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)