This paper provides a nonparametric testing procedure for continuous time volatility models, under minimal assump-tions. In particular, apart from standard regularity conditions, no assumptions are made on the functional forms of either the drift or the variance term. Our test is constructed by comparing two estimators of integrated volatility: one is a kernel estimator of the instantaneous variance, averaged over the sample realization on a fixed time span; the other is realized volatility. Under the hypothesis of the class of endogenous volatility model, the statistics have a mixed normal limiting distribution, while under the alternative hypothesis of stochastic volatility they diverge at an appropriate rate. Versions of the test robust to jumps and to microstructure noise are also provided. The findings from a Monte Carlo study indicate that the suggested tests have good finite sample properties. An empirical illustration to short term rates is included. Finally, the economic effect on bond pricing of choosing a misspecified volatility model is quantified.