Research Paper Series
No. 46

A Term Structure Model of Default-free and Defaultable Interest Rates
with Regime-Switching Properties: Useful Tool for Risk Evaluation

Yukio Muromachi†
March, 2023†

Graduate School of Management, Tokyo Metropolitan University

A Term Structure Model of Default-free and Defaultable Interest Rates
with Regime-Switching Properties: Useful Tool for Risk Evaluation

Yukio Muromachi
Graduate School of Management, Tokyo Metropolitan University
1-1, Minami-Ohsawa, Hachiohji, Tokyo, 192-0397, Japan
muromachi-yukio@tmu.ac.jp
Received (31 March 2023)

Long-term historical data on interest rates and credit spreads can be used to identify
diﬀerent regimes, speciﬁcally, a calm regime with lower default risk and volatility and astressed regime with higher default risk and volatility. We propose a pricing and risk evaluation model of interest rate risk and credit risk with the Markovian regime-switchingproperty. We discuss the dynamics of a regime, the interest rate, and default intensity
under a physical measure and a pricing measure, and propose a simple tractable model.
In our model, the default-free interest rate and the default intensity are dependent on
the regime, and this dependence aﬀects the defaultable zero curve. We propose an appropriate calibration method and demonstrate some numerical examples of the zero curves
with diﬀerent credit ratings, which will show us various types of yield curves in the future.
We hope that such yield curve models will reveal some new methods and perspectives
for ﬁnancial risk management, especially for asset liability management.
Keywords: Risk management; Regime-switching model; Term structure of defaultfree/defaultable interest rates.

1. Introduction
The 21st century has witnessed a few ﬁnancial distresses, with the 2008 worldwide
ﬁnancial crisis being a predominant one. Financial institutions, ﬁnancial regulation agencies, and central banks have discussed ways to control ﬁnancial risks andhave been constructing concepts, tools, and systems for risk management, ﬁnancial
regulations, and ﬁnancial policies. For many ﬁnancial institutions, especially banks,
management of the interest rate and credit risks has been one of the most important
challenges.
From the long-term data on interest rates and credit spreads, a few diﬀerent
regimes have been identiﬁed: typically, a calm regime with lower default risk andvolatility, and a stressed regime with higher credit risk and volatility. Since Hamilton(1989) proposed a regime-switching model and analyzed the economic cycles, many
researchers have adopted such models for analyzing and modeling economic and1

2

Yukio Muromachi

ﬁnancial dataa . Restricted to the studies on interest rates, Inoue & Okimoto (2008)
used regime-switching models for analyzing monetary policy and private sector behavior in Japan. Dai et al. (2007) used a discrete-time regime-switching Gaussian
term structure model, while Wu & Zeng (2005) applied a continuous-time aﬃneterm structure model with regime-dependent parameters. Elliott & Siu (2009) considered a bond valuation as a derivative written on a Markovian regime-switching
instantaneous spot rate, and derived a Markov-modulated exponential-aﬃne bondprice formula. Their results have been applied to the term structures of interest
rates, see Elliott & Nishide (2014).
In credit risk analysis, Gourieroux et al. (2013) derived a defaultable bond pricein a discrete-time setting and analyzed sovereign yield curves, and Monfort & Renne(2013) analyzed the credit spreads of U.S. bonds. Fisher & Stolper (2019) analyzed
the behavior of the credit spreads and their key determinants, and Chun et al.
(2014) proposed a regression model of credit spreads with endogenous regimes, anddemonstrated that the model enhanced the explanatory power of the determinants.
In a continuous-time setting, Hainaut & Le Courtois (2014) proposed a default
intensity model described by regime-switching Lévy process to evaluate the survival
probability, and analyzed the Credit Default Swap (CDS) market. Li & Ma (2013)
discussed pricing options analytically, but the results are limited to the conditional
price considering a sample regime path.
In ﬁnance, most of the available research on regime-switching models involve
analyzing ﬁnancial data and pricing securities. However, there have been limited
research on quantitatively evaluating ﬁnancial risk using these models. If a high-risk
regime is included in the ﬁnancial risk evaluation, it might give us a new assessmentof the future, and one such method is the stress test. In the stress tests, the potential
losses are calculated quantitatively under some given high-risk scenarios. Moreover,
if such a high-risk regime or scenario can be estimated from the observed data, theforward-looking stress tests can be done with some statistical viewpoints.
In this study, we propose a consistent pricing and risk evaluation model of interest rate risk and credit risk under a regime-switching environment in a continuoustime setting. For the risk evaluation of an asset portfolio, we set a risk horizon,
which is a certain future time, and discuss the prices of the surviving securities at
the horizon and the accumulated loss by defaults up to the horizon. Such kinds ofrisk evaluations for a bond portfolio are already discussed in Muromachi (2022).
Contrariwise, in this study, we focus on the construction of the default-free and defaultable yield curves and analyses of the movement of the curves in the future. We
think that these analyses of the regime-switching property are suitable for the evaluation of interest rate risk and credit risk, especially in asset liability management(ALM) because the banks hold various kinds of liabilities sensitive to interest raterisk. In ALM, the synthetic evaluation of the interest rate risk through a common
stochastic model is theoretically desirable, and we suggest that one of the promisinga As an example of review papers, see Ang & Timmermann (2012).

A Term Structure Model of Interest Rates with Regime-Switching Properties

3

tools is the stochastic yield curve model with the regime-switching properties. Our
proposed model can combine such evaluation of liabilities with the risk evaluationof the asset portfolio.
This article is organized as follows: Section 2 describes the construction of asimple pricing and risk evaluation model. Based on a Markovian regime-switchingmodel, the default-free spot rate and the default intensity processes are describedunder the physical measure, and the change of measure is discussed so that theprocesses under the pricing measure are derived. In Section 3, we propose estimation
methods for model parameters and some estimated results. Although our methods
are suboptimal, the basic principles are quite straightforward. In Section 4, we show
some numerical examples of the term structure of default-free and defaultable zerorates, and Section 5 concludes this article.

2. The ModelIn this section, we discuss the construction of a simple pricing and risk evaluation model in a regime-switching environment. Our model follows the framework
proposed by Kijima & Muromachi (2000). They discussed why two probability
measures, the physical and the pricing probability measures, are necessary for risk
evaluation based on market prices. When we calculate risk based on market prices,
we set a risk horizon, which is a certain future time, and discuss the future prices ofthe securities at the horizon, the coupons/dividends, the accumulated loss causedby the defaults, and so on up to the horizon. Given that the accumulated loss
and coupons are evaluated as realized values, they must be measured under thephysical probability measure. By contrast, when we discuss security prices, we use
the no-arbitrage prices not only at present but also at the risk horizon. The noarbitrage prices are given by the expectation of the discounted future cash ﬂows
under the pricing measure equivalent to the physical measure. Therefore, we need
two probability measures. In order to maintain consistency between pricing and risk
evaluation, stochastic modeling is necessary not only in physical measure but alsoin pricing measures. The physical measure part of the model is used for generating
scenarios up to the risk horizon, while the pricing measure part is used for pricing at
the present and the risk horizon. Since the stochastic model in the pricing measure
is obtained from the stochastic model in the physical measure and the scheme of
the change of measure, it is also enough information for describing the system togive the scheme of the change of measure and the stochastic model either in thephysical or the pricing measure.

2.1. Stochastic processes under the physical probability measureConsider a switching regime in a continuous-time and ﬁnite-state Markov chain
model. Let t, t ≥ 0, be time, and t = 0 is present. We consider a ﬁnancial market with a ﬁnite horizon T , 0 < T < ∞, and deﬁne a ﬁltered probability space

4

Yukio Muromachi(Ω, F, (Ft )0≤t≤T , P ) under which all stochastic processes can be described. All probability spaces in this study satisfy the usual conditions.
We consider diﬀerent states K ∈ N , which are called regimes. Let X(t) ∈{e1 , · · · , eK } be a K-dimensional vector and ej , j = 1, · · · , K, are K-dimensional
unit vectors where the i-th component of ej is the Kronecker’s delta δij . X(t)implies an economic state at time t, and is a Markov chain on (Ω, F, (Ft )0≤t≤T , P ).
Let Q ∈ RK ×RK denote the inﬁnitesimal generator matrix of X(t), and we assume
that Q is time-independent, so that we can express
P {X(t + ∆t) = ej |X(t) = ei }, i ̸= j
 lim
∆t→0
∆t
X
Q = (qij )i,j=1,··· ,K , qij =
qkj ,
i=j

−
k̸=j

and we have

Z t
X(t) = X(0) +Q⊤ X(s)ds + M (t),

t≥0

(2.1)

0

where the K-dimensional vector M (t) is a P –martingale. The transition probability
matrix is deﬁned byP(t) = (pij (t))i,j=1,··· ,K , pij (t) = P {X(s + t) = ej |X(s) = ei } , s, t ≥ 0,
and is given by
P(t) = exp {tQ} =

∞ k k
X
t Q
k=0

k!

.

(2.2)

The risk-free instantaneous spot rate at time t, denoted by r(t), follows thestochastic diﬀerential equation (hereinafter, abbreviated as SDE)
dr(t) = µr (t, r(t), X(t))dt + σr (t, r(t), X(t))dzr (t),t ≥ 0,

where zr (t) is a standard Brownian motion under P . Consider a risky security,
subject to its credit risk. Let τ > 0 be the default time of the security. DeﬁneH(t) = 1{τ ≤t} as its default process where 1A is the indicator functionb , and thedefault intensity at time t, denoted by h(t), follows the SDE
dh(t) = µh (t, h(t), X(t))dt + σh (t, h(t), X(t))dzh (t),t ≥ 0,

where zh (t) is a standard Brownian motion under P and dzr (t)dzh (t) = ρ(t)dt. For
any arbitrary time t, 0 ≤ t ≤ T , the ﬁltration generated by the default process
is deﬁned by Ht = σ(H(s) : 0 ≤ s ≤ t) and H = (Ht )0≤t≤T , while the ﬁltration
generated by X(t) is deﬁned by FtX = σ(X(s) : 0 ≤ s ≤ t) and F X = (FtX )0≤t≤T .
The ﬁltrations generated by r(t) and h(t) are Ftr = σ(r(s) : 0 ≤ s ≤ t), F r =(Ftr )0≤t≤T and Fth = σ(h(s) : 0 ≤ s ≤ t), F h = (Fth )0≤t≤T , respectively. The
complete ﬁltration of this system is deﬁned by F = F X ∨ F r ∨ F h ∨ H, that is,Ft = FtX ∨ Ftr ∨ Fth ∨ Ht is satisﬁed for any arbitrary t, 0 ≤ t ≤ T . Additionally,
we use a useful ﬁltration Gt = FTX ∨ Ftr ∨ Fth ∨ Ht , 0 ≤ t ≤ T .
b1

A = 1 when the event A is true, otherwise 1A = 0.

A Term Structure Model of Interest Rates with Regime-Switching Properties

5

2.2. Stochastic processes under a pricing probability measureConsider a probability measure P X equivalent to P . Assume that the inﬁnitesimalX
generator matrix of X(t) under P X , denoted by QX = (qij)i,j=1,··· ,K , is given byi ̸= j,X
qij= (1 + κij )qij ,i, j = 1, · · · , K

where κij > −1 is constant c . Then, the change of probability measure from P to
P X is written asdP X= η X (T ),dP F X
T

where the Radon-Nikodym derivative η X (T ) is deﬁned by




K
K
 Z t X YX
1 +κkℓ ∆H kℓ (u)
η X (t) = exp −κkℓ qkℓ H ℓ (u)du


0
0<u≤t

k,ℓ=1

k,ℓ=1

i

H (t) = 1{X (t) = ei } ,
i = 1, · · · , K,X
H ij (t) =H i (u−)H j (u),i ̸= j,i, j = 1, · · · , K,

0<u≤tand ∆H ij (t) = H ij (t) − H ij (t−). Let P X (t) be the transition probability matrix
under P X . Then, the similar equations such as (2.1) and (2.2) are obtained only
if (Q, P(t), M (t)) are replaced with (QX , P X (t), M X (t)) where M X (t) is a P X martingale.
We follow a standard discussion for modeling credit riskd . Let β(t) be a Ft adapted process and κh (t) > −1 be a Ft -predictable process, and deﬁne a newprobability measure P c equivalent to P X bydP c
= ρh (T ),dP X GTwhereZ t
ρh (t) =(βh (s)dzh (s) + κh (s)dMh (s))
0

and

Z t∧τ
Mh (t) = H(t) −h(s)ds.
0Then, the processZ t
zhc (t) = zh (t) −βh (s)ds0c Generally, κ
ij is a Ft –predictable process.
d See, for example, Kusuoka (1999) and Bielecki & Rutkowski (2002).

6

Yukio Muromachi

becomes a standard Brownian motion under P c , and the processhc (t) = (1 + κh (t))h(t)is regarded as a default intensity under P c , because the process
Z t∧τc
Mh (t) = H(t) −hc (s)ds
0c

becomes a (F, P )-martingale, which corresponds to the fact that Mh (t) is a
(F, P X )-martingale.
The following discussion is based on Elliott et al. (2007). Assuming that theprice of a risky asset at time t, denoted by S(t), follows
dS(t) = µ(t, S(t), X(t))S(t)dt + σ(t, S(t), X(t))S(t)dzSc (t),t≥0where zSc (t) is a standard Brownian motion under P c . Deﬁne the Radon-Nikodym
derivative from P c to its equivalent probability measure P η by e
(Z
)
Z
T
1 T
dP η
2
c
η(s)dzS (s) −(η(s)) ds= exp
dP c GT
2 00
where
η(t) =

r(t) − µ(t, S(t), X(t)),σ(t, S(t), X(t))

then, thanks to the Girsanov’s theorem, the processZ tz η (t) = zSc (t) −η(u)du0

becomes a standard Brownian motion undern P η , and othe relative price of S(t)Rtwith respect to the bank account B(t) = exp 0 r(u)du becomes a (G, P η )–local
martingale. Assuming that the relative price is a (G, P η )–martingale, the price of aEuropean derivative given GT is given by
#
"
( Z
)
M
η
V (t|Gt ) = E P exp −r(u)du G(S(M )) Gt ,twhere G(·) is the payoﬀ function of the derivative at its maturity M, t ≤ M ≤ T ,η
and E P [·|·] is the conditional expectation operator under P η . From the chain ruleof the conditional expectation, given {τ > t}, the price V (t) is given by
"
( Z
)
#
M
η
η
V (t) = E P [ V (t|Gt )| Ft ] = E P exp −r(u)du G(S(M )) Ft .
(2.3)
t

Hereafter, we call P η the risk-neutral probability measure and denote it as Pe, anddenote a standard Brownian motion under Pe as ze. Similarly, hc (t), QX , P X (t) ande This change of measure is called the risk-neutral regime-switching Esscher transform.

A Term Structure Model of Interest Rates with Regime-Switching Properties

7

f (t), respectively. Generally, a regimee P(t)eM X (t) are denoted by eh(t), Q,and M
switching model derives an incomplete market so that many risk-neutral measures
might exist, and P η is one of them. However, according to Elliott et al. (2005), themeasure P η is the minimum entropy martingale measure of P , so that the pricegiven by (2.3) is a reasonable one.
Let Ft -predictable process λr (t) be the market price of risk against zr (t), thenthe stochastic processes describing the market under Pe are as follows:
dr(t) = µer (t, r(t), X(t))dt + σr (t, r(t), X(t))dezr (t),(2.4)

dh(t) = µeh (t, h(t), X(t))dt + σh (t, h(t), X(t))dezh (t)(2.5)eh(t) = (1 + κh (t, X(t)))h(t)

(2.6)

µ
er (t, r(t), X(t)) = µr (t, r(t), X(t)) − λr (t)σr (t, r(t), X(t))

(2.7)

µ
eh (t, h(t), X(t)) = µh (t, h(t), X(t)) + β(t)σh (t, h(t), X(t)).

(2.8)

Consider a default-free discount bond with maturity M, t ≤ M ≤ T . From (2.3),
its price at time t is given by
"
( Z
)
#
M
e
v(t, M, r, X) = E exp −r(u)du r(t) = r, X(t) = X .
t

Next, consider a defaultable discount bond with maturity M , and let τ > 0 be its
default time. Suppose the holder of the bond receives $1 at M if the bond survives
at M , while the holder receives $δ, 0 ≤ δ < 1 at M if default occurs up to M . From
(2.3), given {τ > t}, the price of the bond is given as
#
"
( Z
)
M

e exp −
h(t) = eh, X(t) = X
D(t, M, r, eh, X) = Er(u)du1{τ >M } + δ1{τ ≤M } r(t) = r, et= δv(t, M, r, X) + (1 − δ)p(t, M, r, eh, X),
(2.9)
"
( Z
)
#
M
e
e
e
e
e exp −
p(t, M, r, h, X) = E(r(u) + h(u))du r(t) = r, h(t) = h, X(t) = X .
t(2.10)
Hereafter, we call p(t, M, r, eh, X) the price of the “survival discount bond”, whose
payoﬀ is $1 if and only if the bond survives at M .
2.3. A simple model
Here, we propose a simple model. Under the physical measure P , r(t) follows
dr(t) = a (⟨m, X(t)⟩ − r(t)) dt + ⟨σ, X(t)⟩dzr (t),⊤t ≥ 0,(2.11)⊤

where m = (m1 , · · · , mK ) and σ = (σ1 , · · · , σK ) are K-dimensional constantvectors, a and σk , k = 1, · · · , K are positive constants, and ⟨A, B⟩ is the innerproduct of vectors A and B. Similarly, h(t) follows
dh(t) = ah (⟨mh , X(t)⟩ − h(t)) dt + ⟨σ h , X(t)⟩dzh (t),t ≥ 0,(2.12)

8

Yukio Muromachi

where mh = (mh,1 , · · · , mh,K )⊤ and σ h = (σh,1 , · · · , σh,K )⊤ are K-dimensional
constant vectors, ah and σh,k , k = 1, · · · , K are positive constants, anddzr (t)dzh (t) = ρ(t)dt f . For simplicity, we assume that X(t) is independent of(zr (t), zh (t)), and select the Vasicek model because model parameters can be calibrated easily by using the existing statistical tools such as Matlabg .
To make our model more tractable, we apply the following assumption.
Assumption 2.1. The market price of risk λr (t) is written byλr (t) = ⟨Lr , X(t)⟩ + ψr (t)(2.13)

where Lr = (λr,1 , · · · , λr,K )⊤ is a K-dimensional constant vector and ψr (t) is adeterministic function of time t.
From (2.4), (2.7), (2.11) and (2.13), we havedr(t) = (⟨ϕ(t), X(t)⟩ − ar(t)) dt + ⟨σ, X(t)⟩dezr (t),(2.14)

where the deterministic function vector ϕ(t) = (ϕ1 (t), · · · ϕK (t))⊤ is given as
ϕ(t) = am − σ ⊗ Lr − σψr (t),and A ⊗ B is the Hadamard product of K-dimensional vectors A and B. Then,the discount bond price v(t, M, r, X) is given byv(t, M, r, X) = exp {⟨A(t, M ), X⟩ − B(t, M )r} ,(2.15)

where
B(t, M ) =1 − e−a(M −t)aand A(t, M ) = (A(t, M, e1 ), · · · , A(t, M, eK ))⊤ . Here, deﬁne Āi (t, M ) =exp{A(t, M, ei )} and Ā(t, M ) = (Ā1 (t, M ), · · · , ĀK (t, M ))⊤ , and they are the solution of the following ordinary diﬀerential equations (hereinafter, abbreviated by
ODEs),

1 2 2dĀi (t, M )+σ B (t, M ) − ϕi (t)B(t, M ) Āi (t, M ) + ⟨Ā(t, M ), Q⊤ ei ⟩ = 0,dt
2 i
(2.16)
with the terminal conditions Āi (M, M ) = 1, i = 1, · · · , K. A numerical solution of(2.16) is obtained easily, for example, by using the Runge-Kutta method. The zerorate (yield to maturity of the discount bond) with maturity M at time t is given bylog v(t, M, r(t), X(t)),
0 ≤ t < M.
M −tAdditionally, to manage the default intensity process similarly and to make thecalibration tractable, we introduce the following assumption:
R(t, M, r(t), X(t)) = −f An extension of (a, a ) to the deterministic functions of time t is obvious.
hg In Hatakeyama (2022), a slightly modiﬁed Matlab program was used to estimate parameters.

A Term Structure Model of Interest Rates with Regime-Switching Properties

9

Assumption 2.2. The stochastic process β(t) is written asβ(t) = ⟨Lh , X(t)⟩ + ψh (t)(2.17)⊤

where Lh = (λh,1 , · · · , λh,K ) is a K-dimensional constant vector and ψh (t) is adeterministic function of time t. Further, deﬁneℓ(t) = ⟨ℓ(t), X(t)⟩,ℓ(t) = (ℓ1 (t), · · · , ℓK (t))⊤(2.18)

where ℓj (t) = κh,j (t)h(t), j = 1, · · · , K are diﬀerentiable deterministic functions oftime th .
From (2.5), (2.6), (2.8), (2.12), (2.17) and (2.18), we have idh(t) = (⟨ϕh (t), X(t)⟩ − ah h(t)) dt + ⟨σ h , X(t)⟩dezh (t)(2.19)ϕh (t) = ah mh + σ h ⊗ Lh + σ h ψh (t),(2.20)eh(t) = h(t) + ⟨ℓ(t), X(t)⟩.

(2.21)

Now, consider a survival discount bond with maturity M . From (2.10), the time
t price of the survival discount bond is written asp(t, M, r, e
h, X)
"
( Z
#
)
M
e
e
e
e
= E exp −(r(u) + h(u))du r(t) = r, h(t) = h, X(t) = X .
tFrom Appendix A, we obtainnop(t, M, r, eh, X) = exp ⟨C(t, M ), X⟩ − B(t, M )r − Bh (t, M )eh ,(2.22)

where
Bh (t, M ) =1 − e−ah (M −t)ahand C(t, M ) = (C(t, M, e1 ), · · · , C(t, M, eK ))⊤ . Here, deﬁne C̄i (t, M ) =exp{C(t, M, ei )} and C̄(t, M ) = (C̄1 (t, M ), · · · , C̄K (t, M ))⊤ , and they are the solution of the ODE system:(2σh,idC̄i (t, M )σi2 2+B (t, M ) +B 2 (t, M ) − ϕi (t)B(t, M ) − ah m′h,i (t)Bh (t, M )
dt
2
2 h

+ρ(t)σi σh,i B(t, M )Bh (t, M ) C̄i (t, M ) + ⟨C̄(t, M ), Q⊤ ei ⟩ = 0(2.23)
with the terminal conditions C̄i (M, M ) = 1, i = 1, · · · , K, where
1 dℓi (t)′⊤+ σh,i λh,i + σh,i ψh (t) + ⟨ℓ(t), Q ei ⟩ .
mh,i (t) = mh,i + ℓi (t) +ahdth This diﬀerentiability seems implausible because h(t) is not diﬀerentiable; however, it makes thecalibration easier.
i If we set ℓ(t) as a constant vector ℓ, L has the same eﬀect as ℓ, so that L would be dropped.
h
h

10

Yukio Muromachi

Our valuation is diﬀerent from Li & Ma (2013)’s results because their results are
the conditional price given the sample path of X(·) up to M , while our results are
the unconditional price.
When eh(t) = 0, 0 ≤ t ≤ T , this implies that the bond is default-free,(2.23) reduces
nR to (2.16).o By contrast, consider a discount bond with the payoﬀ
M
1{τ >M } exp t r(u)du at M , then, the price at time t is written as
"

( Z
)
#
M
e
e exp −v̄(t, M, eh, X) = Eh(u)du eh(t) = eh, X(t) = X ,(2.24)tand is given bynov̄(t, M, eh, X) = exp ⟨D(t, M ), X⟩ − Bh (t, M )ehwhere D(t, M ) = (D(t, M, e1 ), · · · , D(t, M, eK ))⊤ . Similarly, deﬁne D̄(t, M ) =(D̄1 (t, M ), · · · , D̄K (t, M ))⊤ where D̄i (t, M ) = exp{D(t, M, ei )}, and they are thesolution of the ODE system:
)
(
2
σh,i
dD̄i (t, M )′2+
B (t, M ) − ah mh,i (t)Bh (t, M ) D̄i (t, M ) + ⟨D̄(t, M ), Q⊤ ei ⟩ = 0dt
2 hwith the terminal conditions D̄i (M, M ) = 1, i = 1, · · · , K. Note that from (2.24),v̄(t, M, eh, X) is the conditional survival probability at M under Pe on {τ > t},that is, v̄(t, M, eh, X) = Pe{τ > M |τ > t, eh(t) = eh, X(t) = X}. From the above
discussion, generally, and even if ρ(t) = 0, it follows thatp(t, M, r, eh, X) ̸= v(t, M, r, X)Pe{τ > M |τ > t, eh(t) = eh, X(t) = X}.
This is because r(t) and eh(t) are dependent through X(t) even if ρ(t) = 0.
A deﬁnite disadvantage of this model is a negative default intensity in the future with a positive probability. To avoid the disadvantage, for example, the CoxIngersoll-Ross (CIR) process or the squared Gaussian process can be proposed.

3. Estimation of model parameters
We discuss on estimating the model parameters used in the simple model in Section
2.3. They are (1) Q, (a, m, σ) and (ah , mh , σ h ) on stochastic processes under P ,e (or κ = (κij )i,j=1,··· ,K ), (Lr , ψr (t)), (Lh , ψh (t)) and ℓ(t) on stochasticand (2) Q
processes under Pe. If the time-series data of r(t) and h(t) can be observed, it isnatural that the parameters in (1) are estimated from the time-series data, while
those in (2) are estimated from the term structures of default-free and defaultableinterest rates at t = 0. However, given that we can only observe the time-series dataof the credit spreads instead of h(t), we have to devise another method.

A Term Structure Model of Interest Rates with Regime-Switching Properties

11

3.1. Estimation procedure
From (2.19) and (2.21), the default intensity eh(t) under P is followed bynodeh(t) = ah ⟨fmh (t), X(t)⟩ − eh(t) dt + ⟨σ h , X(t)⟩dzh (t) + ⟨ℓ(t), dM (t)⟩fh (t) is given by
where the i-th component of m
1 dℓi (t)⊤me h,i (t) = mh,i + ℓi (t) ++ ⟨ℓ(t), Q ei ⟩ ,
ah
dt

(3.1)

i = 1, · · · , K.

We have little information on M (t). So, as the approximated eh(t) process, we usenodeh(t) ≃ ah ⟨fmh (t), X(t)⟩ − eh(t) dt + ⟨e
σ h (t), X(t)⟩dzh (t),
(3.2)
where
(e
σh,i (t))2 = (σh,i )2 +KXqki (ℓi (t) − ℓk (t))2 .

(3.3)

k=1

The second term of the above equation is the inﬂuence of ⟨ℓ(t), dM (t)⟩ on the
instantaneous variance of eh(t) under P .
Then, we propose the following parameter estimation procedure:
fh , σe h (t)) from the time-series data of r(t) and e(1) Estimate (Q, a, m, σ, ah , mh(t).
Here, we assume eh(t) follows the process (3.2) under P .
(2) Estimate ℓ(t) from the observed survival function of τ under P .
e Lr , ψr (t)) in order to ﬁt the present term structure of the default(3) Estimate (Q,free interest rates.
(4) Estimate (Lh , ψh (t)) in order to ﬁt the present term structure of the credit
spreads (or defaultable interest rates).
In Step (1), we can use some existing tools; for example, VAR(1) (1-st order
vector autoregression) model with regimes oﬀered by Matlab j . Moreover, insteadof eh(t) (which is related to the instantaneous credit spread and usually cannot be
observed in the market), we can use the credit spreads of a defaultable discountbond, CS(t), with a certain maturity with some adjustments; for example, eh(t) ≃k
CS(t)/(1 − δ) where δ is the recovery rate of the bond . The main purpose of
Step (2) is ﬁtting the term structure of the default probabilities (the distributionfunction of τ ) in P , and as the same time, the eﬀect of ⟨ℓ(t), dM (t)⟩ on σeh,i (t)is evaluated through (3.3). The derived σ h must be positive. Detailed procedurese Lr , ψr (t)) are calibrated in order toand results are omitted here. l . In Step (3), (Q,e andreproduce the present term structure of the default-free interest rates. First, Q
Lr are calibrated so that the theoretical term structure of the default-free interest
j For details, see Hatakeyama (2022). We are currently preparing to publish his results in English.
k This approximation implies the credit spreads are ﬂat. If several bonds are issued by the same

company, we might estimate more useful information about eh(t).
l We are preparing another paper on the estimation methods and their results.

12

Yukio Muromachi

rates without ψr (t) ﬁts the observed one as much as possible. Here, we use the ridge
regression; the objective function with the ridge parameter λ1 is


K
10
X
X
X
F1 ≡
(R(0, j, r(0), X(0)) − Robs (0, j))2 + λ1  (eqij − qij )2 +L2r,i  (3.4)
j=1

i=1

i̸=j

where Robs (0, j) is the observed zero rate with maturity j years. The residuals
between the observed and theoretical zero rates are compensated by ψr (t), therefore, this simple model can reproduce the present term structures of interest rateseperfectly. Step (4) is similar to Step (3) without Q.
Although these obtained estimates are not optimal, the relations between thedata and the parameters are clear, and each step is simple.
3.2. Setting and estimated parameters
In Step (1) in our estimation procedure, we follow Hatakeyama(2021). He used
the monthly data of the US Treasury Bond Yields and ICE BofA US Corporate
Index OAS (Option Adjusted Spread) from January 2006 to May 2021. Since theinstantaneous default intensity eh(t) is diﬃcult to observe, we assume the recovery
rate δ = 0.4, which is typical for pricing corporate bonds and regard the credit
spread (OAS) divided by (1 − δ) as the default intensity eh(t), then we transformthe parameters obtained by Hatakeyama (2021) into those of eh(t)’s process.
In Step (2), as the observed survival probabilities, we use S&P’s corporate average cumulative default rates, 1981–2019 by Kraemer et al. (2020). Consider defaultfree discount bonds and three kinds of defaultable discount bonds with diﬀerent
levels of credit risk: AAA, BBB, and CCC/C-rated bonds, respectively.
We choose K = 3. In our estimations, statistical information criteria such as
AIC (Akaike’s Information Criterion) and BIC (Bayesian Information Criterion)
decrease as K increases. On the other hand, the estimation of model parameters
becomes more diﬃcult with K because the number of parameters increases by thee (t))
order of K 2 . In time-series data analysis, for simplicity, we assume that (fmh (t), σBBB
CCC
are constant, and assume a = aAAA
=
a
=
a
=
1
becausethesimultaneoushhhestimation of the mean-reverting power and the volatility is diﬃcult. We assume
the present default-free and defaultable zero rate curves as
R(0, M ) = a + bM + cM 2 + dM 3 + eM 4 ,(%)

given in Table 1 where R(0, M ) is the zero rate with maturity M (years) m .
Additionally, for simplicity, we assume ϕr (t) and ϕh (t) are piecewise constant int ∈ [j − 1, j], j = 1, 2, · · · , 10 years, and ℓ(t) is constant in each term. n , t ∈ [0, 1]
m We use these values as examples during these several years.
n It was diﬃcult to reproduce the observed term structures of the default probability under P

on constant ℓ setting, and several trials taught us that the diﬃculty was caused mainly by thediﬀerences in the very short term. We ignore the diﬀerentiability of ℓ(t) at t = 1 year.

A Term Structure Model of Interest Rates with Regime-Switching Properties

13

and t ∈ [1, 10] years. In the above setting, (m, σ) become constant in each term
[0, 1] and [1, 10] years, respectively.

Table 1. Parameters (%) of the present zero rate curves.

default-free
AAA
BBB
CCC/C

a

b

c

d

e

0.00
0.66
0.80
4.20

0.546
0.380
0.740
1.000

−0.0606
0.0036−0.0820−0.1000

0.00233−0.00650
0.00200
0.00300

0.00000
0.000400.00009
0.00003

The estimated parameters on the processes are shown in Table 2, and the estimated generators are given by−1.120 0.967 0.153−1.292 1.291 0.001
e =  1.393 −1.394 0.001  .
Q =  1.659 −2.027 0.368  , Q(3.5)
1.222 1.575 −2.797
1.222 1.576 −2.798
The estimated market prices of risks are shown in Table 3.
ModelMatrix only

Data

2.0%
1.8%
1.6%
1.4%
1.2%
1.0%
0.8%
0.6%
0.4%
0.2%
0.0%
0

1

2

3

4

5

6

7

8

9

10

Fig. 1. Term structures of default-free zero rates. This ﬁgure compares the term structures ofdefault-free zero rates. The horizontal axis represents maturity (years), and the vertical represents
the zero rate. “Model” is the term structure where all calibrated parameters are used, “Matrix
e is used, while “Data” are given data.
Only” is that where the calibrated Q

14

Yukio Muromachi

Table 2. Parameters (%) of default-free interest rate and default intensity processes under P .

default-free
regime k

mk

σk

0
1
2

0.330
1.230
−2.630

0.460
3.380
4.310

AAA
regime k

m
e h,k

σ
eh,k

mh,k
[0, 1]

mh,k[1, 10]σh,k[0, 1]

σh,k[1, 10]ℓk[0, 1]ℓk[1, 10]

0
1
2

0.933
1.833
3.183

0.300
0.550
3.817

−0.2090.935
2.284−0.407
0.517
1.868

0.150
0.531
3.816

0.299
0.550
3.817

0.981
1.121
1.149

1.325
1.338
1.341

regime k

m
e h,k

σ
eh,k

mh,k
[0, 1]

mh,k[1, 10]σh,k[0, 1]

σh,k[1, 10]ℓk[0, 1]ℓk[1, 10]

0
1
2

2.267
4.383
10.533

0.667
1.317
6.317

−0.1752.130
8.273−0.886
1.237
7.392

0.637
1.312
6.316

0.667
1.317
6.317

2.318
2.426
2.445

3.148
3.152
3.155

regime k

m
e h,k

σ
eh,k

mh,k
[0, 1]

mh,k[1, 10]σh,k[0, 1]

σh,k[1, 10]ℓk[0, 1]ℓk[1, 10]0
1
211.30021.58346.5504.050
6.500
26.83329.15042.68667.560−3.0607.211
32.0152.163
6.221
26.8194.049
6.499
26.833−19.994−18.126−17.791

14.39014.37414.295BBB

CCC/C

Figure 1 shows the observed and the theoretical term structures of the defaultfree zero rates, while Figure 2 shows those of the CCC/C. In these ﬁgures, thehorizontal axis represents the maturity, and the vertical represents the zero rates. If
all calibrated parameters are used, all term structures can be reproduced perfectly
due to the time-dependent parameters ψr (t) and ψh (t).
Figure 3 shows the observed and the theoretical survival probabilities of CCC/C
ﬁrms. Notice that in our model the survival probability cannot be reproduced perfectly. This ﬁgure shows that the risk premia adjustments (RPA), ℓ(t), play an

A Term Structure Model of Interest Rates with Regime-Switching Properties

15

Table 3. Estimated market prices of risk (%).

default-free

AAA

BBB

CCC/C

−2.98786
0.425130.00143−0.05383−1.65209−0.00530−2.16136
0.50460
0.00010

Lr,1
Lr,2
Lr,3

−0.1259−0.7595−0.0004

term (years)

ψr (t)ψhAAA (t)ψhBBB (t)ψhCCC (t)[0, 1][1, 2][2, 3][3, 4][4, 5][5, 6][6, 7][7, 8][8, 9][9, 10]−0.05050.1761−0.1546−0.0296−0.1105
0.02680.03390.1020−0.0457−0.0575

0.09766−0.58594
0.488280.29297
0.48828−0.39063−0.68359−0.78125
0.488283.71094−0.12207
0.29297
0.195310.19531−0.17090−0.34180−0.58594−0.58594−0.24414
0.68359

0.21362−0.67749
0.07935−0.12207
0.34180
0.29297
0.64697
0.732421.13525
1.51367

Lh,1
Lh,2
Lh,3

important role on the ﬁtting of the survival probabilities, especially in the lower
credit ratings. Since the ﬁtting between the observed curve and the theoreticalcurve is diﬃcult on the constant ℓ(t) setting, mainly due to the diﬀerence in thevery short term, we use a 2-term model, [0, 1] years and [1, 10] years, and assumeℓ(t) is constant in each term. Here, we show the results for CCC/C rating only, but
we obtain better results in AAA and BBB zero curves.
To our regret, our proposed estimation procedure is not appropriate enough,that is, some estimated values are unreasonable: for example, some mh,0 in Table
1 are negative. More future research will be necessary for robust and sophisticatedestimation methods.
4. Numerical examples
In this section, we show some numerical examples of zero curves at some future
times by using a Monte Carlo simulation. We set δ = 0.4, and ρ(t) = 0.
4.1. Procedure of the Monte Carlo simulation
The Monte Carlo simulation consists of the following steps:
1. Set the initial values (X(0), r(0), h(0)).

16

Yukio Muromachi

Model

Data

without CMPR

8.0%
7.0%
6.0%
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
0

1

2

3

4

5

6

7

8

9

10

Fig. 2. Term structures of CCC/C zero rates. This ﬁgure compares the term structures of CCC/C
zero rates. The horizontal axis represents maturity (years), and the vertical represents the zerorate. “Model” is the term structure where all calibrated parameters are used, “Data” are given
data, and “without CMPR” is that where the calibrated Lh and ψh (t) are not used.

2. Generate a sample path (X(ti ), r(ti ), h(ti )), 0 = t0 < t1 < · · · < tF = 10 (years)
under P . First, generate a path of X(·). Then, given a path X(·), generate apath of (r(·), h(·)).
3. Calculate the prices of the bonds D(ti , ti + M, r(ti ), eh(ti ), X(ti )) for various
maturities M at time ti .
4. Calculate the zero rates from the bond prices.
In Step 2, each sample path is generated from t = 0 to t = TF under P , while in
Step 3 the no-arbitrage price is calculated based on the stochastic processes under
Pe. The input values (r(ti ), h(ti ), X(ti )) in Step 3 are given as the values at t = ti
on each path generated in Step 2. Notice that r(ti ) and X(ti ) are the same values
under P and Pe, while eh(ti ) = h(ti ) + ⟨ℓcr(i) (ti ), X(ti )⟩. We simulate 100,000 runs.
4.2. Distribution of discount bond price in futureFirst, we show the future price distributions of CCC/C discount bond with maturity
M = 10 years. Figure 4 is its histogram at 1-year-after (t = 1); the horizontal axis
represents price DCCC (t = 1, M, r(1), eh(1), X(1)) (bond’s face amount is one), andthe vertical represents frequency. Here, (r(1), eh(1), X(1)) are simulated values. The
peak around 0.34 is the distribution of the recovered price when the default occurs
before 1-year-after, while the distribution above 0.4 is one when the bond survives
at 1-year-after. Two peaks can be seen around 0.54 and 0.50, however, these do notcorrespond with the regime just at 1-year-after directly.

A Term Structure Model of Interest Rates with Regime-Switching Properties

Model

Data

17

no RPAs

100%

90%

80%

70%

60%

50%

40%
0

1

2

3

4

5

6

7

8

9

10

Fig. 3. Survival probabilities of CCC/C ﬁrms under P . This ﬁgure compares the survival probabilities of CCC/C ﬁrms in the physical measure. The horizontal axis represents maturity (years),
and the vertical represents the survival probability. “Model” is the survival probability where allcalibrated parameters are used, “Data” are the given data, and “no RPAs” is where the calibratedℓ(t) are not used.

400
350
300
250
200
150
100
50
0
0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60

Fig. 4. Distribution of CCC/C discount bond price with 10-year maturity. The horizontal axis
represents price, and the vertical axis represents frequency.

Figure 5 is the detailed version of Figure 4; each distribution corresponds toeach regime at 1-year-after, “Regime 0”, “Regime 1” and “Regime 2”, respectively,

18

Yukio Muromachi
400
350
Regime 0300
250Regime 1
Regime 2

200
150
100
50
0
0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60

Fig. 5. Distribution of CCC/C discount bond price with 10-year maturity. The horizontal axis
represents price, and the vertical axis represents frequency. Each distribution corresponds to theregime at 1-year-after.

and when summing up the three distributions, we obtain Figure 4. All distributions
for surviving bond in Figure 5 have two deﬁnite peaks, and the left narrow peakis mainly from parameters in Regime 2, while the right wide peak is mainly fromparameters in Regime 0 and Regime 1. Since the parameters in Regime 1 are notvery diﬀerent from those in Regime 0, they make a distribution with only one
peak. The distributions in AAA and BBB in surviving bond have only one peakbecause the diﬀerences among parameters in diﬀerent regimes are not as evident as
in CCC/C.
4.3. Term structures of zero rates
Figure 6 depicts the intial theoretical (calculated by our model) term structuresof zero rates with diﬀerent credit ratings. The horizontal axis represents maturity(years), and the vertical represents the zero rate (%). In Figure 6, there are four
curves: from below, default-free, AAA, BBB, and CCC/C zero curves, respectively.
Notice that the theoretical curve does not match the given curve at each credit
rating perfectly here because we construct the objective function for calibration
using discrete grid points (0, 1, 2, ...) and are limited to 10 years. There is a tendency
for the curve in the short term to sometimes bend unnaturally, but it does not look
so remarkable. Hereafter, we show some examples of various shapes of theoretical
future zero curves obtained by the Monte Carlo simulations. We assume the initial
regime is “Regime 0” o .
The default-free zero rates at 0-, 2-, 4-, 6-, 8-years-after on a certain sample
path are shown in Figure 7. The zero curve at 2-years-after begins from 2 years too This assumption is consistent with the calibration described in Section 3.2

A Term Structure Model of Interest Rates with Regime-Switching Properties

19

t=0
default-free

AAA

BBB

CCC/C

8.0%
7.0%
6.0%
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
0

1

2

3

4

5

6

7

8

9

10

Fig. 6. Initial term structures of zero rates (initial regime 0). The horizontal axis represents maturity, and the vertical represents the zero rate.

10 years on the horizontal axis; on this curve, the rate on the horizontal 2 years
is the instantaneous spot rate (0-year zero rate) at 2-years-after, and the rate on
the horizontal 10 years is the 8-year zero rate at 2-years-after, and so on. The
regime has much inﬂuence on the shape of the zero curve, but the shape does notcorrespond directly to the regime at the same time because the shape is the resultof the cumulative eﬀect from t = 0.
Defaut-Free
3.5%
3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
-0.5%
-1.0%
0

1

2

3

4

5

6

7

8

9

10

Fig. 7. Future term structures of default-free zero rates on a certain sample path. The horizontal
axis represents maturity, and the vertical represents the zero rate. The curve that begins at 2 years
means the term structures after 2 years, and so on.

Figure 8, 9 and 10 depict the future zero curves with AAA, BBB and CCC/C,

20

Yukio Muromachi

AAA
6.0%

5.0%

4.0%

3.0%

2.0%

1.0%

0.0%
0

1

2

3

4

5

6

7

8

9

10

Fig. 8. Future term structures of AAA zero rates on a certain sample path. The horizontal axis
represents maturity, and the vertical represents zero rate. The curve that begins at 2 years means
the term structures after 2 years, and so on.

BBB
10.0%
9.0%
8.0%
7.0%
6.0%
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
0

1

2

3

4

5

6

7

8

9

10

Fig. 9. Future term structures of BBB zero rates on a certain sample path. The horizontal axis
represents maturity, and the vertical represents the zero rate. The curve that begins at 2 years
means the term structures after 2 years, and so on.

respectively. These ﬁgures are described in the same way as Figure 7. These ﬁgures
have certain common features.
(1) The shapes of these curves are similar at the same future time, especially betweeen the higher credit qualities, default-free and AAA. This is because, as
indicated by (2.9), (2.15) and (2.22), the zero curves at time t are monotonically increasing functions of r = r(t), which is common for all credit ratings.
The intensity eh=eh(t) is diﬀerent from each other, but its volatility is low in

A Term Structure Model of Interest Rates with Regime-Switching Properties

21

CCC/C
30.0%
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
0

1

2

3

4

5

6

7

8

9

10

Fig. 10. Future term structures of CCC/C zero rates on a certain sample path. The horizontal axis
represents maturity, and the vertical represents the zero rate. The curve that begins at 2 years
means the term structures after 2 years, and so on.

higher credit ratings, so that the ﬂuctuation by eh(t) is small. In CCC/C, σh is
so high that the shape of the zero curve becomes diﬀerent from those in higher
credit ratings.
(2) The zero rates in the short term ﬂuctuate widely with time, while the zero rates
in the long term do not ﬂuctuate so much. This is because the instantaneous
zero rate, which is the simulated r(t) + (1 − δ)eh(t), moves stochastically withtime and it has much inﬂuence on shorter zero rates, while the long-term zerorate would approach gradually the limiting distribution so that the ﬂuctuation
would decrease.
(3) The zero curve moves more widely as the credit rating decreases, which corresponds to the increase of σh .
(4) In each moment, the zero rates are usually arranged from lower to upper against
the credit ratings, which can be seen in Figure 11 (zero curves with diﬀerent
credit ratings at 2-year-after p ).
(5) Compared with other credit ratings, future CCC/C zero curves become higher
than the initial curve because eh(t) = h(t) + ⟨ℓ(t), X(t)⟩ in future becomes much
CCC/C
CCC/C
e
higher than h(0) due to high mh,k(t) and ℓk(t).
In general terms, the curves with diﬀerent credit ratings have similar shapes at thesame time, and the main diﬀerence is the level and the ﬂuctuation with time. Moreover, we can see various shapes of zero curves: for example, normal-yield (defaultfree at 2-, 4- and 8-years-after), inverted-yield (CCC/C at 2- and 6-years-after),
weakly humped shape (BBB at 2-years-after), and inverted-humped shape (AAA,p We do not impose any restrictions for the order of the level at all. This is the result of ourcalibration.

22

Yukio Muromachi

BBB and CCC/C at 8-years-after).
t=2
default-free

AAA

BBB

CCC/C

16.0%
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0

1

2

3

4

5

6

7

8

9

10

Fig. 11. Future term structures of zero rates with diﬀerent credit ratings on a certain sample path.
The horizontal axis represents maturity, and the vertical represents the zero rate.

We can show various other shapes. Among them, notably, the normal-yields andthe default-free rate are negative but other rates are positive in the short term, andso on.
4.4. Future distribution of zero rates
Figure 12 shows the distributions of the default-free zero rates with diﬀerent maturities at 1-year-after. The “1-yr zeros” is the distribution of 1-year zero rates (M = 1)
at 1-year-after (t = 1), and “3-yr zeros” is that of 3-year zero rates (M = 3) at thesame time, and so on. The distribution moves upward with maturity, and its width
becomes narrow. This feature corresponds to the second feature (denoted by “(2)”)
of the zero curves described in Section 4.3, that is, the zero rates in the short term
ﬂuctuate widely with time, while the zero rates in the long term do not ﬂuctuate
to that degree. The above feature can be also seen in other credit ratings, Figure
13 and Figure 14.
In general terms, the deﬁnite upward movement of the distribution such as
Figure 12 and 13 implies that the normal-yield curve would be usually seen, but
the inverted yield curve would appear rarely. The inverted-yield curve corresponds
to the curve whose shorter zero rate is higher than the longer zero rate. In Figure
12 and 13, since the distribution of 1-yr zeros is wide but not signiﬁcantly higher
than the distributions of longer zeros, the inverted-yield curve is less predominant.
However, in Figure 14, the distribution of 1-yr zeros is so wide that there exists
some non-negligible probability mass where 1-yr zeros are higher than the longer
zeros probability mass, therefore, the inverted-yield curve often appears in CCC/C
curves.

A Term Structure Model of Interest Rates with Regime-Switching Properties

23

2000
1800
1600
1400
1200
1000
800
600
400
200
0
-1.0%

-0.5%

0.0%

1-yr zeros

0.5%

1.0%

3-yr zeros

1.5%

2.0%

5-yr zeros

2.5%

3.0%

7-yr zeros

3.5%

4.0%

9-yr zeros

Fig. 12. Distributions of the default-free zero rates with diﬀerent maturities at 1-year after.

2000
1800
1600
1400
1200
1000
800
600
400
200
0
-1.0%

-0.5%

0.0%

1-yr zeros

0.5%

1.0%

3-yr zeros

1.5%
5-yr zeros

2.0%

2.5%
7-yr zeros

3.0%

3.5%

4.0%

9-yr zeros

Fig. 13. Distributions of the AAA zero rates with diﬀerent maturities at 1-year after.

5. Concluding remarks
In this study, we propose a term structure model for risk evaluation, sensitive tothe interest rate and the credit risks with Markovian regime-switching property.
First, we discuss the dynamics of a regime, interest rate, and default intensity in thephysical probability measure and the change of measure to an equivalent martingale
measure (pricing measure), and derive the dynamics in the pricing measure and a
pricing formula. Second, we propose and demonstrate a simple model to calculate
the discount bond price, which is a minor extension of model proposed in Elliott &Siu (2009). Third, we propose a calibration method for the simple model. Fourth,
based on the simple model, we present some calibration results and some numerical

24

Yukio Muromachi
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0.0%

2.0%
1-yr zeros

4.0%
3-yr zeros

6.0%

8.0%
5-yr zeros

10.0%
7-yr zeros

12.0%

14.0%

9-yr zeros

Fig. 14. Distributions of the CCC/C zero rates with diﬀerent maturities at 1-year after.

examples of term structures of zero rates with diﬀerent credit ratings at some future
times. As the default-free interest rate and the default intensity are dependent on
X(t), the term structure of the zero rates depends on the regime dynamics. As aresult, we can obtain various shapes of yield curves with diﬀerent credit ratings infuture.
However, considering the existing limitations, additional research is required.
First, more eﬃcient parameter estimation methods should be urgently developed.
In this study, although we propose a simple calibration method, it is apparently
suboptimal because the parameter estimation is performed sequentially. Second, as
we do not have enough knowledge about the regime-switching properties of ﬁnancial
data, more studies on analyzing various kinds of data are necessary. For example,
because of the dependence among the ﬁnancial variables might be dissimilar invarious regimes, the estimation of appropriate copula functions in each regime would
be another plausible research theme. Additionally, some minor changes to the model
are required; we use the Vasicek model as an intensity process, but it is desirable
to replace it with, for example, the CIR model and so on, in order to avoid thenegative intensity.
Finally, there is considerable interest in data analyses of liabilities combinedwith information about regimes. Some kinds of regime-switching properties could
be seen not only in the interest rate risk and credit risk but also in the equity risk,
the currency risk, and so on. Additionally, the dependence of each asset and liability
class on ﬁnancial variables might change in each regime and class. For example, we
consider the growth rates of the liquid deposits’ volumes to be dependent on theinterest rates, however, it is widely known that the dependence disappears under avery low-interest rate environment. We think that, for ﬁnancial institutions such as
banks, the regime-switching models would be promising and useful tools to grasp
the risk precisely in a wide range, especially in the ﬁeld of ALM. By analyzing

A Term Structure Model of Interest Rates with Regime-Switching Properties

25

the properties of assets and liabilities with the information about regimes obtained
through a reasonable regime-switching model, and evaluating the risks of assets
and liabilities synthetically through the model, new features and prospects of therisks might be clariﬁed, and the results could be reﬂected in the practical risk
management. To the best of our knowledge, there are not any studies on assets andliabilities from such a standpoint. We hope that such studies will be encouraged
and this study can be used as a part of the theoretical foundation.
Appendix A. Derivation of (2.23)
Consider the model in Section 2.3. Under Assumption 2.2, we have



dℓ(t)
f (t)⟩
dℓ(t) = d⟨ℓ(t), X(t)⟩ =, X(t) + ⟨ℓ(t), Q⊤ X(t)⟩ dt + ⟨ℓ(t), dM
dt(A.1)
under Pe. The term ⟨ℓ(t), QX(t)⟩ is the contribution of the regime-switching. It
follows from Assumption 2.2, (2.19)–(2.21) and (A.1) thatdeh(t) = dh(t) + dℓ(t)
f (t)⟩,= ah ⟨m′h (t), X(t)⟩ − eh(t) dt + ⟨σ h , X(t)⟩dezh (t) + ⟨ℓ, dM
where A ⊗ B is the Hadamard product of K-dimensional vectors A and B, and
1 dℓ(t)+ σ h ⊗ Lh + ψh (t)σ hm′h (t) = mh + ℓ(t) +ahdt+(⟨ℓ(t), Q⊤ e1 ⟩, · · · , ⟨ℓ(t), Q⊤ eK ⟩)⊤ .
(A.2)
Deﬁne rt,u (r) = r(u|r(t) = r), e
ht,u (eh) = eh(u|eh(t) = eh), 0 ≤ t ≤ u ≤ T , and
"
( Z
)
#
T
et exp −p(t, T, r, eh, X) = Ert,u (r)du 1{τ >T }
"
et
=E

t

( Z
)#
T
exp −
(rt,u (r) + e
ht,u (e
h))du
.

(A.3)

t

It follows from (2.14) that
Z s
Z s
rt,s (r) = r +(⟨ϕ(u), X(u)⟩ − art,u (r)) du +⟨σ, X(u)⟩dezr (u),
tt

so that Dt,s = ∂rt,s (r)/∂r satisﬁes

Z s

Dt,s = 1 − aDt,u du.
t

Diﬀerentiating (A.6) with respect to s, we havedDt,s
= −aDt,s ,ds

Dt,t = 1.

(A.4)

Yukio Muromachi

26

Therefore, we obtainDt,s = e−a(s−t) .

(A.5)

From (A.3) and (A.5), we have!
Z T∂p(t, T, r, eh, X) = −Dt,u du p(t, T, r, eh, X) = −B(t, T )p(t, T, r, eh, X),∂rt
and similarly, we have∂h, X) = −Bh (t, T )p(t, T, r, eh, X).
p(t, T, r, e
∂e
h
Deﬁne

 Z t

pe(t, T, r, eh, X) = exp −(rt,u (r) + e
ht,u (eh))du p(t, T, r, e
h, X)
0
"
( Z
)#
T
e
e
e
= Et exp −(rt,u (r) + ht,u (h))du,0which is a Doob’s martingale under Pe. Applying Ito’s lemma to pe(t, T, r, eh, X) and⊤ee = (eputting p
p1 , · · · , peK ) with pei = pe(t, T, r, h, ei ), i = 1, · · · , K, we havepe(t, T, r, eh, X) = pe(0, T, r(0), eh(0), X(0))
Z t∂ pe 
∂ pe ∂ pe++(⟨ϕ(u), X(u)⟩ − ar(u)) +h(u)⟨ah m′h (u), X(u)⟩ − ah e
∂u ∂r
∂e
h
0

2
2
1
∂ 2 pe∂ pe 1∂ pedu+ ⟨σ, X(u)⟩2 2 + ⟨σ h , X(u)⟩2+ ρ(t)⟨σ ⊗ σ h , X(u)⟩
2
∂r
2
h
∂e
h2
∂r∂ e
Z t
Z t
Z t
∂ pe
∂ pe
e ⊤ X(u)⟩du+⟨σ, X(u)⟩dezr (u) +⟨σ h , X(u)⟩dezh (u) +
⟨e
p, Q
h
0 ∂r
0 ∂e
0

Z t
∂ pe
f
e+
+
p
⊗ ℓ(u), dM (u) .
∂eh0Because of the martingale property of pe, the integrand of du term must be zero, sothat we have the following partial diﬀerential equation (abbreviated as PDE)∂ pe ∂ pe∂ pe 
0=⟨ah m′h (t), X(t)⟩ − ah e+(⟨ϕ(t), X(t)⟩ − ar(t)) +
h(t)
∂t
∂r
∂e
h
2
2
1
1
∂
p
e
∂
pe
∂ 2 pee ⊤ X(t)⟩,+ ⟨σ, X(t)⟩2 2 + ⟨σ h , X(t)⟩2+ ρ(t)⟨σ ⊗ σ h , X(t)⟩
+ ⟨e
p, Q
2
∂r
2
∂e
h2
∂r∂ e
h
which implies that

∂p
∂p
∂p 
⟨ah m′h (t), X(t)⟩ − ah eh(t)− rp − ehp +(⟨ϕ(t), X(t)⟩ − ar(t)) +
∂t
∂r
∂e
h
2
2
1
1
∂2p
2∂ p
2∂ p
e ⊤ X(t)⟩,+ ⟨σ, X(t)⟩+ ⟨σ h , X(t)⟩+ ρ(t)⟨σ ⊗ σ h , X(t)⟩+ ⟨p, Q
2
2
∂r
2
∂e
h2
∂r∂ e
h
(A.6)

0=

A Term Structure Model of Interest Rates with Regime-Switching Properties

27

where p = (p1 , · · · , pK )⊤ with pi = p(t, T, r, eh, ei ), i = 1, · · · , K, with the terminal
condition p(T, T, ·, ·, ·) = 1. (A.6) means a system of K coupled PDEs

∂pi
∂pi
∂pi 
ah m′h,i (t) − ah eh(t)0=
− rpi − ehpi +(ϕi (t) − ar(t)) +
∂t
∂r
∂e
h
2
2
2
1 2 ∂ pi
1 2 ∂ pi
∂ pie ⊤ ei (t)⟩,+ σi+ σh,i+ ρ(t)σi σh,i+ ⟨p, Q0 ≤ t ≤ T,
2 ∂r2
2
h
∂e
h2
∂r∂ e
with pi (T, T, ·, ·) = 1, i = 1, · · · , K. Here, assume thatnopi (t, T, r, eh) = exp Ai (t, T ) − B(t, T )r − Bh (t, T )eh ,i = 1, · · · , K,

where Ai (t, T ) = A(t, T, ei ), then, the following ODEs"#2
σi2 2 σh,iAi dAi′2− ϕi (t)B − ah mh,i (t)Bh +B +B + ρ(t)σi σh,i BBh
0=e
dt
2
2 h
+ ⟨Ā, Q⊤ ei ⟩,

(A.7)

dB
= aB − 1,dtdBh= ah Bh − 1
dt

(A.8)
(A.9)

must be satisﬁed where Ā = (Ā1 (t, T ), · · · , ĀK (t, T ))⊤ with Āi (t, T ) = eAi (t,T ) , i =1, · · · , K. From (A.8) and (A.9), we haveB(t, T ) =1 − e−a(T −t)aandBh (t, T ) =1 − e−ah (T −t),ahand (A.7) is rewritten as(2σh,idĀi (t, T )
σi2 2
0=
+
B (t, T ) +B 2 (t, T ) − ϕi (t)B(t, T ) − ah m′h,i (t)Bh (t, T )
dt
2
2 h

(A.10)+ ρ(t)σi σh,i B(t, T )Bh (t, T ) Āi (t, T ) + ⟨Ā, Q⊤ ei ⟩where
m′h,i (t) = mh,i + ℓi (t) +

1
ah



dℓi (t)+ σh,i λh,i + σh,i ψh (t) + ⟨ℓ, Q⊤ ei ⟩dtwith the terminal conditions Āi (T, T ) = 1, i = 1, · · · , K.
Acknowledgments
The author is grateful to Naoto Hatakeyama, Junpei Igarashi, Prof. Toshinao
Yoshiba, and Prof. Kohta Takehara at Tokyo Metropolitan University for helpful discussions and various kinds of data analyses. The author would also like to
Rongzhu Liu, Toshiaki Seiwa and Masayuki Hamanaka at AFAS Corporation forproviding him with this interesting research theme and continuing the research

28

Yukio Muromachi

project. The author thanks the referees and the editor for their appropriate and insightful comments, and appreciates the research grant funded by the Grant-in-Aid
for Scientiﬁc Research (C) 22K04583.
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