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We study approximate maximum likelihood estimators (MLEs) for the parameters of the widely used Heston Stock price 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 compute the asymptotic biases of these parameter estimators for $$T$$ fixed and $$N \rightarrow \infty $$ , as well as the rate at which these biases vanish when $$T \rightarrow 0$$ . We determine asymptotically consistent explicit modifications of these MLEs. For the Heston volatility SDE, we identify a canonical form determined by two canonical parameters $$0 < \omega < 1$$ and $$\zeta > 1/2$$ which are explicit functions of the original SDE parameters. We analyze theoretically the asymptotic distribution of the MLEs and of their consistent modifications, and we outline their concrete speeds of convergence by numerical simulations. We clarify in terms of $$\zeta $$ the precise dichotomy between asymptotic normality and attraction by stable like distributions with heavy tails. We illustrate numerical model fitting for Heston SDEs by two concrete examples, one for daily data and one for intraday data, both with moderate values of $$N$$ . [ABSTRACT FROM AUTHOR]