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Filtering in Finance

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Filtering in Finance Filtering in Finance 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 Abstract (EKF) as well as the Unscented Kalman Filter (UKF) similar to Kushner’s Nonlinear In this article we present an introduction to various Filtering algorithms and some of Filter. We also tackle the subject of Non-Gaussian filters and describe the Particle their applications to the world of Quantitative Finance. We shall first mention the Filtering (PF) algorithm. Lastly, we will apply the filters to the term structure model of fundamental case of Gaussian noises where we obtain the well-known Kalman Filter. commodity prices and the stochastic volatility model. Because of common nonlinearities, we will be discussing the Extended Kalman Filter 1 Filtering Further, we shall provide a mean to estimate the model parameters via the maximization of the likelihood function. The concept of filtering has long been used in Control Engineering and Signal Processing. Filtering is an iterative process that enables us to esti- 1.1 The Simple and Extended Kalman Filters mate a model’s parameters when the latter relies upon a large quantity 1.1.1 Background and Notations of observable and unobservable data. The Kalman Filter is fast and easy In this section we describe both the traditional Kalman Filter used for lin- to implement, despite the length and noisiness of the input data. ear systems and its extension to nonlinear systems known as the Extended We suppose we have a temporal time-series of observable data z k (e.g. Kalman Filter (EKF). The latter is based upon a first order linearization of stock prices as in Javaheri (2002), Wells (1996), interest rates as in Babbs the transition and measurement equations and therefore would coincide and Nowman (1999), Pennacchi (1991), futures prices as in Lautier (2000), with the traditional filter when the equations are linear. For a detailed Lautier and Galli (2000)) and a model using some unobservable time- introduction, see Harvey (1989) or Welch and Bishop (2002). series x k (e.g. volatility, correlation, convenience yield) where the index k Given a dynamic process x k following a transition equation corresponds to the time-step. This will allow us to construct an algo- = rithm containing a transition equation linking two consecutive unob- x k f(x k−1, wk) (1) servable states, and a measurement equation relating the observed data we suppose we have a measurement z such that to this hidden state. k The idea is to proceed in two steps: first we estimate the hidden state z k = h(x k, u k) (2) 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- where wk and uk are two mutually-uncorrelated sequences of temporally- ditional a posteriori estimation of the state. uncorrelated Normal random-variables with zero means and covariance 4 In what follows we shall first tackle linear and nonlinear equations with matrices Q k, R k respectively . Moreover, wk is uncorrelated with x k−1 and Gaussian noises. We then will extend this idea to the Non-Gaussian case. u k uncorrelated with x k. 2 Wilmott magazine TECHNICAL ARTICLE 1 We denote the dimension of x k as nx, the dimension of w k as nw and Needless to say for a linear system, the function matrices are equal to so on. these Jacobians. This is the case for the simple Kalman Filter. We define the a priori process estimate as 1.1.2 The Algorithm The actual algorithm could be implemented as follows: ˆ − = x k E [x k] (3) (1) Initialization of x0 and P0 which is the estimation at time step k − 1 prior to the step k measurement. For k in 1...N Similarly, we define the a posteriori estimate (2) Time Update (Prediction) equations ˆ = | ˆ − = ˆ x k E [x k z k] (4) x k f(xk−1, 0) (8) which is the estimation at time step k after the measurement. and − − = − ˆ − t t We also have the corresponding estimation errors e k x k x k and = + P k A kPk−1A k W kQ k−1W k (9) ek = x k − xˆ k and the estimate error covariances (3-a) Innovation: We define − − −t P = E e e k k k ˆ− = ˆ − z k h(x k , 0) = t P k E e ke k (5) and = − ˆ− where the superscript t corresponds to the transpose operator. ν k z k z k In order to evaluate the above means and covariances we will need as the innovation process. the conditional densities p(x |z − ) and p(x |z ), which are determined k k 1 k k (3-b) Measurement Update (Filtering) equations iteratively via the Time Update and Measurement Update equations. The basic idea is to find the probability density function corresponding to a ˆ = ˆ − + x k x k K kν k (10) hidden state x k at time step k given all the observations z1:k up to that time. and = − − The Time Update step is based upon the Chapman-Kolmogorov equation P k (I K kH k)P k (11) with − − − p(x k|z1:k−1) = p(x k|xk−1, z1:k−1)p(xk−1|z1:k−1)dxk−1 = t t + t 1 K k P k H k(H kP k H k U kR kU k) (12) (6) = p(x k|xk−1)p(xk−1|z1:k−1)dxk−1 and I the Identity matrix. The above Kalman gain K k corresponds to the mean of the condition- via the Markov property. al distribution of x k upon the observation z k or equivalently, the matrix The Measurement Update step is based upon the Bayes rule that would minimize the mean square error P k within the class of linear estimators. p(zk|x k)p(x k|z1:k−1) This interpretation is based upon the following observation. Having x p(x k|z1:k) = (7) p(z k|z1:k−1) a Normally distributed random-variable with a mean mx and variance Pxx , with and z another Normally distributed random-variable with a mean mz and variance P , and having P = P the covariance between x and z, the con- p(z k|z1:k−1) = p(z k|x k)p(x k|z1:k−1)dx k zz zx xz ditional distribution of x|z is also Normal with A proof of the above is given in the appendix. mx|z = mx + K(z − mz) EKF is based on the linearization of the transition and measurement equations, and uses the conservation of Normal property within the class with = −1 of linear functions. We therefore define the Jacobian matrices of f with K Pxz Pzz respect to the state variable and the system noise as A k and W k respec- which corresponds to our Kalman gain. tively; and similarly for h, as H k and U k respectively. More accurately, for every row i and column j we have 1.1.3 Parameter Estimation For a parameter-set in the model, the calibration could be carried out = ˆ = ˆ Aij ∂fi/∂x j(xk−1, 0), Wij ∂fi/∂wj(xk−1, 0), via a Maximum Likelihood Estimator (MLE) or in case of conditionally ^ = ˆ − = ˆ − Gaussian state variables, a Quasi-Maximum Likelihood (QML) algorithm. Hij ∂hi/∂x j(x k , 0), Uij ∂hi/∂u j(x k , 0) Wilmott magazine 3 N | = 2 + − To do this we need to maximize k=1 p(z k z1:k−1), and given the where the scaling parameters α, β, κ and λ α (na κ) na will be cho- Normal form of this probability density function, taking the logarithm, sen for tuning. changing the signs and ignoring constant terms, this would be equiva- For k in 1...N lent to minimizing over the parameter-set (1-b) State Space Augmentation: As mentioned earlier, we concatenate the state vector with the system noise and the observation noise, and cre- N z − m L = ln(P ) + k z k (13) ate an augmented state vector for each time-step 1:N z k z k P = z k z k k 1 xk−1 a x − = wk−1 For the KF or EKF we have k 1 uk−1 − m = zˆ z k k and therefore xˆ k−1 and a a − xˆ − = E [x − |z k] = 0 P = H P Ht + U R Ut k 1 k 1 z k z k k k k k k k 0 and 1.2 The Unscented Kalman Filter and Kushner’s Pk−1 Pxw (k − 1|k − 1) 0 a = − | − − | − Nonlinear Filter Pk−1 Pxw (k 1 k 1) Pww (k 1 k 1) 0 1.2.1 Background and Notations 00Puu (k − 1|k − 1) Recently Julier and Uhlmann (1997) proposed a new extension of the (1-c) The Unscented Transformation: Following this, in order to use the Kalman Filter to Nonlinear systems, different from the EKF. The new Normal approximation, we need to construct the corresponding Sigma method called the Unscented Kalman Filter (UKF) will calculate the Points through the Unscented Transformation: mean to a higher order of accuracy than the EKF, and the covariance to a = ˆ a the same order of accuracy. χk−1(0) xk−1 Unlike the EKF, this method does not require any Jacobian calculation = ..
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