A Heston calibration returns kappa = 2.0, theta = 0.04, volatility of variance 0.5, rho = -0.7 and v_0 = 0.09. Say what each parameter controls in the shape of the surface, and check whether the fit is admissible.

A Heston calibration returns kappa = 2.0, theta = 0.04, volatility of variance 0.5, rho = -0.7 and v_0 = 0.09. Say what each parameter controls in the shape of the surface, and check whether the fit is admissible.

Approach: Map each parameter onto a feature of the surface, level, slope, curvature and term structure, then test the condition that keeps the variance process strictly positive.

The Feller condition fails, since 2*kappa*theta = 0.16 sits below the squared volatility of variance of 0.25, so the variance process can reach zero. In the Heston model v_0 = 0.09 sets today's level at 30% volatility, theta = 0.04 sets the long-run level at 20%, and kappa = 2.0 sets the speed of mean reversion, so the term structure decays from 30 toward 20 with a half-life of ln(2)/2, about 0.35 years. The correlation rho = -0.7 produces the skew: negative correlation makes down moves coincide with higher variance and tilts the implied distribution left. The volatility of variance at 0.5 produces the smile curvature, and it is the parameter that fights the Feller condition, because a large one is exactly what fitting a steep short-dated smile demands. A calibration that violates Feller is usable for pricing with a scheme that handles the boundary, and it is a warning that the model is being asked to fit short-dated curvature it cannot generate honestly.

Follow-up: The same name has a one-week smile far steeper than this fit produces. What would you add to the model and what does it cost in calibration time?

Key concepts: Heston model, mean reversion, Feller condition, correlation, volatility of variance.