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MIT published a full lecture on risk-neutral pricing and Black-Scholes equation. It covers forward contracts, options valuation, stochastic calculus, and why derivative prices depend on volatility. To all quants out there turn on Notifications, new article will be out tomorrow!

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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

Zyphor

47,157 views • 1 month ago

A Japanese mathematician published a result in 1944 that nobody understood for twenty years. Today it runs inside every options desk on Wall Street. Goldman pays $400K to quants who can derive it from scratch and explain why classical calculus gives the wrong answer without it. His name is Choongbum Lee. MIT, 18.S096, Topics in Mathematics with Applications in Finance. The course that Wall Street watches. This is lecture 17. It derives Ito's Lemma from scratch. He opens with the problem nobody in classical calculus can solve. Then the foundation. Brownian motion is the limit of a random walk taken to infinity. Each trade pushes a price up or down by a tiny amount. A million trades a day. The limit of that process is Brownian motion. Einstein proved this for pollen particles in 1905. The finance world borrowed the math fifty years later. Then three properties that make no sense until you see them derived. Brownian motion crosses zero infinitely often. It never escapes to infinity. And it is nowhere differentiable - with probability one, every path is continuous but has no slope at any point. That last property is why classical calculus breaks completely. Then quadratic variation. For any smooth function, chop an interval into n pieces, square the increments, sum them - the result goes to zero. For Brownian motion it goes to T. The increments are too wild to vanish. That single fact is why Ito's Lemma has a second term that classical calculus does not. Watch the moment he derives it. Taylor expansion applied to a function of Brownian motion. The first term is what you expect. The second term appears precisely because the squared increment does not vanish. Without it, options pricing gives wrong answers. With it, you have Black-Scholes. A quant I know sends this lecture to junior analysts who cannot explain why their pricing model drifts. Says it fixes in ninety minutes what two years of finance courses left open. Free on YouTube, MIT OpenCourseWare, 18.S096. bookmark this and watch later - the math behind every options desk on Wall Street fits on one blackboard, and this is the lecture that shows you why

Lupen

66,176 views • 1 month ago