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 .

When asset prices are modelled by stochastic dynamics, the model parameters are estimated from financial data. We study how estimation errors on model parameters impact the computed option prices, in the case where asset price and volatility follow the classical joint stochastic differential equations (SDEs) parametric model of Heston. Model parameters are estimated by an approximate maximum likelihood approach studied in [4] which presented an implementable computation of the covariance matrix for the errors in model parameters estimation. We then study and compute the sensitivity of optimal option prices to errors on the model parameters. This is achieved by numerically solving the partial differential equations (PDEs) verified by the derivatives of the option price with respect to model parameters. Combining these evaluations of derivatives with the computed covariance matrix of errors on model parameters, we obtain the errors on option price due to parametric estimation errors. We apply our method to the Standard & Poor's (S&P) 500 index options using the implied volatility index (VIX) [29] as a proxy for volatility. IntroductionIt is common practice in finance applications to model the price of securities and other assets using stochastic differential equations. The models are estimated using available data and are then used for forecasting or for computing the price of financial instruments.We study the impact of errors in model estimation on the price of options where the underlying asset is assumed to verify the estimated model. We study how errors on each individual parameter affect option prices. We compute a bound on the error in the option price due to model error. Our study of parametric error sensitivity does not depend on the choice of method used for the calibration of the asset pricing model.Extensions of the Black-Scholes models to local volatility ([8], [9]) and stochastic volatility models ([19], [25]) are often used in practice. Local volatility models allow for the instantaneous volatility to be a

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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 causes that lie behind poor trading performance. It also gives theoretical foundations to a generic framework for real-time trading analysis. The common acronym for investigating the causes of bad and good performance of trading is TCA (Transaction Cost Analysis [Rosenthal, 2009]). Automated algorithms take care of most of the traded flows on electronic markets (more than 70% in the US, 45% in Europe and 35% in Japan in 2012). Academic literature provides different ways to formalize these algorithms and show how optimal they can be from a mean-variance [Almgren and Chriss, 2000], a stochastic control [Guéant et al., 2012], an impulse control [Bouchard et al., 2011] or a statistical learning [Laruelle et al., 2011] 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 (following for instance [Basseville and Nikiforov, 1993]). We then will present an online methodology to quantify how this extended set of factors, at any given time, predicts (i.e. have influence, in the sense of predictive power or information defined in [Cristianini and Shawe-Taylor, 2000], [Shannon, 1948] and [Alkoot and Kittler, 1999]) 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.