We consider the form and sensitivity to risk of the price of perpetual American interest rate derivatives for a broad class of one-factor diffusion models of interest rates. We first present, in terms of the infinitesimal coefficients of the underlying interest rate dynamics, a set of usually satisfied conditions under which the value of the contingent claim is convex, at least on the set where exercising the contract is suboptimal. In line with previous parametrized models considering the valuation of perpetual interest rate derivatives, we find that given our general conditions, the convexity of the exercise payoff is preserved under rational valuation. Consequently, we are able to establish a set of typically satisfied conditions under which increased volatility unambiguously increases the price of the claim and postpones rational exercise by expanding the region where exercising the claim is suboptimal.