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. What a Stop-Loss Costs
In a random walk every stop-loss lowers what you can expect to end with. On the S&P 500 since 1990 a 5% trailing stop cost a quarter of the money, while a 10% one roughly broke even and cut the worst fall from 55% to 34%: the width of the stop decided everything. For a price that comes back, a stop has a stranger effect: it puts a floor under the price you should buy at, and set too tight, it kills the trade.
Read more →2. The Best of 490 Backtests Is Noise
Search a few hundred strategy variants on one year of data and the best one scores a Sharpe near three — with no edge in any of them. The arithmetic, the figure, and what survives it.
Read more →3. Where the Force Vanishes Is Not Where It Rests
Set the arrivals equal to the cancellations, solve for the queue size, and you have found the point where the force on the book vanishes. You have not found where the book sits. The two answers differ by a term nobody writes down, worth 14.2% across three order books that share a balance point exactly.
Read more →4. When Sampling Faster Makes You Wrong
Realized variance should converge to an SDE's quadratic variation as you sample faster — market microstructure noise makes it diverge instead. The volatility signature plot, and the two-scale fix.
Read more →5. The Win Rate That Means Nothing
Put the stop nine times further from the entry than the target and you win 90% of your trades on any stock, and make nothing. The optional stopping theorem shows why a win rate is a setting, not a skill, and what costs and exits do to a day trader's record.
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