filtering_in_finance - Filtering in Finance Alireza...

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2 Wilmott magazine Alireza Javaheri, RBC Capital Markets Delphine Lautier, Université Paris IX, Ecole Nationale Supérieure des Mines de Paris Alain Galli, Ecole Nationale Supérieure des Mines de Paris Filtering in Finance Further, we shall provide a mean to estimate the model parameters via the maximization of the likelihood function. 1.1 The Simple and Extended Kalman Filters 1.1.1 Background and Notations In this section we describe both the traditional Kalman Filter used for lin- ear systems and its extension to nonlinear systems known as the Extended Kalman Filter (EKF). The latter is based upon a first order linearization of the transition and measurement equations and therefore would coincide with the traditional filter when the equations are linear. For a detailed introduction, see Harvey (1989) or Welch and Bishop (2002). Given a dynamic process x k following a transition equation x k = f ( x k 1 , w k ) (1) we suppose we have a measurement z k such that z k = h ( x k , u k ) (2) where w k and u k are two mutually-uncorrelated sequences of temporally- uncorrelated Normal random-variables with zero means and covariance matrices Q k , R k respectively 4 . Moreover, w k is uncorrelated with x k 1 and u k uncorrelated with x k . 1 Filtering The concept of filtering has long been used in Control Engineering and Signal Processing. Filtering is an iterative process that enables us to esti- mate a model’s parameters when the latter relies upon a large quantity of observable and unobservable data. The Kalman Filter is fast and easy to implement, despite the length and noisiness of the input data. We suppose we have a temporal time-series of observable data z k ( e.g. stock prices as in Javaheri (2002), Wells (1996), interest rates as in Babbs and Nowman (1999), Pennacchi (1991), futures prices as in Lautier (2000), Lautier and Galli (2000)) and a model using some unobservable time- series x k ( e.g. volatility, correlation, convenience yield) where the index k corresponds to the time-step. This will allow us to construct an algo- rithm containing a transition equation linking two consecutive unob- servable states, and a measurement equation relating the observed data to this hidden state. The idea is to proceed in two steps: first we estimate the hidden state a priori by using all the information prior to that time-step. Then using this predicted value together with the new observation, we obtain a con- ditional a posteriori estimation of the state. In what follows we shall first tackle linear and nonlinear equations with Gaussian noises. We then will extend this idea to the Non-Gaussian case. Abstract In this article we present an introduction to various Filtering algorithms and some of their applications to the world of Quantitative Finance. We shall first mention the fundamental case of Gaussian noises where we obtain the well-known Kalman Filter.
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filtering_in_finance - Filtering in Finance Alireza...

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