Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Brownian Moving Averages and Applicaions Towards Iners Rae Modelling F. Hubalek, T. Blümmel Ocober 14, 2011
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Table of conens 1 Daa and Observaions 2 Brownian Moving Averages 3 BMA-driven Vasicek-Model 4 Lieraure
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure The Daa For differen ime lags h N, we are ineresed in he (overlapping) incremens of he ineres raes IR ( + h) IR (), N. Ineres Rae Day 1 Day 2 Day 3 Day 4 Day 5... EURIBOR 01M 4.97 4.95 4.96 4.98 5.00... EURIBOR 03M 4.92 4.88 4.90 4.89 4.91... EURIBOR 06M 4.85 4.81 4.83 4.82 4.84... GBP LIBOR 01M 5.86 5.86 5.87 5.90 5.90... GBP LIBOR 03M 5.90 5.90 5.90 5.89 5.89... GBP LIBOR 06M 5.92 5.93 5.93 5.93 5.94............ non-overlapping incremens [IR( h + 1) IR(( 1) h + 1)]
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Graphic I IR.EUR.M01.EURIBOR cbind(y1, y2, y4) 0 1 2 3 4 0 100 200 300 400 500 Figure: overlapping-incremens, non-ol-incremens, sraigh line
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Graphic II IR.EUR.M03.EURIBOR cbind(y1, y2, y4) 0 1 2 3 4 0 100 200 300 400 500 Figure: overlapping-incremens, non-ol-incremens, sraigh line
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Graphic III IR.EUR.M06.EURIBOR cbind(y1, y2, y4) 0 1 2 3 4 0 100 200 300 400 500 Figure: overlapping-incremens, non-ol-incremens, sraigh line
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Definiion and Properies Definiion: Brownian Moving Average Le (B u ) u R be a wo-sided Brownian moion and ϕ a Borel-measurable funcion, which is zero on (0, ) and ϕ(. ) ϕ(.) L 2 (R) for all 0. The Brownian moving average (BMA) concerning ϕ is defined as X ϕ := R (ϕ (u ) ϕ (u)) db u. Properies: Is variance is given by Var ( X ϕ ) = (ϕ (u ) ϕ (u)) 2 du, 0. R X ϕ is a cenered Gaussian process wih saionary incremens.
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Examples Brownian Moving Average X ϕ := (ϕ (u ) ϕ (u)) db u 0 R Brownian Moion (BM): ϕ(u) = 1 {u 0}. Fracional BM (FBM): ϕ(u) = c H ( u) H 1 2 1 {u 0} for H (0, 1). cbind(xf13) 100 50 0 50 100 0 500 1000 1500 2000 2500 g1 Figure: pah of FBM (H = 0.8)
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure BMA-Semimaringales Theorem [Cherny] 1 X ϕ is a ( F B ) -semimaringale if and only if here exis α R and ψ L 2 (R) such ha 0 ϕ (u) = α + u ψ (v) dv, u 0. 2 If X ϕ is a ( F B ) -semimaringale i is coninuous, and is canonical decomposiion is given by X ϕ = (χ (u ) χ (u)) db u + αb, where χ (u) = ϕ (u) α1 {u 0}. R
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Applicaion Fracional Brownian Moion ϕ(u) = c H ( u) H 1 2 1 {u 0} Brownian Moion ϕ(u) = 1 {u 0} Modificaion Regularized FBM (Rogers) ϕ(u) = c H (β u) H 1 2 1 {u 0} ( ) H 1 ϕ(u) = c β u 2 H 1 cu 1 {u 0}
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Pah of BMA Xr11 0 1 2 3 4 0 100 200 300 400 500 g2
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Variance of BMA cbind(z2, Varr1) 0 1 2 3 4 5 6 0 100 200 300 400 500 hh
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure The Dynamics Dynamics of he BMA-driven Vasicek-model dr = (b ar) d + σdx ϕ Remarks: a, b and σ are posiive consans. For ϕ(u) = 1 {u 0} his is he classical Vasicek-model.
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Zero coupon bond prices Due o Gaussianiy we have B(, T ) = E [e ] T r(s)ds F [ [ ] [ T ]] = exp E r(s)ds F 1 T 2 Var r(s)ds F Represenaion of T T r(s)ds = 1 a r(s)ds [ ( b (T ) + 1 e a(t )) ( r () b )] + a + σ a T ( 1 e a(t u)) dx ϕ u
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure dr = (b ar) d + σdx ϕ Condiional Expecaion of X ϕ E [ X ϕ T F ] = X ϕ + R (ϕ(u T ) ϕ(u )) 1 {u } db u =: Y T,ϕ Condiional Variance of X ϕ Var [ X ϕ T F ] = Var X ϕ T Var [ E [ X ϕ T F ]] = Var X ϕ T VarY T,ϕ
Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure dr = (b ar) d + σdx ϕ Condiional Expecaion of T E [ T ( 1 e a(t u) ) dx ϕ u ] ( 1 e a(t u)) T ( dxu ϕ F = 1 e a(t u)) dyu T,ϕ Condiional Variance of T Var [ T = Var T ( 1 e a(t u)) dx ϕ u F ] = ( 1 e a(t u) ) dx ϕ u ( 1 e a(t u)) T ( dxu ϕ Var 1 e a(t u)) dyu T,ϕ
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Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Lieraure Cherny When is a moving average a semimaringale? Rogers Arbirage wih fracional Brownian moion Klüppelberg, e al. Condiional characerisic funcions of processes relaed o fracional Brownian moion Cheridio Regularizing fracional Brownian moion wih a view owards sock price modelling Basse Gaussian moving averages and semimaringales