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Applications in Finance by Ramaprasad Bhar_11

Tham khảo tài liệu 'applications in finance by ramaprasad bhar_11', tài chính - ngân hàng, ngân hàng - tín dụng phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | CDS Options Implied Volatility and Unscented Kalman Filter 239 yt h xt q 9.35 Transition equation xt f xt-i Dt. 9 36 _ T oRt for t t _ EũtũtT K . . 9.37 gD otherwise __T oQt for t t E qqT n 7 . 9.38 go otherwise E lt tT 0 tt 9.39 where f is the process modest is the state of the system at tirtie and Dtis an additive process noise. It is assumed that the only information available about this system is a set of noisy observations These observations are related to the state vector by the nonlinear equation 9.35 h is the observation model that transfers the state space vector into observation space apds an additive measurement noise. It is assumed thait and qare independent uncorrelated white noise. The true states of the system are unknown and need to be estimated. We let t 1 tbe the conditional mean at timteH conditional on all observations up tb that is t i t E t 1 q 94 whereq y1 y2 - - y T- The estimated covariance is I t E xt 1 - t 1 t xt 1- act 1 t qẼ- 9.41 The EKF Extended Kalman Filter is designed to estimate a subclass of nonlinear state-space models with additive Gaussian process and measurement noises. The Extended Kalman Filter assumes that the errors in the state estimates are small. As a consequence the predicted mean is approximated by 240 Stochastic Filtering with Applications in Finance Ei xt Im ffxt qg 9.42 f . That is the predicted mean is equal to the prior mean projected througtf . This estimate does not consider the actual distribution of the errors on the state prediction. Further it is assumed that the state errors propagate through a separate linearized system and the covariance of these errors evolves according to m t f fT Q 9.43 where 7f is the Jacobian matrix tff evaluated aboii t. Similar assumptions are made in predicting t 1 t Fyy t 1 t Unfortunately the EKF suffers two well-known problems first the required Jacobian matrices i.e. the matrix linear approximations of nonlinear functions can be extremely difficult and error-prone to .