James Robilliard
I work on the circangle: replacing π with the complete elliptic integrals K(k) and E(k), and following where that leads, from the ellipse to option pricing and nuclear fusion. Here you’ll find my book and the Robilliard models, which carry the circangle’s two radii into finance.
The Robilliard models
Black–Scholes assumes the market is a circle: one volatility for moves up and down. These models give the two directions their own radii, as the circangle does, and keep closed forms where they can. Each runs in your browser, with charts.
Black–Scholes–Robilliard
Separate upside and downside volatilities in a closed-form option price that is Black–Scholes when they are equal. Prices, Greeks, the skew it produces and the distribution behind it.
Open →The fat-tailed model
A Gaussian upside joined to a heavy-tailed downside whose shape is the circangle modulus k. The skew comes out of the model, with a new Greek, Ξ, for the tail shape.
Open →Two-scale risk and margin
Quantiles, the long against short margin, skewness, and a calibration straight from the mean and variance.
Open →Circangle: π is Obsolete
The book behind the models. It replaces the circle with the circangle, a quarter-ellipse treated as a right-angled triangle, and π with K(k) and E(k), then rewrites formulas across geometry, finance and physics. Part 4 derives the Robilliard models and their Greeks.
Recreational mathematics for the curious, with the hard maths left in.
Play with the geometry at circangle.com: an exact ellipse calculator and the circangle slider.