Feller Positivity Condition for a Discretized Heston Variance Process

Summary

The document asks how to keep variance nonnegative in a discretized mean-reverting Heston-style model. Its response interprets the variance update as a discretization of a square-root diffusion, then states the Feller condition that keeps the continuous-time variance process strictly positive. In the notation given, the condition compares twice the mean-reversion speed times the long-run variance level with the square of the volatility-of-volatility coefficient.

The parameter values in the question do not satisfy that condition when the coefficient on the square-root noise is treated as one. The answer explains that the model should include an explicit volatility-of-volatility parameter and that its size matters for positivity. This condition concerns the continuous process; the document does not provide a discrete-time scheme that guarantees nonnegative values at every step, nor does it discuss alternatives such as truncation or exact simulation. Users should therefore distinguish continuous-time positivity conditions from numerical behavior under a particular discretization.

Key ideas

• The variance equation corresponds to a mean-reverting square-root diffusion in continuous time.
• The Feller condition relates mean reversion, long-run variance, and volatility of volatility.
• The stated parameters fail the condition if the square-root noise coefficient is one.
• Continuous-time positivity does not by itself establish that a chosen discretization stays nonnegative.

Tags

# The non-negativity condition of a discretized mean-reverting Heston model with stochastic violatilities


# The non-negativity condition of a discretized mean-reverting Heston model with stochastic violatilities












I happened to encounter the following discretized mean-reverting Heston model with stochastic volatilities in a paper $$ P(t) = P(t-1) + v_1(u_1-P(t-1))+\sqrt{\sigma(t)}\cdot \epsilon_1(t) \\ \sigma(t) = \sigma(t-1) + v_2(u_2-\sigma(t-1))+\sqrt{\sigma(t-1)}\cdot \epsilon_2(t) $$ where $v_1=v_2=0.1,u_1=100,u_2=0.01$ are pre-set parameters, and $\epsilon_1,\epsilon_2 \sim N(0,1)$ follow the normal distribution IID. Recall that in the original Heston Model formulation, there is a condition (known as the Feller condition) to make sure that the values under the square root is positive. See wiki for more info. But in this case, how can I ensure that the value of $\sigma$ to be positive?

## Answer by Kurt G. (score 1, accepted)

https://quant.stackexchange.com/a/67934

The continuous version of your equation for $\sigma(t)$ reads $$ d\sigma(t)=v_2(u_2-\sigma(t))\,dt+\sqrt{\sigma(t)}\,dW^\sigma_t\,. $$ In this notation, the Feller condition ensuring $\sigma(t)>0$ is $2v_2u_2>1\,.$ This is not the case for the values $v_2=0.1,u_2=0.01$ you have chosen. Note that the Heston model also has a vol-of-vol parameter $\xi$: $$ d\sigma(t)=v_2(u_2-\sigma(t))\,dt+\xi\sqrt{\sigma(t)}\,dW^\sigma_t\, $$ and that the Feller conditon in full glory says $2v_2u_2>\xi^2\,.$ In other words, you should use that vol-of-vol $\xi$ and not make it as large as $\xi=1$.

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