Brownian Moving Averages and Applications Towards Interst Rate Modelling

Transcription

1 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

2 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

3 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 EURIBOR 03M EURIBOR 06M GBP LIBOR 01M GBP LIBOR 03M GBP LIBOR 06M non-overlapping incremens [IR( h + 1) IR(( 1) h + 1)]

4 Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Graphic I IR.EUR.M01.EURIBOR cbind(y1, y2, y4) Figure: overlapping-incremens, non-ol-incremens, sraigh line

5 Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Graphic II IR.EUR.M03.EURIBOR cbind(y1, y2, y4) Figure: overlapping-incremens, non-ol-incremens, sraigh line

6 Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Graphic III IR.EUR.M06.EURIBOR cbind(y1, y2, y4) Figure: overlapping-incremens, non-ol-incremens, sraigh line

7 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.

8 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 {u 0} for H (0, 1). cbind(xf13) g1 Figure: pah of FBM (H = 0.8)

9 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

10 Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Applicaion Fracional Brownian Moion ϕ(u) = c H ( u) H {u 0} Brownian Moion ϕ(u) = 1 {u 0} Modificaion Regularized FBM (Rogers) ϕ(u) = c H (β u) H {u 0} ( ) H 1 ϕ(u) = c β u 2 H 1 cu 1 {u 0}

11 Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Pah of BMA Xr g2

12 Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Variance of BMA cbind(z2, Varr1) hh

13 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.

14 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

15 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

16 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

17 Daa and Observaions Brownian Moving Averages BMA-driven Vasicek-Model Lieraure Thank you for your aenion!

18 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