Accuracy of Maximum Likelihood Parameter Estimators for Heston volatility SDE

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 $N$.Comment: 31 pages, 0 figure

Option Pricing Accuracy for Estimated Heston Models

We consider assets for which price $X_t$ and squared volatility $Y_t$ are jointly driven by Heston joint stochastic differential equations (SDEs). When the parameters of these SDEs are estimated from $N$ sub-sampled data $(X_{nT}, Y_{nT})$, estimation errors do impact the classical option pricing PDEs. We estimate these option pricing errors by combining numerical evaluation of estimation errors for Heston SDEs parameters with the computation of option price partial derivatives with respect to these SDEs parameters. This is achieved by solving six parabolic PDEs with adequate boundary conditions. To implement this approach, we also develop an estimator $\hat \lambda$ for the market price of volatility risk, and we study the sensitivity of option pricing to estimation errors affecting $\hat \lambda$. We illustrate this approach by fitting Heston SDEs to 252 daily joint observations of the S\&P 500 index and of its approximate volatility VIX, and by numerical applications to European options written on the S\&P 500 index

Realtime market microstructure analysis: online Transaction Cost Analysis

Motivated by the practical challenge in monitoring the performance of a large number of algorithmic trading orders, this paper provides a methodology that leads to automatic discovery of the causes that lie behind a poor trading performance. It also gives theoretical foundations to a generic framework for real-time trading analysis. Academic literature provides different ways to formalize these algorithms and show how optimal they can be from a mean-variance, a stochastic control, an impulse control or a statistical learning viewpoint. This paper is agnostic about the way the algorithm has been built and provides a theoretical formalism to identify in real-time the market conditions that influenced its efficiency or inefficiency. For a given set of characteristics describing the market context, selected by a practitioner, we first show how a set of additional derived explanatory factors, called anomaly detectors, can be created for each market order. We then will present an online methodology to quantify how this extended set of factors, at any given time, predicts which of the orders are underperforming while calculating the predictive power of this explanatory factor set. Armed with this information, which we call influence analysis, we intend to empower the order monitoring user to take appropriate action on any affected orders by re-calibrating the trading algorithms working the order through new parameters, pausing their execution or taking over more direct trading control. Also we intend that use of this method in the post trade analysis of algorithms can be taken advantage of to automatically adjust their trading action.Comment: 33 pages, 12 figure