Data Analysis and Machine Learning Lecture 10: Applications of Kalman Filtering

Transcription

1 Data Analysis and Machine Learning Lecture 10: Applications of Kalman Filtering Lecturer, Imperial College London Founder, CEO, Thalesians Ltd

2 The Newtonian system I Let r be the position of a particle, v its velocity, and a its (constant) acceleration. If t is the time, then dr/dt dv/dt da/dt = The state of our system x, is the vector (r, v, a) T. r v a Denote the matrix of ones and zeros above by A. Then the evolution of the state is described by the matrix differential equation dx t dt = Ax t. By analogy with the scalar ordinary differential equation (ODE), its solution is x t = e At x 0, where x t = (r t, v t, a t ) T is the state of the system at time t and the matrix exponential is defined by the (matrix) Taylor series expansion e At := j=0 (At) j. j!.

3 The Newtonian system II Note that, for s t T, x t = e A(t s) x s. Let us now discretise this equation using h k := t s as the time interval between the time ticks k 1 and k (for k N ): where x k = F k x k 1, F k := e Ah (Ah k = k ) j j! j=0 = hk h 2 2 k = 1 h k hk 2/2 0 1 h k, since all the powers of A greater than 2 are zero matrices. We observe that this process model fits the Kalman filter paradigm.

4 The variance-scaled Wiener process with drift Consider the stochastic differential equation (SDE) dx t = µ dt + σ dw t. Integrating both sides, we get hence Note that, for s t T, t 0 t t dx s = µ ds+ σ dw s, 0 0 X t = X 0 + µt + σw t. X t = X s + µ(t s)+σ(w t W s ). Let us now discretise this equation using h k := t s as the time interval between the time ticks k 1 and k (for k N ): x k = F k x k 1 + a k + w k, where F k = I, a k = µh k, w k N(0, σ 2 h k ). We observe that this process model also fits the Kalman filter paradigm.

5 Tutorial Consider the one-dimensional Ornstein-Uhlenbeck (OU) process dx t = θ(µ X t ) dt + σ dw t, where X t R, X 0 = x 0, and θ > 0, µ and σ > 0 are constants. Discretise this SDE and formulate the Kalman filter process model. Write a Python function to generate realisations of this process. Implement a Kalman filter in Python. Apply it to your process model and test on generated process realisations (experiment with adding different levels of white noise to the observations).

6 Tutorial: Hints I First, we must solve the OU process SDE. The solution is well-known, so you may simply quote it. However, this is how you would solve it. Recall Itō s lemma: if dx t = µ t dt + σ t dw t, f(x, t) is a function, and Y t := f(x t, t), then ( f dy t = t + f µ t x + (σ ) 2 2 ) f 2 x 2 dt + σ f t x dw t. We apply the lemma for the OU SDE (so that, in our case, µ t = θ(µ x t ) and σ t = σ) and f(x, t) := xe θt. Then f t = θxe θt, f x = eθt, 2 f = x 2 0, and ( dy t = θx t e θt + θ(µ X t )e θt) dt + σe θt dw t On integrating both sides, we get = θµe θt dt + σe θt dw t. t 0 t dy u = θµe θu ds+ 0 X t e θt X 0 = µ(e θt 1)+ t σe θu dw u, 0 t σe θu dw u. 0

7 Tutorial: Hints II Thus the solution of the OU SDE is It is easy to see that, for s t T, t X t = X 0 e θt + µ(1 e θt )+ σe θ(t u) dw u. 0 t X t = X s e θ(t s) + µ(1 e θ(t s) )+ σe θ(t u) dw u. s It is well known that an Itō integral, t s f(u) dw u, of a deterministic integrand, f(u), is a Gaussian random variable with mean 0 and variance t 0 f 2 (u) du. In our case, f(u) = σe θ(t u), and t 0 f 2 (u) du = σ2 2θ ( 1 e 2θ(t s)).

8 Tutorial: Hints III Let us now discretise the equation for X t using h k := t s as the time interval between the time ticks k 1 and k (for k N ): x k = F k x k 1 + a k + w k, ( ( ) ) where F k = e θh k, a k = µ(1 e θh k), w k N 0, σ2 2θ 1 e 2θh k. We observe that this process model fits the Kalman filter paradigm, and we can apply the Kalman filter equations directly.

9 The loglikelihood function for the Kalman filter We can view the Kalman filter as parameterised by some parameter vector θ R d θ, so that F k, a k, Q k, H k, b k, R k are functions of θ: F k (θ), a k (θ), Q k (θ), H k (θ), b k (θ), R k (θ). Then, for all T N, we can show that the following holds: ln(l(θ)) := ln(f(y 1,...,y T θ)) = T i=1 ln(f(y t θ; y 1,...,y t 1 )) = T 2 ln(2π) 1 T ln(det(s t )) 1 T 2 2 i=1 i=1 ỹ T t S 1 t ỹ. We can use this loglikelihood function as our objective function in the estimation of the parameter vector θ. See, for example, tutorials/xfghtmlnode92.html

10 Outlier detection Note that the predicted observation is distributed as ( ) N H k ˆx k k 1, H k P k k 1 Hk T. Just as we can assign a z-score to y k if y is one-dimensional, we can assign a Mahalanobis distance to it (which corresponds to the z-score if y k is one-dimensional). In general, the Mahalanobis norm of a vector y with respect to N (µ, Σ) is given by y N(µ,Σ) = (y µ) T Σ 1 (y µ). It measures the distance of y R m from the centroid (multidimensional mean) of the distribution. y 2 N(µ,Σ) follows the χ2 -distribution with m degrees of freedom. Thus we can set a cut-off for y k, e.g. on the basis of the 0.975th quantile of the χ 2 -distribution.

11 The extended Kalman filter (EKF) Suppose that our n-dimensional state vector evolves according to the process model x k = f k (x k 1 )+w k, and suppose that our m-dimensional observation vector is related to the state by the observation model y k = h k (x k )+v k where f and h are differentiable functions, w k N(0, Q k ) are R n -valued random variables, and v k N(0, R k ) are R m -valued random variables. {x 0, w 1,...,w k, v 1,...,v k } are mutually independent. The extended Kalman filter [McE66, SSM62], [Hay01, Section 1.6] algorithm is as follows: The prediction step: Predicted (prior) signal state estimate: ˆx k k 1 = f k (ˆx k 1 k 1 ). Predicted (prior) error covariance: P k k 1 = F k P k 1 k 1 F k T + Qk. The update (or correction) step: Innovation, a.k.a. observation residual: ỹ k = y k h k (ˆx k k 1 ). Innovation (observation residual) covariance:s k = H k P k k 1 H k T + Rk. (Optimal) Kalman gain: K k = P k k 1 H T k S 1. k Updated (posterior) signal state estimate: ˆx k k = ˆx k k 1 + K k ỹ k. Updated (posterior) error covariance: P k k = (I K k H k )P k k 1. Here F k = f x and H k = ˆxk 1 k 1 h x. So the idea is simple: linearise the model ˆxk k 1 using Jacobians.

12 Credit risk Credit risk is the risk that an obligor does not honour his payment obligations [Sch03, page 1]. This risk is directly or indirectly associated with some events, called credit events. They can be hard (e.g. bankruptcy) or soft (e.g. restructuring). Credit risk can be categorised into credit deterioration and default risk. A bond that has credit risk associated with it is risky. A bond that has no credit risk associated with it is riskless. The credit spread is the difference between the (promised) yield on the risky bond and the yield on riskless bonds [KN15, pages ].

13 Credit spread How can we quantify the credit spread of a risky bond? One way is to solve for z t0 in the following equation: P t0 = n i=1 ( 1+ cf ti (s i t0 + z t0 ) δ where P t0 is the dirty market price of the bond at time, cf ti is the cash flow generated by the bond at time t i, s i is the zero-coupon swap rate of appropriate maturity for this cashflow, δ is the frequency of the cashflows expressed as a fraction of the year. The resulting z t0 is called the zero volatility spread, or Z-spread for short. For a given issuer, we are interested in modelling the term structure of the Z-spreads across a universe of this issuer s bonds. The Z-spread is then viewed as a function of τ : z t0 (τ); τ could be the time of maturity or, for example, the modified duration. ) i,

14 Modelling the Z-spread The function z t0 (τ) can be written as z t0 (τ) = g(τ; θ t0 ), where θ t0 is a d-dimensional vector of parameters. Its dependence on indicates that the Z-spread curve (as a function of τ) evolves as time progresses. One of our tasks, then, is to keep estimating θ t0 as moves on. Suppose that for a particular issuer we have a universe of K bonds with Z-spreads z (1),...,z (K) and maturities (or modified durations, etc.) τ (1),...,τ (K). Each of these Z-spreads may not lie exactly on the Z-spread curve z t0 (τ) due to the idiosyncracies of that particular bond, so we allow there to be an idiosyncratic spread, λ (k), k {1,...,K}: z (k) = z t0 (τ (k) )+λ (k). As indicated by their dependence on, the idiosyncratic spreads also evolve over time. We will have to keep computing their updated estimates as well.

15 Setting up the EKF: the state and observation Our state at is the vector (θ t0 ) 1. (θ t0 ) d x t0 = λ (1)... λ (K) We are observing individual bond prices, thus our observation at time is y t0 = P (k), the dirty market price of bond k for some k {1,...,K}. NB! Chances are that we our market feeds are providing us with clean prices. We should therefore remember to convert these prices to dirty prices before we proceed.

16 Setting up the EKF: the observation model The function h that maps our state to the corresponding observation is given by h (k) (x t0 ) = P (k) (z (k) ) n (k) = i=1 n (k) = i=1 cf (k) t i ( ( ) ) 1+ st i 0 + z (k) i δ (k) cf (k) t i ( ( ) ) 1+ st i 0 + z t0 (τ (k) )+λ (k) i, δ (k) where z t0 (τ (k) ) is, of couse, our curve model function, whose parameters θ t0 appear in our state, evaluated at the maturity of the bond, and λ (k) is the idiosyncratic spread for this particular bond, which is likewise in our state. Thus, given our state vector, we can evaluate h (k) (x t0 ). Note that the equation above also depends on the appropriate zero-coupon swap rates s i. These are fast-moving and can be provided exogenously. The reason why we need an EKF rather than a KF is that h (k) (x t0 ) is a nonlinear function of the state.

17 Setting up the EKF: the observation Jacobian Let the scalar parameter α t0 be a particular element of our parameter vector θ t0, so it is (θ t0 ) j for some j {1,...,d}. Then h (k) = α t0 xt0 h(k) z = z t0 xt0 α t0 xt0 δ n(k) (k) i=1 i cf (k) t i ( ( ) 1+ st i 0 + z t0 (τ (k) )+λ (k) where z α can be computed analytically for many simple curve models. t0 ) i z τ δ (k) α (k t0 Similarly, for j {1,...,K}, h (k) h (k) λ (j) = z t0 xt0 z λ (j) = δ (k) n(k) i cf (k) t i i=1 ( ( t xt0 xt st i +z t0 (τ (k) )+λ (k) ) i, j = k; 0 t δ (k)) 0 0, otherwise.

18 market depth orders liquidity RFQ, RFS, CAT responses other sources of information venues (economic indicators, news feeds, etc.) (D2C, D2D) trading strategy pricing quoting across the board: beta price alpha price spreading skewing (passive hedging) alerts, circuit breakers, recording and audit, visualisation, control hit-lift protection hedging position service optimal execution strategies risk service

19 Simon Haykin, editor. Kalman Filtering and Neural Networks. John Wiley and Sons, Inc., Robert L. Kosowski and Salih N. Neftci. Principles of Financial Engineering. Academic Press, Bruce A. McElhoe. An assessment of the navigation and course corrections for a manned flyby of mars or venus. IEEE Transactions on Aerospace and Electronic Systems, AES-2(4): , Philipp J. Schönbucher. Credit Derivatives Pricing Models: Models, Pricing and Implementation. Wiley Finance, Gerald L. Smith, Stanley F. Schmidt, and Leonard A. McGee. Application of statistical filter theory to the optimal estimation of position and velocity on board a circumlunar vehicle. NASA technical report R-135, National Aeronautics and Space Administration (NASA), 1962.