Rough Bergomi
Volatility is rough. Gatheral, Jaisson and Rosenbaum (2014) found that it moves like fractional Brownian motion with a Hurst exponent near 0.1, far rougher than classical models assume. Rough Bergomi (Bayer, Friz and Gatheral, 2016) builds on this and fits the steep short-dated skew with a handful of parameters. It is priced by simulation; π enters through its Gaussian drivers and, through Γ(½) = √π, its fractional kernel.
Circangle upgrades
Uses the same H, ρ and random numbers as the classical model.
Brent example: illustrative inputs, not live market data. Equity example: the parameters of Bayer, Friz and Gatheral’s index fit (H = 0.07, η = 1.9, ρ = −0.9).
Volatility smile
Sample volatility paths
The formulas
The rough driver is a Riemann–Liouville fractional Brownian motion, and the variance is its exponential, correlated ρ with the price:
The paths are simulated exactly on a 40-step grid, by Cholesky factorisation of the joint covariance of V and W. With H = ½ the driver is ordinary Brownian motion; small H makes it rough. The variance is unbounded above and never reaches zero.
The experimental circangle version replaces the exponential with the bounded circangle volatility of the book’s stochastic-volatility model, driven by the same rough process:
so the volatility stays between r₂ and r₁ whatever the driver does, and k sets how long it dwells near each. With r₁ = r₂ the price is Black–Scholes. This combination is new: it is not in the book, has not been studied beyond the checks here, and is shown to explore the idea, not as a result.
Both are simulated, so figures carry noise of a few hundredths and change slightly with the inputs. Background on the circangle volatility is in Circangle: π is Obsolete, Part 4.
Use of the models is acceptance that they are for educational purposes only. See the disclaimer.