carr and Madan's 1999 Fourier method priced from a characteristic function without a fresh simulation for every strike.
Why it matters
Bachelier studied at the Sorbonne under Henri Poincaré and wrote on speculation decades before the tools his formulas echo were standard. The lecture says Brownian motion shows up in that work years before Einstein's physics version, and that the dissertation received an honorable mention rather than the higher distinction. It was ignored, then later treated as the start of financial mathematics. The picture is arithmetic Brownian motion: uncertainty tight around the spot at first, then a widening variance as time passes. Bachelier also had an option formula under that arithmetic law, an ancestor of later closed forms. Arithmetic paths can go negative. The lecture notes that when oil prices went negative in the COVID period, some desks reached back to Bachelier because the process can print a negative value. Portfolio replication, a unique risk-neutral measure, and Feynman-Kac were not available to him. The dissertation still built a consistent first model in 1900.
Sharpe's question, after Markowitz's 1950s case that diversification was the only free lunch, was what sets prices when everyone optimizes. The 1964 paper says only systematic risk is paid, because firm-specific risk can be diversified away. Beta and alpha are the descendants. The lecture's animation shows two portfolios, each with a Sharpe ratio near 2.5, that both beat the market. In a perfect CAPM world both would be levered market returns. On the plots, one is highly correlated with the market and is just extra exposure. The other is uncorrelated, so a market drop would hit the first and leave the second as the alpha case. Later factor models, including Fama-French three- and five-factor versions, extend that single market factor.
The idea in plain words
Black-Scholes changed the question from forecasting the spot to replicating the option. With no arbitrage, continuous trading, and no frictions, the option equals the cost of the hedge. If the hedge has cancelled the risk, it must earn the riskless rate. The world of the argument moves from the real-world P measure to a riskless construction. Invert the formula and you get implied volatility, a forward-looking number taken out of today's option price. The model itself uses geometric Brownian motion and one shock, so volatility is constant across strikes and maturities. Markets do not look like that. You can still invert the formula at every strike and maturity, and the surface you get has a skew or a smile and a term structure.
How to use it
Once smiles and stochastic volatility, including Heston, and jump models are available, the tradeoff is effectiveness against speed. Carr and Madan, 1999, move the pricing function to Fourier space. If you know the characteristic function of the spot, you substitute, solve in the transform domain, and invert. The lecture contrasts that with resimulating paths every time you retune a model to liquid options. A Heston surface in the animation is built by the transform route. Later neural approximations of pricing functions are described as a descendant of that speed problem.
What usually goes wrong
Treating Black-Scholes volatility as a flat surface you should see on the screen inverts the lesson. The formula is a quoting language. The surface it implies from real prices is not flat, which is why local volatility exists. Treating Bachelier as obsolete misses the negative-price episode the lecture cites. Treating a Sharpe ratio near 2.5 as alpha misses Sharpe's own split: one of the two animated portfolios is levered market risk.
Questions people ask
Why mention a 1900 dissertation?
It introduced a Brownian model of prices and an early option formula, before replication and Feynman-Kac existed. The lecture also cites it when prices can go negative.
What did Dupire add to Black-Scholes?
A local volatility that depends on strike and maturity, so a model can be calibrated to liquid options and reused for exotics. The paper is Pricing with a Smile, 1994.
Why is the Fourier paper included?
If the characteristic function is known, Carr and Madan price by transforming and inverting instead of simulating again for every calibration. Speed is the reason it is the lecture's favorite of the five.