Accuracy of Maximum Likelihood Parameter Estimators for Heston Stochastic Volatility SDE

Abstract

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, . . . , N T . We study the asymptotic bias of these parameter estimators first for T fixed and N → ∞, as well as when the global observation time S = N T → ∞ and T = S/N → 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 .

Parametric Estimation from Approximate Data: Non-Gaussian Diffusions

“…, X tq . Estimators of this type can for instance be numerically derived by approximate maximum likelihood after time discretization (see, e.g., [1,9,27,33,39,46,58,61]). Under variously formulated sufficient conditions on the diffusion, maximum likelihood estimators of θ become asymptotically consistent for q sufficiently large and for dense enough specific time grids t 1 , .…”

“…Nonparametric approaches for SDE data modeling have used Bayesian methods as in [27,53,54,59,68], exploited the spectral properties of the infinitesimal generator [20,21,37], or have developed maximum likelihood function estimation as in [44,45,65], as well as drift and diffusion estimates by conditional expectations of process dynamics over short time intervals [14,19,31,41,64,67], with potential use of kernel based techniques as in [10,69]. For parametrized SDE models, various moments based parameter estimators (see [32] and references therein) have been implemented, as well as approximate maximum-likelihood parameters estimators after time discretization of the SDEs (see for instance [1,9,17,58]). Minimum-contrast estimators have also been used for parametric estimation of diffusions [22,35].…”

Option Pricing Accuracy for Estimated Heston Models

“…Concretely Ŷt is computed either by squared realized volatilities or by squared implied volatilities. In Azencott and Gadhyan (2015) we have introduced and studied at length explicit discretized maximum likelihood estimators F N (V 1 , . .…”

“…. , VN ) Let us describe how the "non observable" estimator F N is computed from the true volatilities V N in Azencott and Gadhyan (2015), where it is derived by likelihood maximization after formal Euler discretization of the Heston joint SDEs. This approach leads to define the five sufficient statistics a, b, c, d, f a…”

“…To fit Heston model parameters to data, we use discretized maximum likelihood parameter estimators, as developed and studied in Azencott and Gadhyan (2009) and Azencott and Gadhyan (2015). These estimators have explicit closed form expressions, combining sub-sampled data X nT and Y nT .…”