The fat-tailed Robilliard option model

The fat-tailed model

Crash risk lives on the downside. This model keeps a Gaussian upside of scale a and gives the downside a heavy, polynomial tail whose shape is set by the circangle modulus k. The skew is not fitted in: it comes out of the model. Ξ, the sensitivity to the tail shape, is a Greek of its own.

Option

Underlying price

Strike

Time to expiry (years)

Interest rate (%)

Upside volatility a (%)

Downside scale b (%)

Tail shape k² = 0.50

Join the halves smoothly (b follows a and k)

Brent example: underlying 110, at-the-money, three months, rate 4%, upside 22%, k² = 0.5, halves joined, which gives an at-the-money implied volatility near 55%. Illustrative inputs, not live market data.

Fat-tailed model

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Black–Scholes at the upside volatility

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Central differences, in the book’s units: Vega per 1.00 of volatility, Theta per year, Rho per 1.00 of rate, Ξ per 1.00 of k². When the halves are joined, b follows a in Vega and k in Ξ.

The skew it produces

Black–Scholes implied volatility of the model price at each strike.

The tails, on a log scale

Density of the log-return over the option’s life. The downside falls away polynomially, not exponentially.

Strike ladder

Strike	Price	Implied vol	Delta	Gamma	Vega	Theta	Rho	Ξ

The model

The downside comes from the circangle function sc: with u uniform on (−K(k), K(k)), x = sc(u, k) has polynomial tails, and at k = 0 it is the Cauchy distribution. Joined to a Gaussian upside, which keeps E[e^X] finite, the log-return has density

with . The halves meet with the same height, and with they also meet with the same curvature, so the joined curve is smooth to third order and the model has two free parameters, a and k. Prices use S_T = S₀e^rTe^X/E[e^X]; the upside integrals are in closed form and the downside is integrated numerically.

The tail is heavy: with the halves joined at a = 20% over a year, the 5% quantile is a fall of about 91%. The sc downside suits far out-of-the-money puts, where crash risk is priced, more than a whole year’s returns. The derivation and full tables are in Circangle: π is Obsolete, Part 4.

This is a proof of concept. It has not been calibrated to market quotes or checked for arbitrage across a full volatility surface. Use of the model is acceptance that it is for educational purposes only. See the disclaimer.