Financial Econometrics Kalman Filter: some applications to Finance University of Evry - Master 2

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1 Financial Economerics Kalman Filer: some applicaions o Finance Universiy of Evry - Maser 2 Eric Bouyé January 27, 2009 Conens 1 Sae-space models 2 2 The Scalar Kalman Filer 2 21 Presenaion 2 22 Summary of he dieren seps 5 23 Is i an opimal linear esimae? 5 24 Predicion, lering and smoohing 6 3 The Kalman Filer 6 4 Generalized Vasicek Term Srucure Models 9 41 The Model 9 42 Empirical applicaion 10 5 Invesmen Sraegies Tracking problems Innovaion represenaion Empirical applicaion 11 6 Appendix: The condiional gaussian disribuion 13 Universiy of Evry and Sociéé Générale Conac: ericbouye@sgamcom 1

2 1 Sae-space models A general specicaion for he Kalman ler - Kalman 1960, [6] - is provided below The represenaion of Harvey 1991 [5] is used For an excellen presenaion abou lering - and elegan proofs-, we refer o Le Gland 2007 Measuremen equaion y = X β + d + ε = 1,, T 1 Transiion equaion β = T β 1 + c + R η = 1,, T wih β he m 1 sae vecor unobserved ha represens he developmen over ime of he sysem y is he N 1 vecor of he observed variables d and ε are N 1 vecors and X is a N m marix T is an m m marix, R a m g marix, c a m 1 vecor and η a g 1 vecor The disurbances are such ha { ε N 0, H η N 0, Q and ε and η are uncorrelaed Specically, { E ε η s = 0 s, = 1,, T E ε β 0 = 0 for = 1,, T 2 The Scalar Kalman Filer 21 Presenaion For he sake of simpliciy, he scalar model is rs sudied Measuremen equaion y = X β + ε = 1,, T Transiion equaion β = T β 1 + c + η = 1,, T η c β T β 1 Time Can we ler β such ha he eecs of η are minimized? 2

3 η ɛ c β X y T β 1 Time Can we ler y and esimae β such ha he eecs of ε and η are minimized? η ɛ c β X y T β 1 Time β X ŷ T Time K β 1 β In order o predic an esimae of he oupu y, he KF uses an a priori esimae of β noed β such ha β = T β 1 + c Then he dierence beween he esimaed oupu ŷ and acual oupu can be compued y ŷ = y X β and his residual is used o rene he a priori esimae: he a poseriori esimae β is calculaed β = β + K y ŷ = β + K y X β 3

4 Quesion: wha is he appropriae K such ha an opimal esimaor is compued? The errors are dened { e = β β a priori e = β β a poseriori wih respecive variances { p e 2 = E p = E e 2 A Kalman Filer minimizes he a poseriori error variance p Then, le us compue he value of K ha minimizes p ie K = arg min p K [ = arg min E β β ] 2 [ = arg min E β β ] 2 K y X β Then p K K = 0 K = From his resul, one can remark ha if he a priori error is large, hen X p X 2 p + H K 1 X and he condence in he a priori esimae is quie low, if he a priori error is small, hen he correcion by he KF will be small In oher words, if our rs esimaes have been quie good, here is lile need o correc i, if H is large, he condence in he measuremen is quie low The KF has more condence in he previous esimaes 4

5 22 Summary of he dieren seps 1 Model specicaion Measuremen equaion y = X β + ε = 1,, T 2 Predicion sep Transiion equaion β = T β 1 + c + η = 1,, T and he a priori covariance is β = T β 1 + c p = T 2 p 1 + Q 3 Correcion sep Given he addiional informaion on he oupu, one can correc he esimaor wih K he gain: K = and he a poseriori covariance is β = β + K y X β X 2 p X p + H p = p 1 X K 23 Is i an opimal linear esimae? Is he Kalman Filer an opimal linear esimae? ˆβ = a 1ˆβ 1 + a 2 y min p [ min E β β ] 2 { p a 1 = 0 p a 2 = 0 E [β β ] β 1 = 0 E [β β ] y = 0 ha corresponds o he orhogonaliy condiions discussed above Then we obain a 1 = T 1 a 2 X 5

6 Then, by subsiuing in he above equaion ˆβ = a 1ˆβ 1 + a 2 y = T 1 a 2 X ˆβ 1 + a 2 y = T ˆβ 1 + a 2 y T X ˆβ 1 ˆβ = ˆβ + a 2 y X β 24 Predicion, lering and smoohing One disinguishes hree ypes of use of he Kalman ler 1 Predicion 2 Filering 3 Smoohing 3 The Kalman Filer For he sake of simpliciy, le denoe E x = E x y 0,, y 1 and E x = E x y 0,, y The sae-space model is: Measuremen equaion y = X β + d + ε = 1,, T Transiion equaion β = T β 1 + c + R η = 1,, T Theorem The Kalman-Buci ler The Kalman-Buci ler is dened as follows E β = T E β 1 + c 2 and P = T P 1 T + R Q R 3 E β = E β [ + K y X E β ] + d 4 wih K he Kalman gain: P = [I K X ] P 5 K = P X [ X P X + Q ] 1 6 6

7 The iniial sae of he sysem is as follows { E β 0 = E β0 P 0 = Var β 0 = P 0 Proof for 2 E β E β = T E β 1 + c + R E η = T E β 1 + c Proof for 3 P = E [ β E β β E β ] = E {[ T β 1 E β 1 + R E η ] [ T β 1 E β 1 + R E η ] } = T E { β 1 E β 1 β 1 E } β 1 T + R E η η R +T E { β 1 E β 1 η } T + R E { η β 1 E } β 1 T By noing ha E { β 1 E β 1 η } = 0, we ge P = T P 1 T + R Q R Proof for 4 and 5 Le us noe ν he innovaion vecor ν = y E y y 0,, y 1 = y E y We have E β y 0,, y = E β = E β E β = E β + E β β y 0,, y + E β β y 0,, y 1, ν + E β β ν From above, β E β = v E β E β = β E β E β β ν hen, we can calculae P : P = E [β E β β E β ] = E [ β E β E β β ν β E β E β β ν ] 7

8 We need o compue he condiial mean and variance of β E β Firs, we need o compue he variance of ν and is covariance wih β E β We have E ν ν = E [ y E y y E y ] = E [ X β E β + ε X β E β ] + ε = X E [ β E β β E β ] X + E [ ε β E β ] X +X E [ β E β ε ] + E ε ε E ν ν = X P X + H Also he covariance is such as E β E β ν = E β E β X β E β + ε = E [ β E β β E β E ν ν = P X ] X + E [ β E β Then, he join disribuion of β E β and ν is β E β P N 0, P X ν X P X P X + H hen if Q is inverible, hen X P X + H is also inverible, and from a propery of he gaussian condiional disribuion see Appendix 2, ε ] E β = E β + P X [ X P X + H ] 1 ν and P = P P X [ X P X + H ] 1 X P Se of equaions for he Kalman Filer: alernaive noaion E β = T E β 1 + c P = T P 1 T + R Q R E y = X E β + d ν = y E y E V = X P X + Q E β = E β + P X [E V ] 1 ν P = P P X [E V ] 1 X P ˆβ 1 = T ˆβ c ˆP 1 = T ˆP 1 1 T + R Q R ŷ 1 = X ˆβ 1 + d ˆν = y ŷ 1 ˆV = X ˆP 1 X + Q ˆβ = ˆβ 1 + ˆP 1 X 1 ˆV P = ˆP 1 ˆP 1 X ˆV 1 ν X ˆP 1 8

9 4 Generalized Vasicek Term Srucure Models 41 The Model From Babbs and Ben Nowman 1999 Their model is a subclass of Langeieg's linear Gaussian models of he erm srucure: J B M, = exp τ R w τ u ξ j τ X j τ wih R = µ + Q J θ q q=1 j=1 κ jq ξ j 1 2 j=1 Q J q=1 j=1 κ jq ξ j 2 W τ = J j= u ξ j τ J i=1 j=1 Q q=1 θ q κ jq ξ j Q J q=1 i=1 J H Q ξ i + ξ j τ q=1 κ iq κ jq ξ i ξ j κ iq κ jq ξ i ξ j wih τ = M and u x = 1 e x x The heoreical yield curve is such ha R + τ i, = log B + τ i, τ = A 0 τ i A 1 τ i X for i = 1,, N wih A 0 τ i = R w τ i and A 1 τ i = u ξ j τ i a J 1 vecor The ime invarian sae-space model is specied as follows: Measuremen equaion R = Z ψ X + d ψ + ε = 1,, T Transiion equaion X = T ψ X 1 + η = 1,, T wih ε N 0, H ψ and η N 0, V ψ, A 0 τ 1, ψ A 0 τ 2, ψ d ψ =, A 0 τ N, ψ 9

10 Z ψ = A 1 τ 1, ψ A 1 τ 2, ψ A 1 τ N, ψ = u ξ 1 τ 1 u ξ 2 τ 1 u ξ J τ 1 u ξ 1 τ 2 u ξ 2 τ 2 u ξ J τ 2 u ξ 1 τ N u ξ 2 τ N u ξ J τ N and T ψ = e ξ 1 e ξ 2 e ξ J, In pracice, H ψ is assumed o be diagonal wih specic variances h h 2 0 H ψ = 0 0 h N 42 Empirical applicaion 5 Invesmen Sraegies Tracking problems For his applicaion we refer o Roncalli and Weisang 2008 Le assume we wan o replicae he performance of a fund or an index r F for example no invesable wih some liquid insrumens eg fuures ha are called facors Le assume ha we have m facors wih respecive reurns r i for i = 1,, m I is assumed ha he performance of he fund can be linearly explained by he reurns of he facor One can dene he following racking problem: { r F = r w + η 7 w = w 1 + ε In he above TP, he predicion par of he KF algorihm is used in pracice 51 Innovaion represenaion Le us dene he innovaion represenaion of he sae-space model We noe ha ˆβ +1 = T +1 K X ˆβ 1 + K y + c +1 K d = T +1ˆβ 1 + c +1 + K y X ˆβ 1 d 10

11 wih K = T +1 ˆP 1 X 1 ˆV Then, y = d + X ˆβ 1 + ν ˆβ +1 = c +1 + T +1ˆβ 1 + K ν = 1,, T = 1,, T wih Appliying his represenaion o our TP problem above, we ge { r F = r ŵ 1 + ˆν ŵ +1 = ŵ 1 + ˆP 1 r ˆν 8 ˆV ˆν ˆV Commens he normalized racking-error The i h facor is hen ajused as follows 52 Empirical applicaion ŵ+1 i = ŵ+1 i ŵi 1 9 ŵ+1 i ˆν m = ˆP 1 ˆV i,j rj k 10 j=1 11

12 References [1] Babbs Simon H and K Ben Nowman 1999, Kalman Filering of Generalized Vasicek Term Srucure Models, Journal of Financial and Quaniaive Analysis, 34, No 1, March [2] Cheever Erik, Kalman Filer Tuorial, Swarhmore College [3] Davidson Russell and James G MacKinnon 1993, Esimaion and Inference in Economerics, Oxford Universiy Press New York [4] Davidson Russell and James G MacKinnon 2003, Economeric Theory and Mehods, Oxford Universiy Press New York [5] Harvey A C 1991, Forecasing, Srucural Time Series and he Kalman Filer, Cambridge: Cambridge Universiy [6] Kalman RE 1960, A new approach o linear lering and predicion problems, J Basic Eng, 82, 35-45, March [7] Le Gland François 2007, Filre de Kalman e modèles de Markov cachés, Noes de cours [8] Roncalli Thierry 2005, Economérie Financière II, Lecure Noes , Maser Ingénierie Economique e Saisique Appliquée, Universié d'evry [9] Roncalli Thierry and Guillaume Weisang 2008, Tracking Problems, Hedge Fund Replicaion and Alernaive Bea, Working Paper [10] Welch Greg and Gary Bishop 2006, An Inroducion o he Kalman Filer, Working Paper 12

13 6 Appendix: The condiional gaussian disribuion Le assume a gaussian vecor X 1, X 2 such as X 1, X 2 N µ1 µ 2 Σ1 Σ, 12 Σ 21 Σ 2 If Σ 2 is inverible, he condiional densiy of he random vecor X 1 given X 2 = x 2, is gaussian wih mean µ 1 2 and covariance Σ 1 2 such ha µ 1 2 = µ 1 + Σ 12 Σ 1 2 x 2 µ 2 and Σ 1 2 = Σ 1 Σ 12 Σ 1 2 Σ 21 13