MIT FILMED A PROFESSOR WHOSE EXPLANATIONS ARE SO CLEAR AND INTUITIVE THAT THE MOST COMPLEX CALCULUS ON WALL STREET BECOMES OBVIOUS - AND IT PROVES WHY NEWTON'S VERSION GIVES THE WRONG ANSWER FOR EVERY OPTION PRICE This is an MIT lecture on Itô calculus, course 18.S096. The professor opens with one equation that breaks classical calculus - dB squared equals dt. In normal calculus, dt squared vanishes when you take limits. But the quadratic variation of Brownian motion forces the second derivative term to survive. This one fact changes every formula in calculus. Then Itô's lemma. In classical calculus, if you apply a smooth function to a path, you only need the first derivative term. In Itô calculus you need an extra term - half the second derivative times dt. He derives it in 10 lines of Taylor expansion, showing exactly which terms survive and which vanish. The correction term is not a small adjustment. It is the difference between the right answer and the wrong one. Then geometric Brownian motion. The natural guess for modeling stock prices is to take the exponential of a Brownian motion. It fails - the Itô correction introduces an unwanted drift term that makes the expected value grow over time. The fix is to subtract exactly half of sigma squared from the exponent. This is the formula that runs every Black-Scholes calculation on earth. Then adapted processes and Itô integrals. In classical integration it doesn't matter which point in each interval you use to form Riemann sums - the limit is always the same. In Itô calculus left endpoints and right endpoints give different answers. Itô's choice of left endpoints is not arbitrary - it encodes the fundamental constraint that financial decisions must be made using only past information. Watch the moment he explains the Girsanov theorem - that a Brownian motion with drift and a Brownian motion without drift are equivalent probability measures. Two processes whose paths look completely different in the long run can be converted into each other by multiplication. This is how quants transform non-martingale stock price processes into martingales for pricing. A derivatives trader I know rewatched this lecture before his first day at a quantitative hedge fund. Said it was the first time Itô's lemma felt like a theorem with a reason rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare, Creative Commons license. bookmark this and watch later - after this lecture every option price you see will feel like a solution to a stochastic differential equation waiting to be written down