Local-Momentum Autoregression and the Modeling of Interest Rate Term - PowerPoint PPT Presentation

Local-Momentum Autoregression and the Modeling of Interest Rate Term Structure Jin-Chuan Duan Business School, Risk Management Institute, and Department of Economics (www.rmi.nus.edu.sg/duanjc) National University of Singapore (April 2015) (The 7 th Annual NYU-Stern School Volatility Institute Conference) Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 1 / 27

Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 2 / 27

Standard mean-reversion ∆ X t = κ x ( µ − X t − 1 ) + σ x ε t ε t | F t − 1 ∼ D ( 0 , 1 ) Is the AR(1) model of this type suitable for modeling interest rates or other highly persistent systems with long periods of directional moves? Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 3 / 27

How about adding a latent stochastic central tendency factor? ∆ X t = κ x ( µ t − X t − 1 ) + σ x ε t ∆ µ t = κ µ (¯ µ − µ t − 1 ) + σ µ ǫ t ε t | G t − 1 ∼ D ( 0 , 1 ) Note: Adding a latent stochastic central tendency factor is an idea in Balduzzi, Das and Foresi (1998, Review of Economics and Statistics ), because empirical evidence suggests that the short-term interest rate tends to move towards the long term interest rate. Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 4 / 27

Local-momentum with latent central tendency (LM-CTAR) � ¯ � ∆ X t = κ x ( µ t − X t − 1 ) + ω X ( t − 1 ) | n − X t − 1 + σ x ε t ∆ µ t = κ µ (¯ µ − µ t − 1 ) + σ µ ǫ t t − 1 ¯ � = X ( t − 1 ) | n b t − i X i i = t − n ε t | G t − 1 ∼ D ( 0 , 1 ) , ǫ t | G t − 1 ∼ D ( 0 , 1 ) where � n i = 1 b i = 1 with b i ≥ 0 for i = 1 , 2 , · · · , n . ¯ X ( t − 1 ) | n is meant to be some sort of moving weighted sample mean. (Note: Exponentially decaying weights with n being set to ∞ lead to a 3-dimensional Markov system, i.e., ( X t , ¯ X t |∞ , µ t ) , with one extra decaying parameter and an additional latent factor, ¯ X t |∞ .) Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 5 / 27

Local-momentum without latent central tendency (LM-AR) � ¯ � ∆ X t = κ x (¯ µ − X t − 1 ) + ω X ( t − 1 ) | n − X t − 1 + σ x ε t t − 1 ¯ � X ( t − 1 ) | n = b t − i X i i = t − n ε t | G t − 1 ∼ D ( 0 , 1 ) Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 6 / 27

Interesting features Local momentum building: ω < 0 Local momentum preserving: ω > 0 Basic properties Stationarity and ergodicity of LM-CTAR can be characterized by recognizing it as ARMA( n , ∞ ). The spectral radius of the AR coefficient matrix less than 1 is both sufficient and necessary, because the MA( ∞ ) coefficients are absolutely summable. Easy to verify sufficiency conditions are given in the paper. Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 7 / 27

LM-CTAR model in a matrix form X t = A + BX t − 1 + Z t       X t κ x ( µ t − ¯ µ ) + σ x ε t κ x ¯ µ X t − 1 0 0       X t = Z t = A =  .   .   .  . . .       . . .       X t − n + 1 0 0   1 − κ x − ω ( 1 − b 1 ) ω b 2 . . . ω b n − 1 ω b n . . . 1 0 0 0     0 1 . . . 0 0 B =    . . . . .  . . . . .   . . . . .   0 0 . . . 1 0 Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 8 / 27

A simulated sample path for the LM-AR with 5 lags. The parameters are: µ = 0 , κ x = 0 . 001 , ω = − 0 . 5 , σ x = 0 . 002 and σ µ = 0. ¯ Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 9 / 27

A simulated sample path for the LM-AR with 10 lags. The parameters are: µ = 0 , κ x = 0 . 001 , ω = − 0 . 2 , σ x = 0 . 02 and σ µ = 0. ¯ Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 10 / 27

Continuous-time LM-CTAR � ¯ � �� dX t = κ x ( µ t − X t ) + ω X t ( τ ) − X t dt + σ x dW xt d µ t = κ µ (¯ µ − µ t − 1 ) dt + σ µ dW µ t � t ¯ X t ( τ ) = b ( t − s ) X s ds t − τ � τ where κ µ > 0, σ µ > 0, κ x ≥ 0, σ x > 0, and 0 b ( s ) ds = 1 with b ( s ) ≥ 0 for 0 ≤ s ≤ τ ; and W xt and W µ t are two independent Wiener processes. (Note: Again exponentially decaying weights with n being set to ∞ can lead to a 3-dimensional Markov system, i.e., ( X t , ¯ X t |∞ , µ t ) , with one extra decaying parameter and an additional latent factor, ¯ X t |∞ .) Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 11 / 27

US Treasury rates (constant maturity, continuously compounded) (Every Wednesday in the period of Jan 4, 1954 to December 31, 2013) Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 12 / 27

Using the weekly series of one interest rate (3-month) (Both versions of the local-momentum model use a 7-week moving-window average.) Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 13 / 27

A 3-factor term structure model The base interest rate dynamic has two components: global driver ( X t ) and local variation ( v t ): r t = X t + v t (Note: The base rate is driven by 3 latent factors, because X t is latent, X t ’s central tendency is also latent, and v t is latent.) The local variation is a standard AR(1) process: ∆ v t = − κ v v t − 1 + σ v ξ t ξ t | ( G t ∪ v t − 1 ) ∼ N ( 0 , 1 ) where 0 < κ v < 2, and ( G t ∪ v t − 1 ) denotes the minimum σ -algebra generated by G t and v t − 1 . Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 14 / 27

The 3-factor model in a matrix form H ′ X ∗ r t = t SX ∗ C + DX ∗ = t − 1 + W t t where     1 σ x ε t     X t A 0 n − 1 0 n − 1     X ∗ µ t − ¯ H = t = µ W t = C = 0     0     σ µ ǫ t     v t 0 1 σ v ξ t  1 0 − κ x 0    B 0 0 0 I ( n − 1 ) × ( n − 1 ) 0 0   D = 0 1 − κ µ 0 S =     0 0 1 0 0 0 1 − κ v   0 0 0 1 σ 2  0 0 0  x 0 0 ( n − 1 ) × ( n − 1 ) 0 0   U =  σ 2  0 0 0   µ σ 2 0 0 0 v Note: U is the covariance matrix for W t , and X t , A and B have been previously defined. Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 15 / 27

Stochastic discount factor and term structure h : the length of one period measured as the fraction of a year. The stochastic discount factor from time t + τ back to time t is assumed to be exp [ − r t ( τ ) τ h ] M t , t + τ where for s ≥ t , s � M t , s = α ( t , s ) exp { ( λ 0 + λ 1 X j − 1 ) ε j + ( ψ µ 0 + ψ µ 1 µ j − 1 ) ǫ j j = t + 1 +( ψ v 0 + ψ v 1 v j − 1 ) ξ j } Note that α ( t , s ) is the factor that makes M t , s a martingale for s ≥ t . Define a martingale measure Q t , T by setting dQ t , T / dP = M t , T . Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 16 / 27

Forward and spot interest rates f t ( τ ) : the one-period forward rate at time t starting at time t + τ , where each of the τ periods is of length h . H t : the filtration generated by { ( X s , µ s , v s ); s ≤ t } . It follows that, for τ ≥ 1, f t ( τ ) = − ln E Q � e − r t + τ h |H t � h Note: The base interest rate r t equals f t ( 0 ) . Forward rates for different forward starting times can in turn be used to compute spot interest rate rates such as, for τ ≥ 1, τ − 1 r t ( τ ) = 1 � f t ( j ) . τ j = 0 Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 17 / 27

Risk-neutral system H ′ X ∗ r t = t C ∗ + D ∗ X ∗ SX ∗ t − 1 + W ∗ = t t where σ x ε Q     κ x ¯ µ + σ x λ 0 t 0 n − 1 0 n − 1 C ∗ =     W ∗ t =    σ µ ǫ Q  σ µ ψ µ 0     t σ v ξ Q σ v ψ v 0 t d ω b 2 ω b n − 1 ω b n 0 0  . . .  1 0 . . . 0 0 0 0   0 1 0 0 0 0  . . .     . . . . . . .  D ∗ = . . . . . . .   . . . . . . .     0 0 1 0 0 0 . . .     0 0 . . . 0 0 1 − κ µ + σ µ ψ µ 1 0   0 0 0 0 0 1 − κ v + σ v ψ v 1 . . . and d = 1 − κ x − ω ( 1 − b 1 ) + σ x λ 1 . Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 18 / 27

The term structure formula For τ ≥ 1, r t ( τ ) = Φ 1 ( τ ) + Φ 2 ( τ ) X ∗ t where   τ − 1  I − 1 H ′ ( I − S − 1 D ∗ ) − 1 � ( S − 1 D ∗ ) j  S − 1 C ∗ Φ 1 ( τ ) = τ j = 0   j − 1 τ − 1  h � � ( S − 1 D ∗ ) i S − 1 U ( S − 1 ) ′ [( S − 1 D ∗ ) i ] ′  H − H ′ 2 τ j = 0 i = 0   τ − 1  1 � ( S − 1 D ∗ ) j H ′ Φ 2 ( τ ) =  τ j = 0 Duan&Miao (NUS) Local-Momentum Autoregression ... (04/2015) 19 / 27