  We study approximate maximum likelihood estimators (MLEs) for the parameters
of the widely used Heston stock and volatility stochastic differential
equations (SDEs). We compute explicit closed form estimators maximizing the
discretized log-likelihood of $N$ observations recorded at times $T,2T, \ldots,
NT$. We study the asymptotic bias of these parameter estimators first for $T$
fixed and $N \to \infty$, as well as when the global observation time $S= NT
\to \infty$ and $T = S/N \to 0$. We identify two explicit key functions of the
parameters which control the type of asymptotic distribution of these
estimators, and we analyze the dichotomy between asymptotic normality and
attraction by stable like distributions with heavy tails. \\ We present two
examples of model fitting for Heston SDEs, one for daily data and one for
intraday data, with moderate values of $N$.
