Black–Scholes–Robilliard option model

Black–Scholes–Robilliard

Black–Scholes gives the market one volatility for moves up and down: a circle. Black–Scholes–Robilliard gives the upside and the downside their own scales, a and b, and keeps a closed-form price. With a = b it is Black–Scholes exactly. The defaults are an illustrative Brent crude option, where falls tend to be sharper than rises.

Option

Underlying price

Strike

Time to expiry (years)

Interest rate (%)

Upside volatility a (%)

Downside volatility b (%)

Brent example: underlying 110, at-the-money, three months, rate 4%, upside 48%, downside 62%. Illustrative inputs, not live market data.

Black–Scholes–Robilliard

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Black–Scholes at the average volatility

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Greeks of the Black–Scholes–Robilliard price by central differences. Vega moves both volatilities together and is per 1.00 of volatility (divide by 100 for one point); Theta is per year; Rho is per 1.00 of rate.

What each model expects at expiry

Probability density of the underlying price at expiry; the dashed line is the strike.

The skew it produces

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

Value today and at expiry

Option value against the underlying price, with the chosen strike and time.

Strike ladder

Strike	BSR	Black–Scholes	Difference	Implied vol	Delta

The model

The log-return over the option’s life has a half-bell of scale a above the peak and a half-bell of scale b below, joined at the peak with the same height:

Here the scales are the annual volatilities times √T, and √(4K(0)) is the Gaussian integral √(2π). The price at expiry is S_T = Fe^x/M, with F = S₀e^rT, so that its expected value is the forward. With w = 2/(a + b) and N the normal distribution function,

The put follows by put–call parity. At a = b = σ√T, a − κ/a is d₁ and −κ/a is d₂, and the call is the Black–Scholes price. The derivation, a fat-tailed variant and worked figures are in Circangle: π is Obsolete, Part 4.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.