Research Paper Series
No. 34

Pricing and Risk Evaluation of Interest Rate Risk
and Credit Risk under Regime-Switching Environment

Yukio Muromachi†
March, 2022

†Tokyo Metropolitan University, Graduate School of Management

Pricing and Risk Evaluation of Interest Rate Risk
and Credit Risk under Regime-Switching Environment
Yukio Muromachi∗

March 16, 2022
Abstract
Long term historical data of interest rates and credit spreads imply that there exist
some diﬀerent regimes; a calm regime with lower default risk and volatility, and astressed regime with higher default risk and volatility. In this article, we propose apricing and risk evaluation model of interest rate risk and credit risk with Markovian
regime-switching property. We discuss the dynamics of regime, interest rate and default
intensity under the physical probability and the change of measure to an equivalentmartingale one, and we propose a simple tractable model. In our model, the defaultfree interest rate and the default intensity are dependent through the regime, and thedependence aﬀects the price of the defaultable bond. We calculate distributions of abond portfolio’s price at a risk horizon, which reflects the actual default loss up to thehorizon and the decrease of market prices due to the transition to a stressed regime.
Numerical examples show that the price distribution has a short right tail and a longleft tail, and that the distribution depends strongly on the present regime. Such results
would be applicable to the financial risk management, especially on the stress tests.

Keywords: financial risk management, regime-switching model, interest rate risk, credit
risk, term structure of interest rates, change of measure
JEL Classification Numbers: G12, G13, G17.

∗

Tokyo Metropolitan University, Graduate School of Management

Email address: muromachi-yukio@tmu.ac.jp1

1

Introduction

In the 21st century some financial distresses have already happened, and one of them was
the worldwide financial crisis. Many financial institutions, financial regulation agencies andcentral banks have discussed to control financial risks, and have been constructing concepts,
tools and systems for risk management, financial regulations, and financial policies. For
many financial institutions, especially for banks, management of the interest rate risk andthe credit risk has been one of the most important challenges.
From the long term data of interest rates and credit spreads, some feels that there exist
some diﬀerent regimes; typically, a calm regime with lower default risk and volatility, and astressed regime with higher credit risk and volatility. Since Hamilton [12] proposed a regimeswitching model and analyzed the economic cycles, such models have been used by many
researchers, mainly for analyzing and modeling economic and financial data1 . Restricted to
the researches on interest rates, Inoue and Okimoto [14] used Markovian regime-switchingmodels for analyzing monetary policy and private sector behavior in Japan. Dai et al. [4] useda discrete-time regime-switching Gaussian term structure model, while Wu and Zeng [20]applied a continuous-time aﬃne term structure model with regime-dependent parameters.
Elliott and Siu [7] considered a bond valuation as a derivative written on a Markovian regimeswitching 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 and Nishide [6] and so on.
On credit risk analyses, Gourieroux et al. [10] derived a defaultable bond price in a
discrete-time setting and analyzed sovereign yield curves, and Monfort and Renne [19] analyzed the credit spreads of U.S. bonds. Fischer and Stolper [9] analyzed the behavior of thecredit spreads and their key determinants, and Chun et al. [3] proposed a regression modelof credit spreads with endogenous regimes, and showed the model enhanced the explanatory
power of the determinants. In a continuous-time setting, Hainaut and Le Courtois [11] proposed a default intensity model described by regime-switching Lévy process to evaluate thesurvival probability, and analyzed the CDS market. Li and Ma [18] discussed pricing options
analytically, but their results are limited to the conditional price given a sample path of theregime.
Many research papers have already existed on the regime-switching models. However,
we do not know papers in which such models are used for evaluating the financial risk
1

As an example of review papers, see Ang and Timmermann [1].

2quantitatively. If an extremely high-risk regime is included in the risk evaluation model, it
might give us fresh views against the future risk. Moreover, if such a high-risk regime canbe estimated from the observed data, another kind of forward-looking stress tests could beobtained statistically.
In this article, we propose a pricing and risk evaluation model of interest rate risk andcredit risk with regime-switching property, under a continuous-time setting. For evaluatingrisk, we set a risk horizon which is a certain future time, and discuss the price at the horizon
and the accumulated loss up to the horizon totally. For pricing, we use the no-arbitrage pricenot only at present but also at the risk horizon because the no-arbitrage price at the horizon
can reflect the influence of the loss which could occur after the horizon, while the loss up
to the horizon can be evaluated as a realized loss on a simulation path. In order to keep
the consistency between pricing and risk evaluation, stochastic modelings are necessary not
only under the physical measure but also under the pricing measure such as the risk-neutral
measure. The physical measure part of the model is used for generating scenarios up to therisk horizon, while the pricing measure part is used for pricing at the risk horizon.
This article is organized as follows. In Section 2, we describe constructions of a pricing
and risk evaluation model, and propose a simple model. Based on a Markovian regimeswitching model, the default-free spot rate and the default intensity processes are describedunder the physical measure, and the change of measure to the equivalent one is discussed,
so that the processes under the pricing measure are derived explicitly. In Section 3, we show
some numerical examples of the term structure of default-free and defaultable zero rates, andexplain the eﬀect of the existence of regimes. In Section 4, the price distributions of a bondportfolio are shown by the Monte Carlo simulation, and some implications are discussed.
Section 5 concludes this article.

2

The Model

In Section 2.1 and 2.2, constructions of a pricing and risk evaluation model with a regimeswitching property is discussed, while in Section 2.3, a simple tractable model is proposed.
If pricing securities is the only one object, discussion in Section 2.1 is not necessary, however,since we would like to proceed the risk evaluation which is consistent with pricing, we begin to
discuss from Section 2.1. See Kijima and Muromachi [16] for detail, in which constructions ofa risk evaluation model consistent with pricing was discussed. One of their main discussions3

is the reason why two probability measures, the physical measure and the pricing measure,must be needed. Throughout this article, we use a simple homogeneous model as a regimeswitching part.

2.1

Stochastic processes under physical probability measureConsider a switching regime under a continuous-time and finite-state Markov chain model.
Let t 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 (Ω, F, (Ft )0≤t≤T , P ) under which allstochastic processes can be described. All probability spaces appeared in this article satisfy
the usual conditions.
There exist K, K ∈ N , diﬀerent states 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 thei-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 steady, so that we can express
P {X(t + ∆t) = ej |X(t) = ei }, j ̸= i
 lim
∆t→0
∆t
∑
Q = (qij )i,j=1,··· ,K , qij =

−
qik ,
j=i



(2.1)

k̸=i

and
∫ t
QX(s)ds + M (t),X(t) = X(0) +

t≥0

(2.2)

0

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

(2.3)

is the solution of the following diﬀerential equationdP(t)= QP(t),dt

P(0) = IK ,(2.4)

where IK is the K-dimensional indentical matrix, and is given by
P(t) = exp {tQ} =

∞ k k
∑
t Q
k=0

4

k!

.

(2.5)

The risk-free instantaneous spot rate at time t, denoted by r(t), follows the stochastic
diﬀerential equation (hereafter, abbreviated by SDE)
dr(t) = µr (t, r(t), X(t−))dt + σr (t, r(t), X(t−))dzr (t),t ≥ 0,(2.6)

where zr (t) is the standard Brownian motion under P . Consider a risky security subject toits credit risk. Let τ > 0 be the default time of the issuer of the security or the counterpartyof a contract of the security. Define H(t) = 1{τ ≤t} as its default process where 1A is theindicator function2 , and the default 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,(2.7)

where zh (t) is a standard Brownian motion under P and dzr (t)dzh (t) = ρ(t)dt.
The filtrations in this article are as follows; for any arbitrary time t, 0 ≤ t ≤ T , thefiltration generated by the default process is defined by Ht = σ(H(s) : 0 ≤ s ≤ t) andH = (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 . Similarly, 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 .

2.2

Stochastic processes under risk-neutral probability measureConsider a probability measure P X equivalent to P . Assume that the infinitesimal generator
matrix of X(t) under P X , denoted by QX = (qijX )i,j=1,··· ,K , is given byi ̸= j,qijX = (1 + κij )qij ,i, j = 1, · · · , K(2.8)

where κij > −1 are constant. Then, the change of probability measure from P to P X iswritten by
dP X= η X (T ),dP F X
T

2

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

5

(2.9)

where the Radon-Nykodym derivative η X (T ) is defined by
{ ∫ K
}
(
)
K
t ∑
∏
∑
η X (t) = exp −κkℓ qkℓ H k (u)du1+κkℓ ∆H kℓ (u)0 k,ℓ=10<u≤t

H i (t) = 1{X (t) = ei } ,
i = 1, · · · , K,∑
H ij (t) =H i (u−)H j (u),i ̸= j,(2.10)k,ℓ=1(2.11)
i, j = 1, · · · , K,

(2.12)

0<u≤tand ∆H ij (u) = H ij (u)−H ij (u−). Let P X (t) be the transition probability matrix under P X .
Then, the similar equations such as (2.2), (2.4) and (2.5) 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 in the credit risk model3 . Let β(t) be a Ft -adapted
process and κh (t) > −1 be a Ft -predictable process, and define∫ tρh (t) =(βh (s)dzh (s) + κh (s)dMh (s))(2.13)

0and a new probability measure P c equivalent to P X bydP c
= ρh (T ).
dP X GT

(2.14)

Then, the process∫ tzhc (t) = zh (t) −βh (s)ds(2.15)

0

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∫ t∧τc
Mh (t) = H(t) −hc (s)ds(2.16)(2.17)

0

becomes a (F, P c )-martingale, which corresponds to the fact that the process∫ t∧τMh (t) = H(t) −h(s)ds,0

is a (F, P X )-martingale.
3

See Kusuoka [17], Bielecki and Rutkovski [2], and so on.

6(2.18)

The following discussion is according to Elliott et al.[8]. Assuming that the time t priceof a risky asset, 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≥0(2.19)

where zSc (t) is a standard Brownian motion under P c . Define the Radon-Nikodym derivativefrom P c to its equivalent probability measure P η by 4
}
{∫ T
∫
dP η
1 T
2
c
(η(s)) ds= exp
η(s)dzS (s) −dP c GT2 00(2.20)

where
η(t) =

r(t) − µ(t, S(t), X(t)),σ(t, S(t), X(t))(2.21)

then, thanks to Girsanov’s theorem, the process∫ tz (t) = zSc (t) −ηη(u)du(2.22)

0

becomes a standard Brownian motion under P η , and the relative price of S(t) with respect}{∫t
to the bank account B(t) = exp 0 r(u)du becomes a (G, P η )–local martingale. Assumingthat the relative price is a (G, P η )–martingale, the price of a European derivarive given GT
is given by[
V (t|Gt ) = E

Pη

}
]
{ ∫ M
r(u)du G(S(M )) Gt ,exp −(2.23)twhere G(·) is the payoﬀ function of the derivative at its maturity M, t ≤ M ≤ T , and E P [·|·]
ηis the conditional expectation operator under P η . From the chain rule of the conditional
expectation, given {τ > t}, the price V (t) is given by
}
[
{ ∫ M
]
Pη
Pη
V (t) = E [V (t|Gt )| Ft ] = E
r(u)du G(S(M )) Ft .
exp −(2.24)t

Hereafter, we call P η the risk-neutral probability measure, and denote it Pe, and a standard
Brownian motion under Pe is denoted by ze. Similarly, hc (t), QX , P X (t) and M X (t) aref (t), respectively.
e P(t)e and Mdenoted by eh(t), Q,
Generally speaking, a regime-switching model derives an incomplete market so that manyrisk-neutral measures might exist, and P η is one of them. However, according to Elliott et
4This change of measure is called the risk-neutral regime-switching Esscher transform.

7al.[5], since the measure P η is the minimum entropy martingale measure of P , the pricegiven by (2.24) might be a reasonable one.
Let Ft -predictable process λr (t) be the market price of risk for zr (t), then the 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),(2.25)

dh(t) = µeh (t, h(t), X(t−))dt + σh (t, h(t), X(t−))dezh (t)(2.26)eh(t) = (1 + κh (t, X(t−)))h(t)(2.27)µer (t, r(t), X(t−)) = µr (t, r(t), X(t−)) − λr (t)σr (t, r(t), X(t−))(2.28)µeh (t, h(t), X(t−)) = µh (t, h(t), X(t−)) + β(t)σh (t, h(t), X(t−)).

(2.29)

Consider a default-free discount bond with maturity M, t ≤ M ≤ T . From (2.24), its
price at time t is given by
]
[
{ ∫ M
}
e
v(t, M, r, X) = E exp −r(u)du r(t) = r, X(t) = X .

(2.30)t

Next, consider a defaultable discount bond with maturity M , and let τ > 0 its default time.
Suppose the holder of the bond receives $1 at M if the bond survives at M , while he/she
receives $δ, 0 ≤ δ < 1, at M if default occurs up to M . From (2.24), given {τ > t}, theprice of the bond is given by
D(t, M, r, e
h, X)
]
}
[
{ ∫ M
{
}
e exp −
h(t) = eh, X(t) = X
r(u)du 1{τ >M } + δ1{τ ≤M } r(t) = r, e=Et= δv(t, M, r, X) + (1 − δ)p(t, M, r, eh, X),(2.31)

p(t, M, r, e
h, X)
[
{ ∫ M
}
]
e exp −
(r(u) + eh(u))du r(t) = r, eh(t) = eh, X(t) = X .
=E(2.32)t

We call p(t, M, r, eh, X) the price of a “survival discount bond”, whose payoﬀ is $1 if andonly if the bond survives at M .

2.3

A simple model

Here, we propose a simple tractable model. Under the physical measure P , r(t) follows
dr(t) = a (⟨m, X(t)⟩ − r(t)) dt + ⟨σ, X(t)⟩dzr (t),8t ≥ 0,(2.33)

where m = (m1 , · · · , mK )⊤ and σ = (σ1 , · · · , σK )⊤ are K-dimensional constant vectors, aand σk , k = 1, · · · , K are positive constants, ⟨A, B⟩ is the inner product of vectors A andB, and zr (t) is a standard Brownian motion under P . Similarly, h(t) follows
dh(t) = ah (⟨mh , X(t)⟩ − h(t)) dt + ⟨σ h , X(t)⟩dzh (t),t ≥ 0,(2.34)

where mh = (mh,1 , · · · , mh,K )⊤ and σ h = (σh,1 , · · · , σh,K )⊤ are K-dimensional constantvectors, ah and σh,k , k = 1, · · · , K, are positive constants, zh (t) is a standard Brownian
motion under P and dzr (t)dzh (t) = ρ(t)dt 5 . For simplicity, we assume that X(t) is independent of (zr (t), zh (t)). As you see, we select the Vasicek model because model parameters
can be calibrated easily by using the existing statistical tools such as Matlab6 .
In order to simplify the mode, we use the following assumption.
Assumption 2.1. The market price of risk λr (t) against zr (t) is written byλr (t) = ⟨Lr , X(t)⟩ + ψr (t)(2.35)

where Lr = (λr,1 , · · · , λr,K )⊤ is a K-dimensional constant vector and ψr (t) is a deterministic
function of time t.
From (2.25), (2.28), (2.33) and Assumption 2.1, we havedr(t) = (⟨ϕ(t), X(t)⟩ − ar(t)) dt + ⟨σ, X(t)⟩dezr (t),(2.36)

where the deterministic function vector ϕ(t) = (ϕ1 (t), · · · ϕK (t))⊤ is given by
ϕ(t) = am − σ ⊗ Lr − σψr (t),(2.37)and A⊗B is the Hadamard product of K-dimensional vectors A and B. Then, the discountbond price v(t, M, r, X) is given byv(t, M, r, X) = exp {⟨A(t, M ), X⟩ − B(t, M )r} ,(2.38)

where
B(t, M ) =5
6

1 − e−a(M −t),a(2.39)The extension of (a, ah ) to the deterministic functions of time t is obvious, not essential.
In Hatakeyama [13], a slightly modified program based on Matlab was used, which released our labor

much.

9

and Ā(t, M ) = (Ā1 (t, M ), · · · , ĀK (t, M ))⊤ , Āi (t, M ) = exp{A(t, M, ei )}, i = 1, · · · , K, isthe solution of the following ordinary diﬀerential equations (hereafter, abbreviated by ODEs,
and these are Kolmogorov’s backward equations){}
K∑dĀi (t, M )
1 2 2+
σ B (t, M ) − ϕi (t)B(t, M ) Āi (t, M ) +qeij Āj (t, M ) = 0,
dt
2 i
j=1

(2.40)

with the terminal conditions Āi (M, M ) = 1, i = 1, · · · , K. A numerical solution of (2.40)
is obtained easily, for example, by using the Runge-Kutta method. The zero rate (Yield To
Maturity of the discount bond) at time t is given by
R(t, M, r(t), X(t)) = −log v(t, M, r(t), X(t)),
M −t

0 ≤ t < M.

(2.41)In order to treat the default intensity process similarly above, and in order to make thecalibration tractable, we introduce the following assumption.
Assumption 2.2. The stochastic process β(t) is written asβ(t) = ⟨Lh , X(t)⟩ + ψh (t)(2.42)

where Lh = (λh,1 , · · · , λh,K )⊤ is a K-dimensional constant vector and ψh (t) is a deterministic
function of time t. And, defineℓ(t) = ⟨ℓ(t), X(t)⟩,ℓ(t) = (ℓ1 (t), · · · , ℓK (t))⊤(2.43)

where ℓj (t) = κh,j (t)h(t), j = 1, · · · , K, are diﬀerentiable deterministic functions7 of time t.

From (2.26), (2.27), (2.29), (2.34) and Assumption 2.2, we havedh(t) = (⟨ϕh (t), X(t)⟩ − ah h(t)) dt + ⟨σ h , X(t)⟩dezh (t)(2.44)ϕh (t) = ah mh + σ h ⊗ Lh + σ h ψh (t),(2.45)eh(t) = h(t) + ⟨ℓ(t), X(t)⟩.

(2.46)

Now, consider a survival discount bond with the maturity M . From (2.32), the time tprice of the bond is written byp(t, M, r, e
h, X)
[
{ ∫ M
}
]
e
e
e
e
= E exp −(r(u) + h(u))du r(t) = r, h(t) = h, X(t) = X ,(2.47)

t7This seems impossible because h(t) cannot be diﬀerentiable, however, it is very useful for calibration.

10and from the results in Appendix A, it is given by{}p(t, M, r, eh, X) = exp ⟨A(t, M ), X⟩ − B(t, M )r − Bh (t, M )eh ,(2.48)

where
Bh (t, M ) =1 − e−ah (M −t),ah(2.49)and Ā(t, M ) = (A1 (t, M ), · · · , AK (t, M ))⊤ , Āi (t, M ) = exp{Ai (t, M, ei )}, i = 1, · · · , K, isthe solution of the following ODE system:
{ 2
2
σh,i
dĀi (t, M )σi 2+B (t, M ) +Bh2 (t, M ) − ϕi (t)B(t, M )
dt
2
2
}
′′
−ah mh,i (t)Bh (t, M ) + ρ(t)σi σh,i B(t, M )Bh (t, M ) Āi (t, M )K∑qeij Āj (t, M ) = 0.

(2.50))
K∑dℓi (t)+ σh,i λh,i + σh,i ψh (t) +qeij ℓj (t)dt
j=1(2.51)

+

j=1

where
1
m′′h,i (t) = mh,i + ℓi (t) +ah(with the terminal condition Āi (M, M ) = 1, i = 1, · · · , K. Since our valuation method
considers the regime-dependence of r(t) and h(t) directly, it is diﬀerent from Li and Ma’s
result [18].
When eh(t) = 0 for 0 ≤ t ≤ T , which implies that the bond is default-free, (2.50) reduces}{∫Mto (2.40). In contrast, consider a discount bond with the payoﬀ 1{τ >M } exp t r(u)du at
M , then, its time t price is written as 8
[
{ ∫ M
}
]
e
e
e
e
e
v̄(t, M, h, X) = E exp −h(u)du h(t) = h, X(t) = X ,(2.52)tand is given by{}v̄(t, M, eh, X) = exp ⟨A(t, M ), X⟩ − Bh (t, M )eh

(2.53)

where Ā(t, M ) is the solution of the following ODE system: for i = 1, · · · , K,dĀi (t, M )
+
dt
8

{ 2
}
K
∑
σh,i 2′B (t, M ) − ah mh,i (t)Bh (t, M ) Āi (t, M ) +qeij Āj (t, M ) = 0. (2.54)
2 h
j=1

The price does not depend on the short rate r(t).

11Notice that from (2.52), v̄(t, M, eh, X) is the conditional survival probability at M on {τ > t},that is, v̄(t, M, eh, X) = Pe{τ > M |τ > t}. From the above discussion, generally, and even if
ρ(t) = 0, it follow thatp(t, M, r, eh, X) ̸= v(t, M, r, X)Pe{τ > M |τ > t}.

(2.55)

This is because even if ρ(t) = 0, r(s) and eh(s), t ≤ s ≤ T , are F X -conditionally independent,
which does not mean independent.
The necessary model parameters are (1) Q, (a, m, σ) and (ah , mh , σ h ) on stochastice (or κ = (κij )i,j=1,··· ,K ), (Lr , ψr (t)), (Lh , ψh (t)) and ℓ(t) on
processes under P , and (2) Q
stochastic processes under Pe (or, change of measure). The parameters in (1) are estimatedfrom the time-series data of interest rates and credit spreads, while those in (2) are estimatedfrom the term structures of default-free and defaultable intererest rates at t = 0.
A definite disadvantage of this model is that the distribution of the future intensityh(t), t > 0, is constructed basically from normal distributions, which means that a negative
intensity appears with a positive probability. In order to avoid it, the CIR process or thesquared Gaussian process can be proposed. However, since more than 99% of the credit
spreads in future becomes positive in our preliminary results, we might not worry so muchagainst negative h(t).

3Term structures of interest rates

In this section we show some numerical examples of the term structures of default-free anddefaultable zero rates (we call “zero curves”). Since we have been trying to estimate allmodel parameters from observed data but not succeeded yet, some parameters used below
are estimates from observed data, but others are not. Many parameters are estimated fromthe time-series data of credit spreads of bonds in the United States, but the details areomitted here.

3.1

Model parameters

First, we set K = 3. In all our preliminary estimations, statistical information criteria,
AIC and SBIC, decrease as K increases, however, the cases with K ≥ 4 are diﬃcult for us
because the number of parameters increases by the order of K 2 . Consider three defaultablediscount bonds with diﬀerent levels of credit risk; here we call AAA-, BBB-, and CCC- rated
12

bond, respectively. The mean-reverting powers of the default-free interest rate process andthe default intensity processes are assumed a = aAAA
= aBBB= aCCC= 1.0, and otherhhhparameters of the processes are listed in Table 1. The infinitesimal generator Q is shownin (3.56). These parameters are estimated from the time-series data9 . The present defaultfree zero rate curve is given by R(0, M ) = a + bM + cM 2 + dM 3 (%) where (a, b, c, d) =(0, 0.546, −0.0606, 0.00233), and assume that Lr = 0 and the deterministic function ψr (t)e and the term
is piecewise constant. The infinitesimal generator under Pe, denoted by Q,
structure of ψr (t) are calibrated from the above conditions, and are shown in (3.56) andTable 2.


−1.5


1.3


Q=
 2.6

−2.6

0.0

2.5



0.2



e 
0.0 
, Q = −2.5−6.6086 6.2546


0.35402.576−2.5760.0

2.5005 −2.5005

0.0





(3.56)

Other parameters are not from observed data. For simplicity, we set ρ(t) = 0, and therisk-premia adjustments are assumed to be constant with time, and are as follows10 ;
0.0042 0.0102 0.0508
( AAA BBB CCC ) 
.
ℓ
ℓ
ℓ
=
0.00820.0197
0.0971
0.0143 0.0474 0.2095

3.2

Term structures of zero rates with diﬀerent credit ratings

We show some numerical examples of zero curves. The recovery rate is δ = 0, and the initial
cr(i)value of default intensity for i-th bond is hi (0) = mh,k(0) where k(0) is the initial regimenumber and cr(i) is the credit rating of i-th bond, that is, each initial value is typical for its
initial regime. The initial value of the short rate must be r(0) = R(0, 0) = 0.0%.
Figure 1 and 2 show the default-free and defaultable (AAA, BBB and CCC) zero curves
with the initial regime 0 and 2, respectively. The horizontal axis is the maturity (years), andthe vertical axis is the zero rate (%). In Figure 1, since each initial value of the intensity
for i-th bond, ehi (0), is lower than the average value on the steady state of the credit rating
cr(i), each zero curve goes upwards gradually. On the other hand, since BBB and CCC
910

If you would like to see details and can read Japanese, see Hatakeyama [13].
In our preliminary results, we estimated the mean-reverting levels of the default intensities under Pe.

Roughly, one-fourth of the estimated level is allocated as the mean-reverting level under P (in Table 1), andthe residual (three-fourth) is allocated as the risk-premia adjustments.

13

initial values in Figure 2 are higher than the average values on the steady state, these zero
curves go downwards.
Next, compare the same credit zero curves with diﬀerent initial regimes. Figure 3 and4 show the CCC and AAA curves with diﬀerent initial regimes. Figure 3 seems natural because the zero curve becomes higher as the initial regime is higher (“higher” means“more risky”). On the other hand, in Figure 4, except in the short maturities, the zero
curve with initial regime 2 is lower than those with initial regime 0 and 1. Such an unnatural feature can be explained by the regime-dependence of r(t) and eh(t). See Figure
5, which is the same figure as Figure 4 in the case where the default intensity eh(t) only
has a regime-switching property11 . Figure 5 seems natural because the zero curve becomes higher as the initial regime is higher, likely to Figure 3. The diﬀerence between
Figure 4 and Figure 5 is caused by the joint regime movement of r(t) and eh(t) throughX(t). In the case of Figure 5, since r(t) and eh(t) are independent, the following equationp(0, M, r(0), eh(0), X(0)) = v(0, M, r(0), X(0))Pe{τ > M } is always satisfies, so that the relative diﬀerence v(0, M, r(0), X(0))Pe{τ > M }/p(0, M, r(0), eh(0), X(0)) − 1 is always zero.
Figure 6 shows the term structure of the relative diﬀerence in the same case as Figure 4(AAA). The horizontal axis is the maturity (years), and the vertical axis is the relative
diﬀerence. Figure 6 says that the regime-dependence of r(t) and eh(t) decreases the bondprices (increases the zero rates) for the initial regime 1 case, while increases the bond prices(decreases the zero rates) for the initial regime 2 case at most 1% compared to the independent case. The eﬀect of regime-dependence is consistent with the diﬀerence between Figure
3 and Figure 4 quantitatively12 . Also in Figure 3, the same eﬀect has similar influence on
the zero curves, however, an apparent change cannot be seen because of the large diﬀerence
between eh(t) with diﬀerent regimes. Changing the correlation ρ(t) has a little influence on
the relative diﬀerence curve. When ρ(t) = −0.7, the curves in Figure 6 move upwards andhave positive slopes, but since the diﬀerence is at most 0.03%, the zero curves change little.

4

Risk evaluation of bond portfolio

In this section we consider a risk evaluation of a bond portfolio consisting of N discountbonds by the Monte Carlo simulation. The risk horizon is denoted by TR , 0 < TR ≤ M .
Here, we consider TR = 1 year, M = 5 years and N = 20 discount bonds; 5 bonds of
11
12

r(t) follows the Vasicek model with regime-1 parameters.
Roughly speaking, the duration eﬀect on the zero rate is ∆p/p ≃ −t∆R(0, t) where p is the bond price.

14

them are AAA, other 5 bonds are BBB, and the residual 10 are CCC. Most of the model
parameters are the same in Section 3.2. And, all the correlations between the Brownian
motions including ρ(t) are assumed to be zero. As initial values, we use r(0) = 0, and theinitial value of i-th bond’s default intensity, hi (0), is set to be the mean-reverting level ofthe regime 0 for each credit rating cr(i)13 . The number of simulation runs is 50,000.

4.1

Monte Carlo simulation for risk evaluation

We run the Monte Carlo simulation by using our simple model, and its procedure consistsof the following steps. Here, we denote h(t) = (h1 (t), · · · , hN (t)).
1. Set the initial values (X(0), r(0), h(0)).
2. Generate a sample path (X(ti ), r(ti ), h(ti )), 0 = t0 < t1 < · · · < tF = TR under P .
3. Under P , judge whether each bond survives or has defaulted at ti independently. The
default probability of j-th bond during (ti , ti+1 ] is given by hj (ti )(ti+1 − ti ) if ti+1 − tiis small enough.
4. Calculate the price of each bond at time TR ; for i-th bond, D(TR , M, r(TR ), ehi (TR ), X(TR ))if the bond survives at TR , or δv(TR , M, r(TR ), X(TR )) if the bond has defaulted up to
TR 14 .
5. Calculate the portfolio price at TR .
6. Repeat the above steps until enough scenarios have been obtained in order to calculate
necessary statistics accurately.
In Step 2, each sample path is generated from t = 0 to t = TR under P , while in Step
4 the no-arbitrage price is calculated based on the stochastic processes under Pe. Noticethat the input (r(TR ), ehi (TR ), X(TR )) in Step 4 is given based on the values at t = TR on
the sample path generated in Step 3. r(TR ) and X(TR ) are the same under P and Pe, butehi (TR ) = hi (TR ) + ⟨ℓcr(i) (TR ), X(TR )⟩.

13
14

It is diﬀerent from the setting in Section 3.2.
This recovery setting is consistent with that used in Jarrow and Turnbull [15].

15

4.2

Price distribution at risk horizon

Figure 7 depicts the distributions of the CCC discount bond prices at TR . The horizontal
axis is the price, and the vertical axis is the frequency. As the number of initial regime
becomes high, the distribution moves left and becomes wider because the default probability
in future tends to increase as the initial regime number is higher. The numbers of defaulted
scenarios are 974, 1,504 and 3,268 for initial regime 0, 1, and 2, respectively. These numbers
reflect the mean values of the initial regime, mCCC
h,k , k = 0, 1, 2.
Figure 8 depicts the distributions of the portfolio price at TR with diﬀerent initial regimes.
All distributions have short right tails and long left tails, and each has two or three peaks;
the highest peak is the distribution consisting of no-default scenarios, and the second highest
peak is the distribution of one-default scenarios. The third peak can be seen slightly only inthe distribution with initial regime 2. This figure shows that even if the initial observablevalues (r(0), h(0)) and model parameters are the same, the distribution of the future pricedepends strongly on the initial regime X(0), which can not be observed.
Table 3 summarizes some basic statistics for the distributions in Figure 8. In this table,
100α%-VaR is defined by the diﬀerence between the average and the 100(1 − α)-pecentile for
0 < α < 1. The results clearly show that each distribution has a long left tail, which becomes
longer as the initial regime number is higher, and imply that the existence of regimes might
be important for risk evaluation. Although we cannot observe the regime directly, we could
use the probability distribution of regimes estimated from time-series data. When the initial
distribution is assumed, for example, (p1 , p2 , p3 ) = (0.5, 0, 3, 0.2) where pi is the probabilityof regime i, a corresponding distribution is obtained in Figure 9, and the risk measures such
as VaR (Value at Risk) and ES (Expected Shortfall) are calculated based on the distribution.
Such information might be useful for risk management.
The results such as Table 3 could be applied to stress tests. For example, the resultslabelled as “initial regime 2” in Table 3 can be regarded as a case study when the regimechanges suddenly from regime 0 to regime 2. Since our model includes the dynamics ofvariables, we can evaluate the time-evolution of the cumulative loss after the regime change;
we can see the enlarging aggregate loss and the recovering process from the stressed regime
with diﬀerent risk horizons. Such kinds of dynamic stress tests will give new information
about risk management to financial institutions.
Samples of the time-evolution of the zero curves are shown in Figure 10 (default-free)and Figure 11 (CCC). The vertical axes are the zero rates, while the horizontal axes are the16times. In each figure, a zero curve at future time t1 ≥ 0 starts from t1 on the horizontal
axis, and the zero rate on t1 + 1 on the curve shows one-year zero rate at future t1 , andgenerally, the zero rate on t1 + n on the curve shows n-year zero rate at t1 . So, Figure 10 and
Figure 11 show the time-evolution of default-free and CCC zero curves on a certain futurescenario, respectively. You can see various kinds of curves (increasing ones, humped ones,
and decreasing ones), and various curves appear on a future scenario. If you use our model,
you can evaluate the time-evolving properties of your portfolio on time-evolving interest ratescenarios with various shapes.

5Concluding remarks

In this article we propose a pricing and risk evaluation model of interest rate risk and credit
risk with Markovian regime-switching property. First, we discuss the dynamics of regime,
interest rate and default intensity under the physical probability measure and the change ofmeasure to an equivalent martingale measure (pricing measure), and derive the dynamics
under the pricing measure. Second, we propose a simple model, and derive a method to
calculate the discount bond price, which is a slight extension of the existing papers suchas Elliott and Siu [7]. Based on the simple model, we show some numerical examples for
pricing, and clarify the eﬀects of regime-switching on pricing. Since the default-free interest
rate and the default intensity are dependent through regimes, the term structure of thezero rates depends on the regime dynamics. And we calculate the distribution of a bondportfolio’s price at the horizon, which reflects the actual default loss up to the horizon
and the decrease of market prices due to the transition to a stressed regime. Numericalexamples show that the price distribution has a short right tail and a long left tail, andthat the distribution depends strongly on the present regime. The former result is derived
from a general feature of regimes; the existence of a calm regime with low default risk andvolatility and a stressed regime with high default risk and volatility. The latter result seems
applicable to the financial risk management, especially on the stress tests. We think thatstochastic models with regime-switching property would be promising and useful for pricing
and financial risk management in future.
However, there remain necessary future works. First of all, it is desired to develop an
eﬃcient and stable parameter estimation method. Since we have not developed yet, we usedsome existing packages such as Matlab in order to analyse the time-series data, therefore,

17the stochastic processes are limited to the Vasicek model, and the number of regimes arelimited up to 3. Additionally, a stable calibration method to fit theoretical prices to market
prices is also necessary. Second, since we do not have enough knowledge about the regimeswitching properties on financial and economic data, more researches on various kinds ofdata are needed. For example, since the dependence between the financial variables might
be diﬀerent in diﬀerent regimes, such analyses would be fruitful. The dependence describedby a copula, which is frequently used in credit risk models, would be diﬀerent in eachregime, More knowledge about regime-switching properties will make models sophisticatedto be more useful for the financial risk management.

A

Derivation of (2.50)

Consider the model in Section 2.3. Under Assumption 2.2, we have
⟩
)
(⟨
dℓ(t)
f (t)⟩, X(t) + ⟨ℓ(t), QX(t)⟩ dt + ⟨ℓ(t), dM
dℓ(t) = d⟨ℓ(t), X(t)⟩ =
dt

(A.1)

under Pe. The term ⟨ℓ(t), QX(t)⟩ means the jump due to the regime-switching. It follows
from Assumption 2.2, (2.44)–(2.46) and (A.1) thatdeh(t) = dh(t) + dℓ(t) = ah (⟨mh , X(t)⟩ − h(t)) dt + ⟨σ h , X(t)⟩dzh (t) + d⟨ℓ(t), X(t)⟩()= ⟨ah mh , X(t)⟩ − ah (eh(t) − ⟨ℓ(t), X(t)⟩) dt + ⟨σ h , X(t)⟩ (dezh (t) + (⟨Lh , X(t)⟩ + ψh (t))dt)
(⟨
⟩
)
dℓ(t)
f (t)⟩+, X(t) + ⟨ℓ(t), QX(t)⟩ dt + ⟨ℓ(t), dM
dt
(⟨
⟩
)
dℓ(t)
e
=
ah mh + ah ℓ(t) + σ h ⊗ Lh + ψh (t)σ h +, X(t) + ⟨ℓ(t), QX(t)⟩ − ah h(t) dt
dt
f (t)⟩+ ⟨σ h , X(t)⟩dezh (t) + ⟨ℓ(t), dM()
f (t)⟩,= ah ⟨m′h (t), X(t)⟩ − eh(t) dt + ⟨σ h , X(t)⟩dezh (t) + ⟨ℓ, dM(A.2)
where A ⊗ B is the Hadamard product of K-dimensional vectors A and B, and)(1 dℓ(t)⊤′+ σ h ⊗ Lh + ψh (t)σ h + (⟨ℓ(t), Qe1 ⟩, · · · , ⟨ℓ(t), QeK ⟩) .
mh (t) = mh + ℓ(t) +
ah
dt
(A.3)
We define rt,u (r) ≡ r(u|r(t) = r), e
ht,u (eh) ≡ eh(u|eh(t) = eh) for 0 ≤ t ≤ u ≤ T , and
}]
}
]
[
{ ∫ T
[
{ ∫ T
e
e
e
e
e
(rt,u (r) + ht,u (h))du .
rt,u (r)du 1{τ >T } = Et exp −p(t, T, r, h, X) = Et exp −
t

t

(A.4)
18

It follows from (2.36) that∫ s
∫ srt,s (r) = r +(⟨ϕ(u), X(u)⟩ − art,u (r)) du +⟨σ, X(u)⟩dezr (u),
t

(A.5)

t

so that Dt,s = ∂rt,s (r)/∂r satisfies∫ sDt,s = 1 − aDt,u du.

(A.6)

t

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

Dt,t = 1.

(A.7)

Therefore, we obtainDt,s = e−a(s−t) .
From (A.4) and (A.8), we have)(∫ T∂
Dt,u du p(t, T, r, eh, X) = −B(t, T )p(t, T, r, eh, X),
p(t, T, r, eh, X) = −
∂r
t

(A.8)

(A.9)

and similarly, we have∂p(t, T, r, eh, X) = −Bh (t, T )p(t, T, r, eh, X).
∂ehDefine

{ ∫ t
}
e
e
e
pe(t, T, r, h, X) ≡ exp − (rt,u (r) + ht,u (h))du p(t, T, r, e
h, X)
0
}]
[
{ ∫ T
et exp −=E(rt,u (r) + e
ht,u (eh))du ,

(A.10)(A.11)

0which is a Doob’s martingale under Pe. Applying Ito’s lemma to pe(t, T, r, eh, X) and puttinge = (epp1 , · · · , peK )⊤ with pei = pe(t, T, r, eh, ei ), i = 1, · · · , K, we havepe(t, T, r, eh, X) = pe(0, T, r(0), eh(0), X(0))∫ t[)∂ pe ∂ pe∂ pe (++(⟨ϕ(u), X(u)⟩ − ar(u)) +⟨ah m′h (u), X(u)⟩ − ahe
h(u)
∂u ∂r
∂e
h
0
]
2
2
∂
p
e
∂
p
e
1
1
∂ 2 pe22du+ ⟨σ, X(u)⟩+ ⟨σ h , X(u)⟩+ ρ(t)⟨σ ⊗ σ h , X(u)⟩
2
∂r2 2
∂e
h2
∂r∂ e
h
∫ t
∫ t
∫ t
∂ pe
∂ pe
e
+
⟨σ, X(u)⟩dezr (u) +⟨σ h , X(u)⟩dezh (u) +⟨ep, QX(u)⟩du
h
0 ∂r
0 ∂e
0
⟩
∫ t⟨
∂ pe
f (u) .
e++⊗ ℓ(u), dM(A.12)p∂eh
019Because of the martingale property of pe, the integrand of du term must be zero, that is, wehave the following partial diﬀerential equation (hereafter, abbreviated by PDE))
∂ pe ∂ pe∂ pe (
0=+(⟨ϕ(t), X(t)⟩ − ar(t)) +⟨ah m′h (t), X(t)⟩ − ahe
h(t)
∂t ∂r
∂e
h
2
2
∂
p
e
∂
pe
1
1
∂ 2 pee+ ⟨σ, X(t)⟩2 2 + ⟨σ h , X(t)⟩2+ ρ(t)⟨σ ⊗ σ h , X(t)⟩+ ⟨ep, QX(t)⟩,
2
e
e
2
∂r
2
∂h
∂r∂ h
which implies that
)
∂p
∂p
∂p (
′
e
e
0=
⟨ah mh (t), X(t)⟩ − ah h(t)− rp − hp +(⟨ϕ(t), X(t)⟩ − ar(t)) +
∂t
∂r
∂e
h
2
2
1
1
∂ 2p
2∂ p
2∂ p
e
+ ⟨σ, X(t)⟩+ ⟨σ h , X(t)⟩+ ρ(t)⟨σ ⊗ σ h , X(t)⟩+ ⟨p, QX(t)⟩,
2
∂r2 2
∂e
h2
∂r∂ e
h
(A.13)
where p = (p1 , · · · , pK )⊤ with pi = p(t, T, r, eh, ei ), i = 1, · · · , K, with the terminal conditionp(T, T, r, eh, ·) = 1. (A.13) means a system of K coupled PDEs:
)
∂pi
∂pi (
∂pi
− rpi − ehpi +(ϕi (t) − ar(t)) +0=
ah m′h,i (t) − ahe
h(t)
∂t
∂r
∂e
h
2
2
2
1 2 ∂ pi 1 2 ∂ pi
∂ pi
e i (t)⟩,+ σi 2 + σh,i+ ρ(t)σi σh,i+ ⟨p, Qe0 ≤ t ≤ T,
2
e
e
2 ∂r
2
∂h
∂r∂ h
with pi (T, T, r, eh) = 1, i = 1, · · · , K. Here, assume that{}eepi (t, T, r, h) = exp Ai (t, T ) − B(t, T )r − Bh (t, T )h ,i = 1, · · · , K,

(A.14)

where Ai (t, T ) = A(t, T, ei ), then, Ai (t, T ) satisfies the following ODEs:][2
σi2 2 σh,i′2
Ai dAi− ϕi (t)B − ah mh,i (t)Bh + B +B + ρ(t)σi σh,i BBh
0=e
dt
2
2 h
+ ⟨A, Qei (t)⟩,(A.15)dB= aB − 1,dtdBh= ah Bh − 1.
dt
where A = (eA1 (t,T ) , · · · , eAK (t,T ) )⊤ . From (A.16) and (A.17), we have(A.17)

1 − e−a(T −t)a(A.18)

B(t, T ) =and Bh (t, T ) =1 − e−ah (T −t).
ah(A.16)Let Āi (t, T ) = eAi (t,T ) , i = 1, · · · , K, then (A.15) can be rewritten as
{ 2
2
σh,i
dĀi (t, T )
σi 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
}
K
∑
+ ρ(t)σi σh,i B(t, T )Bh (t, T ) Āi (t, T ) +Āj (t, T )eqji .
(A.19)j=1

20

In practice, we will solve numerically the following Kolmogorov’s backward equations:
{ 2
2
σh,i
dĀi (t, T )
σi 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
}
K
∑
+ ρ(t)σi σh,i B(t, T )Bh (t, T ) Āi (t, T ) +qeij Āj (t, T ), i = 1, · · · , K,
(A.20)
j=1

where15m′′h,i (t) = mh,i + ℓi (t) +

1
ah

(

K
∑
dℓi (t)+ σh,i λh,i + σh,i ψh (t) +qeij ℓjdt
j=1)(A.21)

with the terminal conditions Āi (T, T ) = 1, i = 1, · · · , K.
Acknowledgements
The author is grateful to Naoto Hatakeyama, Junpei Igarashi, Prof. Toshinao Yoshiba andProf. Kohta Takehara at Tokyo Metropolitan University for helpful discussions and various
kinds of data analyses, and is also grateful to Rongzhu Liu, Toshiaki Seiwa and Masayuki
Hamanaka at AFAS Corporation for giving me this interesting research theme and continuing
the research project.

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21

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[17] Kusuoka, S. (1999)，“A remark on default risk models,” Advances in Mathematical
Economics, 1, pp.69–82.
[18] Li, J. and S. Ma (2013), “Pricing options with credit risk in Markovian regime-switchingmarkets,” Journal of Applied Mathematics, 2013(2), pp.1–9.
[19] Monfort, A. and J.-P. Renne (2013), “Default, liquidity, and crises: an econometricframework,” Journal of Financial Econometrics, 11(2), pp.221–262.
[20] Wu, S. and Y. Zeng (2005), “A general equilibrium model of the term structure ofinterest rates under regime-switching risk,” International Journal of Theoretical andApplied Finance, 8 (7), pp.839–869.

22

Table 1: Parameters of default-free interest rate and default intensity processes under P .
regime k

mk

σk

mAAA
h,k

AAA
σh,k

mBBB
h,k

BBB
σh,k

mCCC
h,k

CCC
σh,k

0

0.33 %

0.46 %

0.14 %

0.18 %

0.34 %

0.40 %

1.70 %

2.43 %

1

2.73 %

1.08 %

0.28 %

0.33 %

0.66 %

0.79 %

3.24 %

3.90 %

2−1.13 %

1.51 %

0.48 %

2.29 %

1.58 %

3.79 %

6.98 %

16.10 %

23

Table 2: Term structure of market prices of risk for default-free interest rate.
t(0, 1)(1, 2)

ψr

0.385 0.143(2, 3)(3, 4)(4, 5)(5, 6)(6, 7)−0.253 −0.212 −0.226 −0.054 0.031

24(7, 8)(8, 9)(9, 10)0.116 0.038 −0.192

Table 3: Basic statistics of portfolio’s price distributions with diﬀerent initial regimes.
initial regime
0

1

2

average

14.57 14.29

14.04

standard deviation

0.33

0.40

0.56

50.00%

14.68 14.54

14.10

10.00%

14.07 13.86

13.26

5.00%

13.91 13.60

12.99

1.00%

13.43 13.15

12.38

0.50%

13.26 12.94

12.15

0.10%

12.83 12.50

11.65

percentile

VaR = average − 100(1 − α)-percentile
10.00%

0.49

0.53

0.78

5.00%

0.66

0.79

1.05

1.00%

1.14

1.24

1.66

0.50%

1.31

1.46

1.89

0.10%

1.73

1.90

2.39

25

default-free

AAA

BBB

CCC

14%
12%
10%
8%
6%
4%
2%
0%
0

1

2

3

4

5

6

7

8

9

10

Figure 1: Term structures of zero rates (initial regime 0).

26

default-free

AAA

BBB

CCC

30%

25%

20%

15%
10%

5%

0%
0

1

2

3

4

5

6

7

8

9

10

Figure 2: Term structures of zero rates (initial regime 2).

27

initial regime = 0

initial regime = 1

initial regime = 2

4

7

30.0%
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
0

1

2

3

5

6

8

9

10

Figure 3: CCC zero rate curves with diﬀerent initial regime.

28

initial regime = 0

initial regime = 1

initial regime = 2

1

4

7

3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
0

2

3

5

6

8

9

10

Figure 4: AAA zero rate curves with diﬀerent initial regime.

29

initial regime = 0

initial regime = 1

initial regime = 2

1

4

7

3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
0

2

3

5

6

8

9

10

Figure 5: AAA zero rate curves with diﬀerent initial regime (r(t) is not regime-switching).

30

initial regime = 0

initial regime = 1

initial regime = 2

4

7

0.4%
0.2%
0.0%
-0.2%
-0.4%
-0.6%
-0.8%
-1.0%
-1.2%
0

Figure

1

2

3

5

6:

Term structure of relativeM }/p(0, M, r(0), eh(0), X(0)) − 1 (AAA bond).

31

6

diﬀerence

8

9

10

v(0, M, r(0), X(0))Pe{τ>1,600initial regime = 0

1,400

initial regime = 1
1,200

initial regime = 2

1,000
800
600
400
200
0
0.45

0.50

0.55

0.60

0.65

0.70

Figure 7: Distributions of CCC bond price with diﬀerent initial regime after 1 year.

32

2,800initial regime = 0

2,400

initial regime = 1
2,000
initial regime = 2
1,600
1,200
800
400
0
12.00

12.50

13.00

13.50

14.00

14.50

15.00

15.50

Figure 8: Distributions of portfolio price with diﬀerent initial regime after 1 year.

33

2,000
1,800
1,600
1,400
1,200
1,000
800
600
400
200
0
12

12.5

13

13.5

14

14.5

15

15.5

Figure 9: Hybrid distribution of portfolio price after 1 year.

34

3.5%
3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
0

1

2

3

4

5

6

7

8

9

10

Figure 10: Evolution of default-free zero curve on a scenario.

35

25%

20%

15%

10%

5%

0%
0

1

2

3

4

5

6

7

8

9

10

Figure 11: Evolution of CCC zero curve on a scenario.

36