This paper is concerned with the optimal selection of the smoothing parameter $h$ in kernel estimation of a trend in nonparametric regression models with (dependent) stochastic volatility errors $\epsilon_i= \sigma_i Z_i$, $i=1,\cdots, n$, where $(\sigma_i)_i$ is referred as the volatility sequences and $(Z_i)_i$ a sequence of i.i.d random variables. We consider three types of volatility sequences; the log-normal volatility, the Gamma volatility and the log-linear volatility with Bernoulli innovations. In fact, based on three criteria for deriving optimal smoothing parameters, namely the average squared error, the mean average squared error and an adjusted Mallows-type criterion to the dependent case, we show that these three minimizers are first-order equivalent in probability. Moreover, we derive the normal asymptotic distribution of the difference between the minimizer of the average squared error and the minimizer based on the Mallows-type criterion. A Monte-Carlo simulation is conducted for a log-normal stochastic volatility model.