For the variance process dv = kappa(theta - v) dt + xi sqrt(v) dW, state the Feller condition and what it controls. Check it for kappa = 1.5, theta = 0.04 and xi = 0.5, and give E[v_t] and the transition law.

For the variance process dv = kappa(theta - v) dt + xi sqrt(v) dW, state the Feller condition and what it controls. Check it for kappa = 1.5, theta = 0.04 and xi = 0.5, and give E[v_t] and the transition law.

Approach: Compare the strength of the mean reverting pull near the origin with the size of the diffusion there, then quote the known affine mean and the exact transition law of the square root diffusion.

The Feller condition is 2 kappa theta ≥ xi^2, and it fails here since 2(1.5)(0.04) = 0.12 < 0.25 = xi^2, so the variance reaches zero with positive probability. In the square root diffusion the diffusion coefficient xi sqrt(v) vanishes at the origin at rate sqrt(v) while the mean reversion drift there is kappa theta, and the condition is the exact statement that the upward pull dominates strongly enough to make zero unattainable. The mean is unaffected by the square root: E[v_t] = theta + (v_0 - theta) e^{-kappa t}, since the stochastic term has zero mean whatever the state dependence. The transition law is noncentral chi-squared: 2 c v_t given v_0 is noncentral chi-squared with 4 kappa theta / xi^2 degrees of freedom, where c = 2 kappa / (xi^2 (1 - e^{-kappa t})), and the degrees of freedom being below 2 is exactly the failure of the Feller condition.

Follow-up: With these parameters, what does the attainability of zero do to the implied volatility smile for short maturities?

Key concepts: Feller condition, square root diffusion, noncentral chi-squared, mean reversion.