Feller condition | Glossary | Derivatives Compass

The Feller condition is a condition on the parameters of a CIR (square-root) process, as used in the Heston model, under which the variance never reaches zero and stays strictly positive. With κ the speed of mean reversion, v̄ the long-run level and ξ the coefficient on the random moves of the variance, it reads 2κv̄ ≥ ξ².

The condition is also written as 4κv̄/ξ² ≥ 2. The left side is the dimension of a related process (the Bessel process) that is linked to the CIR process.

Consider a hypothetical example in which κ = 2 and v̄ = 0.04. Then 2κv̄ = 0.16. If ξ = 0.3 (ξ² = 0.09), the condition holds, and if ξ = 0.5 (ξ² = 0.25), it does not.

Even when the condition fails, the variance touches zero only for a very short time and does not stay there. However, volatility repeatedly reaches low levels that are not observed in the market, and a Monte Carlo simulation needs care so that the variance does not turn negative. When the model is fitted to market prices, parameters that violate the condition are not unusual.

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