Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Brownian Moving Averages and Applications Towards Interst Rate Modelling F . Hubalek, T. Bl¨ ummel October 14, 2011
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Table of contents Data and Observations 1 Brownian Moving Averages 2 BMA-driven Vasicek-Model 3 Literature 4
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature The Data For different time lags h ∈ N , we are interested in the (overlapping) increments of the interest rates IR ( t + h ) − IR ( t ) , t ∈ N . Interest Rate 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 increments [ IR ( t ∗ h + 1 ) − IR (( t − 1 ) ∗ h + 1 ) ]
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Graphic I IR.EUR.M01.EURIBOR 4 3 cbind(y1, y2, y4) 2 1 0 0 100 200 300 400 500 Figure: overlapping-increments, non-ol-increments, straight line
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Graphic II IR.EUR.M03.EURIBOR 4 3 cbind(y1, y2, y4) 2 1 0 0 100 200 300 400 500 Figure: overlapping-increments, non-ol-increments, straight line
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Graphic III IR.EUR.M06.EURIBOR 4 3 cbind(y1, y2, y4) 2 1 0 0 100 200 300 400 500 Figure: overlapping-increments, non-ol-increments, straight line
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Definition and Properties Definition: Brownian Moving Average Let ( B u ) u ∈ R be a two-sided Brownian motion and ϕ a Borel-measurable function, which is zero on ( 0 , ∞ ) and ϕ ( . − t ) − ϕ ( . ) ∈ L 2 ( R ) for all t ≥ 0. The Brownian moving average (BMA) concerning ϕ is defined as � X ϕ t := ( ϕ ( u − t ) − ϕ ( u )) dB u . R Properties: Its variance is given by � ( ϕ ( u − t ) − ϕ ( u )) 2 du , X ϕ � � Var = t ≥ 0. t R X ϕ is a centered Gaussian process with stationary increments.
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Examples Brownian Moving Average � X ϕ t := ( ϕ ( u − t ) − ϕ ( u )) dB u t ≥ 0 R 100 Brownian Motion (BM): 50 ϕ ( u ) = 1 { u ≤ 0 } . cbind(Xf13) 0 Fractional BM (FBM): ϕ ( u ) = c H ( − u ) H − 1 2 1 { u ≤ 0 } −50 for H ∈ ( 0 , 1 ) . −100 0 500 1000 1500 2000 2500 tg1 Figure: path of FBM ( H = 0 . 8)
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature BMA-Semimartingales Theorem [Cherny] X ϕ is a � F B � -semimartingale if and only if there exist 1 t α ∈ R and ψ ∈ L 2 ( R ) such that 0 � ϕ ( u ) = α + ψ ( v ) dv , u ≤ 0. u If X ϕ is a F B � � -semimartingale it is continuous, and its 2 t canonical decomposition is given by � X ϕ t = ( χ ( u − t ) − χ ( u )) dB u + α B t , R where χ ( u ) = ϕ ( u ) − α 1 { u ≤ 0 } .
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Application Fractional Brownian Motion ϕ ( u ) = c H ( − u ) H − 1 2 1 { u ≤ 0 } Brownian Motion Modification ϕ ( u ) = 1 { u ≤ 0 } Regularized FBM (Rogers) ϕ ( u ) = c H ( β − u ) H − 1 2 1 { u ≤ 0 } � H − 1 2 1 { u ≤ 0 } � β − u ϕ ( u ) = c H 1 − cu
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Path of BMA 4 3 Xr11 2 1 0 0 100 200 300 400 500 tg2
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Variance of BMA 6 5 4 cbind(Z2, Varr1) 3 2 1 0 0 100 200 300 400 500 hh
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature The Dynamics Dynamics of the BMA-driven Vasicek-model dr = ( b − ar ) dt + σ dX ϕ Remarks: a , b and σ are positive constants. For ϕ ( u ) = 1 { u ≤ 0 } this is the classical Vasicek-model.
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Zero coupon bond prices Due to Gaussianity we have � T � � e − r ( s ) ds |F t B ( t , T ) = E t �� T �� T � � �� − 1 = exp E r ( s ) ds |F t 2 Var r ( s ) ds |F t t t � T Representation of t r ( s ) ds � T r ( s ) ds = 1 � 1 − e − a ( T − t ) � � r ( t ) − b �� � b ( T − t ) + + a a t T + σ � � 1 − e − a ( T − u ) � dX ϕ u a t
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature dr = ( b − ar ) dt + σ dX ϕ Conditional Expectation of X ϕ � X ϕ = X ϕ ( ϕ ( u − T ) − ϕ ( u − t )) 1 { u ≤ t } dB u =: Y T ,ϕ � � E T |F t t + t R Conditional Variance of X ϕ X ϕ = Var X ϕ X ϕ � � � � �� Var T |F t T − Var E T |F t = Var X ϕ T − VarY T ,ϕ t
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature dr = ( b − ar ) dt + σ dX ϕ � T dX ϕ 1 − e − a ( T − u ) � � Conditional Expectation of u t �� T � T � � 1 − e − a ( T − u ) � � 1 − e − a ( T − u ) � dY T ,ϕ dX ϕ E u |F t = u t t � T dX ϕ 1 − e − a ( T − u ) � � Conditional Variance of u t �� T � � 1 − e − a ( T − u ) � dX ϕ Var u |F t = t � T � T � 1 − e − a ( T − u ) � � 1 − e − a ( T − u ) � dY T ,ϕ dX ϕ = Var u − Var u t t
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Thank you for your attention!
Data and Observations Brownian Moving Averages BMA-driven Vasicek-Model Literature Literature Cherny ” When is a moving average a semimartingale? ” Rogers ” Arbitrage with fractional Brownian motion ” Kl¨ uppelberg, et al. ” Conditional characteristic functions of processes related to fractional Brownian motion ” Cheridito ” Regularizing fractional Brownian motion with a view towards stock price modelling ” Basse ” Gaussian moving averages and semimartingales ”
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