Heston
Heston (1993) lets the variance itself wander, as a square-root process that reverts to a long-run level. Prices come from Fourier inversion of its characteristic function, with a 1/π in front. It is the standard stochastic-volatility model, but its variance can touch zero unless the Feller condition holds, and nothing caps it from above. The circangle upgrade reads volatility off a circangle instead, so the two radii become a floor and a ceiling on every path.
Circangle upgrades
Uses the Heston correlation ρ between the price and Y.
Brent example: illustrative inputs, not live market data. Book example: the figures in Circangle: π is Obsolete, Part 4, where Heston gives 9.2425.
Volatility smile
Sample volatility paths
The formulas
Heston’s variance follows , correlated ρ with the price, and the call is with
where fj are Heston’s two characteristic functions. The variance stays above zero only when 2κθ ≥ ξ², the Feller condition.
The circangle model replaces the square root with a circangle: , with Y an Ornstein–Uhlenbeck process, . Since sn lies between −1 and 1, the volatility stays between the two radii on every path at any parameters. The level it settles to is in closed form, through K(k), E(k) and the nome q:
with Ω = 2K(0)/K(k) and s² = η²/2κ. The model is not affine, so it is priced by simulation (16,000 antithetic pairs, 100 steps), and its figures carry simulation noise of a few hundredths. The ceiling cuts both ways: calibrated on calm data, it will understate volatility in a crisis. Derivations and worked figures are in Circangle: π is Obsolete, Part 4.
Use of the models is acceptance that they are for educational purposes only. See the disclaimer.