$\sigma$-indepencene in affine multi-factor model for interest rate derivatives

The model here is affine two-factor model for interest rates.

Let $p = p(r, \sigma)$ denote bond prices which take the usual exponential form.

Let $r$ have some $Q$ dynamics, and let $\sigma$ be the stochastic (!) volatility.

• Are there known conditions one can impose on the dynamics such that $p$ does not depend on $\sigma$?
• Given those conditions, would $\sigma$ still make have its say in the determination of option prices (e.g., a call), and if so, why?

For the first, I am thinking something there may be some particular unspanned stochastic volatility model? For the second, I am not sure.

• For your first question, I do not think that is possible. Even in the simple Ho-Lee model, the bond price depends on the volatility. See this question for details. – Gordon Mar 6 '17 at 16:44