estimation of multi-factor term structure model on japanese interest rates by using monte carlo...
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Estimation of Multi-factor Term Structure Model on
Japanese Interest Rates by Using Monte Carlo Filter
Akihiko Takahashi (Tokyo Univ.)
and
Seisho Sato (ISM)
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Observational Data (interest rates)
Estimated Factors (State Variables)
•Monte Carlo Filter
•State Space Model
Multi-Factor Model
EstimatedTerm Structure
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Term structure model of interest rates
State variables: Y (k-dimensional)
W : n dimensional Brownian motion
Short-term interest rate : ),( tYrr
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Price of zero coupon bonds : P(t,T)
Q : Risk neutral measure
Under Q
T : maturity
),),((),( TttYBTtP
* *
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General State Space Model
System model
Observational model
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General case :
System Model:
Linear case :
SS
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Observational Model
Price of a zero coupon bond
General case :
Additive case :
Examples of H( ・ ) :
(LIBOR)
(Swap rate)
)](| tY
),);(( TttYB
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Monte Carlo Filter : ( Kitagawa [1996] )
Initial distribution
Prediction
~ likelihood
Re-sampling by
Filter
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Log-Likelihood
AIC (Akaike Information Criterion)
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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)
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Swap Data
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Model : (Hull and White [1994] )
Y: 3-dimentional State vector
ttt vFYY 1
System:
v: Normal
(Linear case)
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Observation:
where
Avoid negative interest rate!
tu ,1
tu ,2
u: Normal
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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
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P(t,T)
T
An example of numerical simulation
0001.0 (1bsp)
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