Essays
These are our essays. They showcase applied mathematics across a wide span of subjects, from physics and engineering to finance, history and medicine: the models, the methods that solve them, the code that runs them and the algorithms behind them, often with a live model you can try for yourself. Each essay starts from a question and works it through to a number. Nothing here is a product or advice. The code we publish as Jupyter notebooks is under Notebooks, the live boards that each answer one practical question are under Tools, and the research papers are under Research Papers. Every essay belongs to one of our five research desks; the buttons below show one desk's essays at a time.
1. Four Hundred Dollars or Nine Cents
A dollar in TQQQ, the 3x Nasdaq-100 fund, became about 408 dollars between 2010 and 2026 while the plain fund made 20. The same daily-reset machine run on the S&P 500 from 1998 to 2009 turned a dollar into 9 cents while the index, with dividends, went nowhere. One line of Itô calculus explains both: a leveraged fund is a power of the index taxed by its variance, and it wins only when its leverage is below the market's best, (μ − r)/σ².
Read more →2. The Man Who Could Have Won the Nobel Twice
In 1965 Paul Samuelson wrote down an option formula with two unknowns, and admitted he had no theory to pin them down. Set both to the interest rate and it is the Black–Scholes–Merton formula, term for term, eight years early. The missing piece was not mathematics but an argument, and his own student helped supply it.
Read more →3. Kelly, the Growth-Optimal Portfolio, and the Benchmark Approach to Option Pricing
The Kelly criterion and Black-Scholes option pricing are the same mathematical object viewed from different angles. Both lead to the growth-optimal portfolio — Platen's benchmark — which prices derivatives under the real-world measure without requiring a risk-neutral one.
Read more →4. Langevin Dynamics and Why They Matter in Finance
The Langevin equation was written to describe a particle in a fluid. It turns out to be exactly the right language for interest rates, volatility, and forward curves.
Read more →5. Two Worlds, One Price: Entropy and the Risk-Neutral Measure
The change of measure at the heart of derivative pricing is an exponential tilt of the real-world probability measure — identical in structure to the Boltzmann distribution. The cost of that tilt is relative entropy.
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