Post on 20-Jan-2016
Estimation of Multi-factor Term Structure Model on
Japanese Interest Rates by Using Monte Carlo Filter
Akihiko Takahashi (Tokyo Univ.)
and
Seisho Sato (ISM)
Observational Data (interest rates)
Estimated Factors (State Variables)
•Monte Carlo Filter
•State Space Model
Multi-Factor Model
EstimatedTerm Structure
Term structure model of interest rates
State variables: Y (k-dimensional)
W : n dimensional Brownian motion
Short-term interest rate : ),( tYrr
Price of zero coupon bonds : P(t,T)
Q : Risk neutral measure
Under Q
T : maturity
),),((),( TttYBTtP
* *
General State Space Model
System model
Observational model
General case :
System Model:
Linear case :
SS
Observational Model
Price of a zero coupon bond
General case :
Additive case :
Examples of H( ・ ) :
(LIBOR)
(Swap rate)
)](| tY
),);(( TttYB
Monte Carlo Filter : ( Kitagawa [1996] )
Initial distribution
Prediction
~ likelihood
Re-sampling by
Filter
Log-Likelihood
AIC (Akaike Information Criterion)
Example : Interest rate of Japanese Yen
LIBOR Data
8-dimensional dataData: • LIBOR - 6M & 1Yr• Swap rates - 2,3,4,5,7,10Yr(Jan. 1st, 1997 - Jul. 22nd, 1999)
Swap Data
Model : (Hull and White [1994] )
Y: 3-dimentional State vector
ttt vFYY 1
System:
v: Normal
(Linear case)
Observation:
where
Avoid negative interest rate!
tu ,1
tu ,2
u: Normal
In this case, we cannot obtain the closed form of
Simulation Method
Evaluated by the numerical simulation!
For
Generate },,{ ))(())(( jiTt
jitt YY Under Q.
Calculate
T
ts
jis
jiT YgP )(exp ))((
,1))((
Expectation
M
j
jiT
i PM
TtP1
))(()( 1),(
M=300,using antithetic variables method
)()( it
it pY
P(t,T)
T
An example of numerical simulation
0001.0 (1bsp)