Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 621371, 9 pages
http://dx.doi.org/10.1155/2013/621371
Research Article
Pricing Options with Credit Risk in Markovian
Regime-Switching Markets
Jinzhi Li1 and Shixia Ma2
1
2
College of Sciences, Minzu University of China, Beijing 100081, China
College of Sciences, Hebei University of Technology, Tianjin 300401, China
Correspondence should be addressed to Jinzhi Li; lijinzhi118@gmail.com
Received 16 February 2013; Revised 16 May 2013; Accepted 20 May 2013
Academic Editor: Shan Zhao
Copyright © 2013 J. Li and S. Ma. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper investigates the valuation of European option with credit risk in a reduced form model when the stock price is driven by
the so-called Markov-modulated jump-diffusion process, in which the arrival rate of rare events and the volatility rate of stock are
controlled by a continuous-time Markov chain. We also assume that the interest rate and the default intensity follow the Vasicek
models whose parameters are governed by the same Markov chain. We study the pricing of European option and present numerical
illustrations.
1. Introduction
Pricing options with credit risk is an important topic in
finance from both theoretical and practical perspectives.
Credit risk refers to an investor’s risk that a borrower will
default on making payments as promised. There are basically
two kinds of models to describe the default: structural models
and reduced form models. The structural approach was firstly
introduced by Merton [1] who investigated European option
pricing for modeling single corporate default. The approach
is further extended by recent literature: see Ammann [2] and
Klein and Inglis [3]. Another tractable approach is called
reduced form model, which models the intensity of arrival of
default events directly. The reduced form models can be seen
in Artzner and Freddy [4], Duffie and Singleton [5], Duffie
and Gârleanu [6], and Leung and Kwok [7] and are developed
extensively by Su and Wang [8].
Recently, Markovian regime-switching models have
attracted attention among researchers and practitioners
in economics and mathematics. Elliott et al. [9] introduce
a self-calibrating model for short-term interest rate by
assuming that the short rate is governed by a finite state space
Markov process. Elliott et al. [10] and Elliott and Osakwe
[11] use Markov-modulated market parameters to capture
the time inhomogeneity generated by the financial market.
Elliott et al. [12] perform the valuation of option under
a generalized Markov-modulated jump-diffusion model.
Siu et al. [13] consider the pricing currency options under
two-factor Markov-modulated stochastic volatility models.
Bo et al. [14] derive the valuation of currency option when
the spot foreign exchange rates follow Markov-modulated
jump-diffusion model.
In this paper, we investigate the valuation of European
option with credit risk in a reduced form model in Markovian
regime-switching markets. We assume that the recovery rate
is constant; that is, when the writer of the option defaults,
a specified constant fraction times the payoff will be paid
at maturity. In order to incorporate both rare events and
time-inhomogeneity in finance market, we model the stock
price by the so-called Markov-modulated jump-diffusion
process, in which rare events are described as a compound
Poisson process and the arrival rate of Poisson process and
the volatility rate of stock are governed by a continuous-time
Markov chain. The states of Markov chain can be interpreted
as the states of the market. The transitions of the states of the
market may describe changes of economy, finance, business
cycles, and other conditions. In addition, we assume that the
interest rate and the default intensity both follow the Vasicek
models and the parameters of models are correlated with the
same Markov chain. By the method of changing measures,
2
Journal of Applied Mathematics
we obtain the closed form formula for the valuation of the
European option.
The rest of this paper is organized as follows. Section 2
presents the model description. In Section 3, by applying
Girsanov’s measure changing theorem, we derive the formula
of the pricing of European-style call option. We provide
numerical analysis in Section 4. The concluding remarks are
contained in Section 5.
2. The Model Description
Let (Ω, F, 𝑃) be a complete probability space, where 𝑃 is
a neutral-risk probability measure. Define 𝜉 = {𝜉𝑡 , 𝑡 ≥ 0}
on (Ω, F, 𝑃) as a continuous-time, finite state Markov chain
with 𝑛-state space 𝐸. We interpret the state of 𝜉 as the states
of the economy as follows (Elliott et al. [10] and Elliott and
Osakwe [11]). Without loss of generality, we take the state
space of 𝜉 to be a finite set of unite vectors {𝑒1 , 𝑒2 , . . . , 𝑒𝑛 } with
𝑒𝑖 = (0, . . . , 1, . . . , 0) ∈ 𝑅𝑛 . And 𝜉 has the following
semimartingale representation:
𝑑𝜉𝑡 = 𝑄𝜉𝑡 𝑑𝑡 + 𝑑𝑀𝑡 ,
(1)
where 𝑄 = (𝑞𝑖𝑗 )𝑖,𝑗=1,2,...,𝑛 is 𝑄-matrix of 𝜉 and 𝑀 = {𝑀𝑡 }0≤𝑡≤𝑇
is an 𝑅𝑛 -valued martingale with respect to the filtration
generated by {𝜉𝑡 , 0 ≤ 𝑡 ≤ 𝑇} under 𝑃. Suppose that the stock
price 𝑆𝑡 and interest rate 𝑟𝑡 satisfy the following stochastic
differential equations (SDE):
𝑑𝑆𝑡
= (𝑟𝑡 − 𝑘]𝑡 ) 𝑑𝑡 + 𝜎1𝑡 𝑑𝑊1𝑡 + (𝑒𝑌𝑡− − 1) 𝑑𝑁 (𝑡) ,
𝑆𝑡−
(2)
where 𝑊3𝑡 is standard Brownian motion and 𝜎3𝑡 is the
stochastic volatility of default intensity. 𝜎3𝑡 , 𝛼𝑡 , and 𝛽𝑡 are also
controlled by 𝜉𝑡 and satisfy
𝛼𝑡 = ⟨𝛼, 𝜉𝑡 ⟩,
𝛼 = (𝛼1 , 𝛼2 , . . . , 𝛼𝑛 ) ∈ (0, ∞)𝑛 ,
𝛽𝑡 = ⟨𝛽, 𝜉𝑡 ⟩,
𝛽 = (𝛽1 , 𝛽2 , . . . , 𝛽𝑛 ) ∈ (0, ∞)𝑛 .
1 𝜌12 𝜌13
(𝜌12 1 𝜌23 ) 𝑡.
𝜌13 𝜌23 1
3. Pricing Options with Credit Risk
We consider the case of a European call option with credit
risk. Assume that the recovery rate is a constant 𝑤. When the
seller of option defaults, the payoff is given by 𝑤 times the
payoff of the default-free option at maturity. Therefore, by the
risk neutral pricing theorem, the valuation of the European
call option at time 𝑡, with strike price 𝐾 and maturity 𝑇, is
given by
𝐶 (𝑡, 𝑇)
𝑇
= 𝐸 [𝑒− ∫𝑡
𝜎1 = (𝜎11 , 𝜎12 , . . . , 𝜎1𝑛 ) ∈ (0, ∞) ,
𝜎2𝑡 = ⟨𝜎2 , 𝜉𝑡 ⟩,
𝜎2 = (𝜎21 , 𝜎22 , . . . , 𝜎2𝑛 ) ∈ (0, ∞)𝑛 ,
𝑛
]𝑡 = ⟨], 𝜉𝑡 ⟩,
] = (]1 , ]2 , . . . , ]𝑛 ) ∈ (0, ∞) ,
𝑎𝑡 = ⟨𝑎, 𝜉𝑡 ⟩,
𝑎 = (𝑎1 , 𝑎2 , . . . , 𝑎𝑛 ) ∈ (0, ∞)𝑛 ,
𝑏𝑡 = ⟨𝑏, 𝜉𝑡 ⟩,
𝑏 = (𝑏1 , 𝑏2 , . . . , 𝑏𝑛 ) ∈ (0, ∞)𝑛 ,
+
(𝑤(𝑆𝑇 − 𝐾) 1(𝜏≤𝑇) + (𝑆𝑇 − 𝐾) 1(𝜏>𝑇) ) | F𝑡 ]
𝑟𝑠 𝑑𝑠
(𝑤(𝑆𝑇 −𝐾) +(1−𝑤) (𝑆𝑇 −𝐾) 1(𝜏>𝑇) ) | F𝑡 ] .
+
+
Following Lando [15],
𝑇
𝐸 [𝑒− ∫𝑡
𝑟𝑠 𝑑𝑠
+
(𝑆𝑇 − 𝐾) 1(𝜏>𝑇) | F𝑡 ]
𝑇
− ∫𝑡 (𝑟𝑠 +𝜆 𝑠 )𝑑𝑠
= 1(𝜏>𝑡) 𝐸 [𝑒
(8)
+
(𝑆𝑇 − 𝐾) | F𝑡 ] .
We can obtain the following expression:
𝑇
𝐶 (𝑡, 𝑇) = 𝑤𝐸 [𝑒− ∫𝑡
𝑟𝑠 𝑑𝑠
(3)
+
(𝑆𝑇 − 𝐾) | F𝑡 ]
𝑇
+ (1 − 𝑤) 1(𝜏>𝑡) 𝐸 [𝑒− ∫𝑡
(𝑟𝑠 +𝜆 𝑠 )𝑑𝑠
+
(𝑆𝑇 − 𝐾) | F𝑡 ]
= 𝐼1 + 𝐼2 .
where ⟨⋅, ⋅⟩ denotes the inner product in 𝑅𝑛 . Let 𝜏 denote the
default time of the writer of the option with default intensity
process 𝜆 𝑡 , and 𝜆 𝑡 is given by
𝑑𝜆 𝑡 = (𝛼𝑡 − 𝛽𝑡 𝜆 𝑡 ) 𝑑𝑡 + 𝜎3𝑡 𝑑𝑊3𝑡 ,
+
𝑟𝑠 𝑑𝑠
(7)
𝑛
𝜎1𝑡 = ⟨𝜎1 , 𝜉𝑡 ⟩,
(6)
The filtration F𝑡 is generated by F𝑡 = F𝑆𝑡 ∨F𝑟𝑡 ∨F𝜆𝑡 ∨F𝜉𝑇 ∨
H𝑡 , where F𝑆𝑡 = 𝜎(𝑆𝑠 , 0 ≤ 𝑠 ≤ 𝑡), F𝑟𝑡 = 𝜎(𝑟𝑠 , 0 ≤ 𝑠 ≤ 𝑡),
F𝜆𝑡 = 𝜎(𝜆 𝑠 , 0 ≤ 𝑠 ≤ 𝑡), F𝜉𝑡 = 𝜎(𝜉𝑠 , 0 ≤ 𝑠 ≤ 𝑡), and H𝑡 =
𝜎(1(𝜏≤𝑠) , 𝑠 ≤ 𝑡).
𝑇
where 𝑊1𝑡 , 𝑊2𝑡 are standard Brownian motions and 𝜎1𝑡 , 𝜎2𝑡
are the stochastic volatility of the stock and the interest rate,
respectively. {𝑁𝑡 }0≤𝑡≤𝑇 is Poisson process with the stochastic
jump intensity {]𝑡 }0≤𝑡≤𝑇 , and the jump amplitude is controlled
by {𝑌𝑡 }. 𝑌𝑠 and 𝑌𝑡 for 𝑠 ≠ 𝑡 independently identify distribution,
and write 𝑘 = 𝐸𝑒𝑌𝑖 − 1. Moreover, 𝑌𝑡 , 𝑁𝑡 are assumed to be
mutually independent. 𝜎1𝑡 , 𝜎2𝑡 , V𝑡 , 𝑎𝑡 , 𝑏𝑡 are controlled by 𝜉𝑡 ,
that is,
(5)
Moreover, we assume that 𝑁𝑡 , 𝑌𝑡 are independent of 𝑊1𝑡 , 𝑊2𝑡 ,
and 𝑊3𝑡 and the covariance matrix of the Brownian motion
(𝑊1𝑡 , 𝑊2𝑡 , 𝑊3𝑡 ) is
= 𝐸 [𝑒− ∫𝑡
𝑑𝑟𝑡 = (𝑎𝑡 − 𝑏𝑡 𝑟𝑡 ) 𝑑𝑡 + 𝜎2𝑡 𝑑𝑊2𝑡 ,
𝜎3 = (𝜎31 , 𝜎32 , . . . , 𝜎3𝑛 ) ∈ (0, ∞)𝑛 ,
𝜎3𝑡 = ⟨𝜎3 , 𝜉𝑡 ⟩,
(4)
(9)
Next, we calculate 𝐼1 and 𝐼2 , respectively. For 𝑠 > 𝑡, we have
𝑠
𝑠
𝑠
𝑠
𝑠
𝑟𝑠 = 𝑒− ∫𝑡 𝑏𝑢 𝑑𝑢 𝑟𝑡 + ∫ 𝑒− ∫V 𝑏𝑢 𝑑𝑢 𝑎V 𝑑V + ∫ 𝑒− ∫V 𝑏𝑢 𝑑𝑢 𝜎2V 𝑑𝑊2V .
𝑡
𝑡
(10)
Journal of Applied Mathematics
3
Integrated from 𝑡 to 𝑇 in both sides of (10),
Thus, we can define the probability measure 𝑄 by
𝑇
𝑇
𝑇
𝑡
𝑡
𝑠
𝑇
𝑇
𝑡
V
exp {− ∫𝑡 (𝑟𝑠 + 𝜆 𝑠 ) 𝑑𝑠}
𝑑𝑄 󵄨󵄨󵄨󵄨
󵄨󵄨 =
𝑑𝑃 󵄨󵄨F𝑡 𝐸 [exp {− ∫𝑇 (𝑟 + 𝜆 ) 𝑑𝑠} | F ]
𝑠
𝑠
𝑡
𝑡
𝑠
∫ 𝑟𝑠 𝑑𝑠 = ∫ 𝑒− ∫𝑡 𝑏𝑢 𝑑𝑢 𝑑𝑠𝑟𝑡 + ∫ 𝑎V 𝑑V ∫ 𝑒− ∫V 𝑏𝑢 𝑑𝑢 𝑑𝑠
𝑇
𝑇
𝑡
V
𝑠
− ∫V 𝑏𝑢 𝑑𝑢
+ ∫ 𝜎2V 𝑑𝑊2V ∫ 𝑒
𝑇 − ∫𝑠 𝑥𝑢 𝑑𝑢
𝑦
Let 𝑀(𝑥, 𝑦, 𝑇) = ∫𝑦 𝑒
(11)
1 𝑇 2 2
= exp {− ∫ 𝜎2𝑢
𝑀 (𝑏, 𝑢, 𝑇) 𝑑𝑢
2 𝑡
𝑑𝑠.
1 𝑇 2 2
− ∫ 𝜎3𝑢
𝑀 (𝛽, 𝑢, 𝑇) 𝑑𝑢}
2 𝑡
𝑇
𝑑𝑠; then
× exp {−𝜌23 ∫ 𝜎2𝑢 𝜎3𝑢 𝑀 (𝑏, 𝑢, 𝑇) 𝑀 (𝛽, 𝑢, 𝑇) 𝑑𝑢}
𝑡
𝑇
𝑇
𝑇
𝑡
𝑡
× exp {− ∫ 𝜎2V 𝑀 (𝑏, V, 𝑇) 𝑑𝑊2V
𝑡
∫ 𝑟𝑠 𝑑𝑠 = 𝑀 (𝑏, 𝑡, 𝑇) 𝑟𝑡 + ∫ 𝑎V 𝑀 (𝑏, V, 𝑇) 𝑑V
𝑇
𝑇
(12)
− ∫ 𝜎3V 𝑀 (𝛽, V, 𝑇) 𝑑𝑊3V } .
𝑡
+ ∫ 𝜎2V 𝑀 (𝑏, V, 𝑇) 𝑑𝑊2V .
(15)
𝑡
By Girsanov’s theorem,
Similarly,
𝑑𝑊2𝑡𝑄 = 𝑑𝑊2𝑡 + 𝜎2𝑡 𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡 + 𝜌23 𝜎3𝑡 𝑀 (𝛽, 𝑡, 𝑇) 𝑑𝑡,
𝑑𝑊3𝑡𝑄 = 𝑑𝑊3𝑡 + 𝜎3𝑡 𝑀 (𝛽, 𝑡, 𝑇) 𝑑𝑡 + 𝜌23 𝜎2𝑡 𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡,
𝑇
𝑇
𝑡
𝑡
𝑑𝑊1𝑡𝑄 = 𝑑𝑊1𝑡 + 𝜌12 𝑀1 (𝑡) 𝑑𝑡 + 𝜌13 𝑀2 (𝑡) 𝑑𝑡,
∫ 𝜆 𝑠 𝑑𝑠 = 𝑀 (𝛽, 𝑡, 𝑇) 𝜆 𝑡 + ∫ 𝛼V 𝑀 (𝛽, V, 𝑇) 𝑑V
𝑇
(13)
+ ∫ 𝜎3V 𝑀 (𝛽, V, 𝑇) 𝑑𝑊3V .
(16)
where
𝑡
𝑀1 (𝑡) = 𝜎2𝑡 𝑀 (𝑏, 𝑡, 𝑇) + 𝜌23 𝜎3𝑡 𝑀 (𝛽, 𝑡, 𝑇) ,
𝑀2 (𝑡) = 𝜎3𝑡 𝑀 (𝛽, 𝑡, 𝑇) + 𝜌23 𝜎2𝑡 𝑀 (𝑏, 𝑡, 𝑇) .
From (12) and (13), we have that
(17)
𝑊1𝑡𝑄, 𝑊2𝑡𝑄, and 𝑊3𝑡𝑄 are standard Brownian motions under
probability measure 𝑄, and (𝑊1𝑡𝑄, 𝑊2𝑡𝑄, 𝑊3𝑡𝑄) has the same
covariance matrix as (𝑊1𝑡 , 𝑊2𝑡 , 𝑊3𝑡 ). Therefore,
𝑇
𝑍 (𝑡, 𝑇) := 𝐸 [exp {− ∫ (𝑟𝑠 + 𝜆 𝑠 ) 𝑑𝑠} | F𝑡 ]
𝑡
𝑇
𝐸 [𝑒− ∫𝑡
= exp {−𝑀 (𝑏, 𝑡, 𝑇) 𝑟𝑡 − 𝑀 (𝛽, 𝑡, 𝑇) 𝜆 𝑡
(𝑟𝑠 +𝜆 𝑠 )𝑑𝑠
+
(𝑆𝑇 − 𝐾) | F𝑡 ]
+
𝑄
(18)
= 𝑍 (𝑡, 𝑇) 𝐸 ((𝑆𝑇 − 𝐾) | F𝑡 ) .
𝑇
− ∫ 𝑎𝑢 𝑀 (𝑏, 𝑢, 𝑇) 𝑑𝑢
𝑡
In addition,
𝑇
+
𝐸𝑄 ((𝑆𝑇 − 𝐾) | F𝑡 )
− ∫ 𝛼𝑢 𝑀 (𝛽, 𝑢, 𝑇) 𝑑𝑢}
𝑡
= 𝐸𝑄 (𝑆𝑇 1(𝑆𝑇 ≥𝐾) | F𝑡 ) − 𝐾𝐸𝑄 (1(𝑆𝑇 ≥𝐾) | F𝑡 ) .
1 𝑇 2 2
𝑀 (𝑏, 𝑢, 𝑇) 𝑑𝑢
× exp { ∫ 𝜎2𝑢
2 𝑡
By the solution of SDE (2),
1 𝑇 2 2
+ ∫ 𝜎3𝑢
𝑀 (𝛽, 𝑢, 𝑇) 𝑑𝑢}
2 𝑡
𝑇
1 2
𝑆𝑇 = 𝑆𝑡 exp {∫ (𝑟𝑠 − 𝑘]𝑠 − 𝜎1𝑠
) 𝑑𝑠
2
𝑡
𝑇
× exp {𝜌23 ∫ 𝜎2𝑢 𝜎3𝑢 𝑀 (𝑏, 𝑢, 𝑇) 𝑀 (𝛽, 𝑢, 𝑇) 𝑑𝑢} .
𝑡
(19)
(14)
𝑇
𝑇
𝑡
𝑡
+ ∫ 𝜎1𝑠 𝑑𝑊1𝑠 + ∫ 𝑌𝑠− 𝑑𝑁𝑠 } .
(20)
4
Journal of Applied Mathematics
Under 𝑄,
𝐸(⋅) is the expectation of 𝑌𝑗 under 𝑃, and 𝑁(⋅) is distribution
function of standard normal distribution. On the other hand,
let
𝑇
𝑆𝑇 = 𝑆𝑡 exp {Λ (𝑡, 𝑇) + ∫ 𝜎1𝑠 𝑑𝑊1𝑠𝑄
𝑡
+∫
𝑇
𝑡
𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑄
(21)
𝑇
+ ∫ 𝑌𝑠− 𝑑𝑁𝑠 } ,
L (𝑡, 𝑇)
:= 𝐸𝑄 (𝑆𝑇 | F𝑡 )
𝑡
1 𝑇 2
1 𝑇 2 2
𝑑𝑠 + ∫ 𝜎2𝑠
𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠}
= 𝑆𝑡 𝑒Λ(𝑡,𝑇) exp { ∫ 𝜎1𝑠
2 𝑡
2 𝑡
where
1 2
Λ (𝑡, 𝑇) = − ∫ (𝑘]𝑠 + 𝜎1𝑠
) 𝑑𝑠 + 𝑀 (𝑏, 𝑡, 𝑇) 𝑟𝑡
2
𝑡
𝑇
𝑇
𝑇
𝑡
𝑡
× exp {𝜌12 ∫ 𝜎1𝑠 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠 + 𝑘 ∫ ]𝑠 𝑑𝑠} .
𝑇
𝑇
2
+ ∫ 𝑎𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠 − ∫ 𝜎2𝑠
𝑀2 (𝑏, 𝑠, 𝑇) 𝑑𝑠
𝑡
𝑡
𝑇
𝑇
𝑡
𝑡
− 𝜌12 ∫ 𝜎1𝑠 𝑀1 (𝑠) 𝑑𝑠 − 𝜌13 ∫ 𝜎1𝑠 𝑀2 (𝑠) 𝑑𝑠
𝑇
− 𝜌23 ∫ 𝜎2𝑠 𝜎3𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑀 (𝛽, 𝑠, 𝑇) 𝑑𝑠.
𝑡
(22)
We can define the probability measure 𝑅 as follows:
𝑆𝑇
𝑑𝑅 󵄨󵄨󵄨󵄨
󵄨 =
𝑑𝑄 󵄨󵄨󵄨F𝑡 𝐸𝑄 (𝑆𝑇 | F𝑡 )
𝑇
𝑇
𝑡
𝑡
= exp {∫ 𝜎1𝑠 𝑑𝑊1𝑠𝑄 + ∫ 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑄
1 𝑇 2
− ∫ 𝜎1𝑠
𝑑𝑠}
2 𝑡
So,
𝑃 (𝑆𝑇 ≥ 𝐾 | F𝑡 )
𝑇
𝑇
𝑡
𝑡
= 𝑃𝑄 (∫ 𝜎1𝑠 𝑑𝑊1𝑠𝑄 + ∫ 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑄
𝑇
−𝜌12 ∫ 𝜎1𝑠 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠}
𝑡
𝑇
≥ ln (𝐾/𝑆𝑡 ) − Λ (𝑡, 𝑇) − ∫ 𝑌𝑠 𝑑𝑁𝑠 )
∫𝑡 𝜎1𝑠 𝑑𝑊1𝑠𝑄
+
𝑇
∫𝑡 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑄
𝑇
𝑇
𝑡
𝑡
× exp {∫ 𝑌𝑠 𝑑𝑁𝑠 − 𝑘 ∫ ]𝑠 𝑑𝑠} .
𝑡
𝑇
(26)
1 𝑇 2 2
𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠
× exp {− ∫ 𝜎2𝑠
2 𝑡
𝑄
= 𝑃𝑄 (
(25)
(23)
√Δ
By Girsanov’s theorem, we have that
𝑇
≤
𝑇
∞
=∑
ln (𝑆𝑡 /𝐾) + Λ (𝑡, 𝑇) + ∫𝑡 𝑌𝑠 𝑑𝑁𝑠
(∫𝑡 ]𝑠 𝑑𝑠)
𝑛=0
𝑛!
√Δ
𝑑𝑊1𝑡𝑅 = 𝑑𝑊1𝑡𝑄 − 𝜎1𝑡 𝑑𝑡 − 𝜌12 𝜎2𝑡 𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡,
)
𝑑𝑊2𝑡𝑅 = 𝑑𝑊2𝑡𝑄 − 𝜎2𝑡 𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡 − 𝜌12 𝜎1𝑡 𝑑𝑡,
𝑛
𝑇
𝑒− ∫𝑡
]𝑠 𝑑𝑠
𝐸 (𝑁 (𝑑1 )) ,
where, under 𝑅, 𝑊1𝑡𝑅 , 𝑊2𝑡𝑅 are standard Brownian motions
and their correlation coefficient is 𝜌12 , 𝑁𝑡 is Poisson process
with intensity (𝑘 + 1)]𝑡 , and the density function of 𝑌1 is
𝑒𝑦 𝑓(𝑦)/(𝑘 + 1), where 𝑓(⋅) is the density function of 𝑌1 under
𝑃. Since, under 𝑅,
where
𝑇
𝑇
2
2
Δ = ∫ 𝜎1𝑠
𝑑𝑠 + ∫ 𝜎2𝑠
𝑀2 (𝑏, 𝑠, 𝑇) 𝑑𝑠
𝑡
𝑡
𝑇
𝑇
+ 2𝜌12 ∫ 𝜎1𝑠 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠,
(24)
𝑡
𝑑1 =
ln (𝑆𝑡 /𝐾) + Λ (𝑡, 𝑇) + ∑𝑛𝑗=1 𝑌𝑗
√Δ
(27)
.
𝑆𝑇 = 𝑆𝑡 exp {Λ (𝑡, 𝑇) + Δ + ∫ 𝜎1𝑠 𝑑𝑊1𝑠𝑅
𝑡
𝑇
𝑇
𝑡
𝑡
+ ∫ 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑅 + ∫ 𝑌𝑠− 𝑑𝑁𝑠 } ,
(28)
Journal of Applied Mathematics
5
Denote by 𝑃(𝑡, 𝑇) the value at time 𝑡 of a 𝑇 maturity zero
coupon bond whose face value is 1. Then
then
𝐸𝑄 (𝑆𝑇 1(𝑆𝑇 ≥𝐾) | F𝑡 )
𝑇
𝑃 (𝑡, 𝑇) = 𝐸 [𝑒− ∫𝑡
= L (𝑡, 𝑇) 𝑃𝑅 (𝑆𝑇 ≥ 𝐾 | F𝑡 )
𝑇
𝑇
𝑡
𝑡
= L (𝑡, 𝑇) 𝑃𝑅 (∫ 𝜎1𝑠 𝑑𝑊1𝑠𝑅 + ∫ 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑅
≥ ln (
𝑅
= L (𝑡, 𝑇) 𝑃 (
∫𝑡 𝜎1𝑠 𝑑𝑊1𝑠𝑅
+
𝑇
𝑡
𝑇
∫𝑡 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑅
1
2
𝑀2 (𝑏, V, 𝑇) 𝑑V} ,
+ ∫ 𝜎2V
2 𝑡
ln (𝑆𝑡 /𝐾) + Λ (𝑡, 𝑇) + Δ + ∫𝑡 𝑌𝑠 𝑑𝑁𝑠
𝑇
∞
= L (𝑡, 𝑇) ∑
(∫𝑡 ]𝑠 𝑑𝑠)
𝑛!
𝑛=0
(34)
𝑇
√Δ
√Δ
(33)
𝑃 (𝑡, 𝑇) = exp {−𝑀 (𝑏, 𝑡, 𝑇) 𝑟𝑡 − ∫ 𝑎V 𝑀 (𝑏, V, 𝑇) 𝑑V
𝑑𝑃 (𝑡, 𝑇)
= 𝑟𝑡 𝑑𝑡 − 𝑀 (𝑏, 𝑡, 𝑇) 𝜎2𝑡 𝑑𝑊2𝑡 .
𝑃 (𝑡, 𝑇)
𝑇
≤
| F𝑡 ] .
From (12),
𝑇
𝐾
) − Λ (𝑡, 𝑇) − Δ − ∫ 𝑌𝑠 𝑑𝑁𝑠 )
𝑆𝑡
𝑡
𝑇
𝑟𝑠 𝑑𝑠
)
(35)
Define the Radon-Nikodym derivative given by
𝑛
𝑇
− ∫𝑡 ]𝑠 𝑑𝑠
𝑒
𝐸𝑅 (𝑁 (𝑑2 )) ,
𝑇
𝑑𝑄𝑇
𝑒− ∫𝑡 𝑟𝑠 𝑑𝑠
=
𝑇
𝑑𝑃
𝐸 [𝑒− ∫𝑡 𝑟𝑠 𝑑𝑠 | F𝑡 ]
(29)
where
𝑇
𝑑2 =
ln (𝑆𝑡 /𝐾) + Λ (𝑡, 𝑇) + Δ +
∑𝑛𝑗=1
𝑌𝑗
√Δ
𝑡
1 𝑇 2 2
− ∫ 𝜎2V
𝑀 (𝑏, V, 𝑇) 𝑑V} ,
2 𝑡
(30)
= 𝑑1 + √Δ
and 𝐸𝑅 (⋅) is the expectation of 𝑌𝑗 under 𝑅. By (23) and (29),
we have the following lemma.
and by Girsanov’s theorem, under 𝑄𝑇 ,
𝑇
𝑑𝑊2𝑡𝑄 = 𝑑𝑊2𝑡 + 𝜎2𝑡 𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡
Lemma 1. Consider the Following:
𝑇
− ∫𝑡 (𝑟𝑠 +𝜆 𝑠 )𝑑𝑠
𝐸 [𝑒
𝑇
𝑑𝑊1𝑡𝑄
+
(𝑆𝑇 − 𝐾) | F𝑡 ]
×∑
(37)
= 𝑑𝑊1𝑡 + 𝜌12 𝜎2𝑡 𝑀 (𝑏, 𝑡, 𝑇) 𝑑𝑡.
Thus 𝑆𝑇 can be rewritten as
= 𝑍 (𝑡, 𝑇)
∞
(36)
= exp {− ∫ 𝑀 (𝑏, V, 𝑇) 𝜎2V 𝑑𝑊2V
𝑇
(∫𝑡 ]𝑠 𝑑𝑠)
𝑛
𝑇
𝑛!
𝑛=0
𝑡
𝑇
− ∫𝑡 ]𝑠
×𝑒
𝑇
𝑄
𝑆𝑇 = 𝑆𝑡 exp {A (𝑡, 𝑇) + ∫ 𝑀 (𝑏, V, 𝑇) 𝜎2V 𝑑𝑊2V
𝑑𝑠
𝑇
𝑟𝑠 𝑑𝑠
𝑟𝑠 𝑑𝑠
𝑆𝑇 1(𝑆𝑇 ≥𝐾) | F𝑡 ]
𝑇
− ∫𝑡 𝑟𝑠 𝑑𝑠
− 𝐾𝐸 [𝑒
= 𝐼3 − 𝐼4 .
1(𝑆𝑇 ≥𝐾) | F𝑡 ]
(38)
𝑡
𝑡
𝑇
1 2
A (𝑡, 𝑇) = − ∫ (𝑘]𝑠 + 𝜎1𝑠
) 𝑑𝑠 + 𝑀 (𝑏, 𝑡, 𝑇) 𝑟𝑡
2
𝑡
+
𝑇
𝑇
where
(𝑆𝑇 − 𝐾) | F𝑡 ]
= 𝐸 [𝑒− ∫𝑡
𝑇
𝑄
+ ∫ 𝜎1V 𝑑𝑊1V
+ ∫ 𝑌𝑠− 𝑑𝑁𝑠 } ,
[L (𝑡, 𝑇) 𝐸𝑅 (𝑁 (𝑑2 )) − 𝐾𝐸 (𝑁 (𝑑1 ))] .
(31)
In the following; in order to calculate 𝐼1 , we write
𝐸 [𝑒− ∫𝑡
𝑇
(32)
𝑇
𝑇
𝑡
𝑡
2
+ ∫ 𝑎𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠 − ∫ 𝜎2𝑠
𝑀2 (𝑏, 𝑠, 𝑇) 𝑑𝑠
𝑇
− 𝜌12 ∫ 𝜎1𝑠 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑠.
𝑡
(39)
6
Journal of Applied Mathematics
and, under 𝑄𝑆 , 𝑁𝑡 is Poisson process with intensity (𝑘 + 1)]𝑡 ,
and the density function of 𝑌1 is 𝑒𝑦 𝑓(𝑦)/(𝑘 + 1), where 𝑓(⋅) is
the density function of 𝑌1 under 𝑃. Since, under 𝑄𝑆 ,
Then
𝑇
𝑃𝑄 (𝑆𝑇 ≥ 𝐾 | F𝑡 )
𝑇
𝑇
𝑇
𝑇
𝑇
= 𝑃𝑄 (∫ 𝜎1𝑠 𝑑𝑊1𝑠𝑄 + ∫ 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑄
𝑡
𝑇
𝑡
𝑇
𝐾
≥ ln ( ) − A (𝑡, 𝑇) − ∫ 𝑌𝑠− 𝑑𝑁𝑠 )
𝑆𝑡
𝑡
𝑇
𝑄𝑇
=𝑃
(
𝑇
∫𝑡 𝜎1𝑠 𝑑𝑊1𝑠𝑄
+
+∫
√Δ
∞
=∑
𝑛=0
√Δ
𝑇
𝑆
)
𝑒
𝑛!
𝑇
𝑆
𝐸 (𝑁 (𝑑3 )) ,
ln (𝑆𝑡 /𝐾) + A (𝑡, 𝑇) +
𝑌𝑗
.
=𝑃
(41)
𝑄𝑇
(
∞
𝑇
= 𝐾𝑃 (𝑡, 𝑇) ∑
ln (𝑆𝑡 /𝐾) + A (𝑡, 𝑇) + Δ + ∫𝑡 𝑌𝑠− 𝑑𝑁𝑠
√Δ
(1(𝑆𝑇 ≥𝐾) | F𝑡 )
(∫𝑡 ]𝑠 𝑑𝑠)
𝑛!
𝑛=0
𝑆
√Δ
≤
𝑇
𝑇
𝑆
∫𝑡 𝜎1𝑠 𝑑𝑊1𝑠𝑄 + ∫𝑡 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑄
Therefore,
𝐼4 = 𝐾𝑃 (𝑡, 𝑇) 𝐸
𝑇
𝐾
) − A (𝑡, 𝑇) − Δ − ∫ 𝑌𝑠− 𝑑𝑁𝑠 )
𝑆𝑡
𝑡
𝑇
∑𝑛𝑗=1
√Δ
𝑄𝑇
𝑆
𝑡
≥ ln (
where 𝐸(⋅) is the expectation of 𝑌𝑗 under 𝑃,
𝑑3 =
𝑇
𝑆
= 𝑃𝑄 (∫ 𝜎1𝑠 𝑑𝑊1𝑠𝑄 + ∫ 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑄
𝑡
𝑇
𝑡
𝑃𝑄 (𝑆𝑇 ≥ 𝐾 | F𝑡 )
𝑛
− ∫𝑡 ]𝑠 𝑑𝑠
+ ∫ 𝑌𝑠− 𝑑𝑁𝑠 } ,
then
ln (𝑆𝑡 /𝐾) + A (𝑡, 𝑇) + ∫𝑡 𝑌𝑠− 𝑑𝑁𝑠
(∫𝑡 ]𝑠 𝑑𝑠)
(46)
𝑇
𝑄𝑆
𝜎1V 𝑑𝑊1V
(40)
𝑇
≤
𝑇
𝑡
𝑇
𝑇
∫𝑡 𝜎2𝑠 𝑀 (𝑏, 𝑠, 𝑇) 𝑑𝑊2𝑠𝑄
𝑆
𝑄
𝑆𝑇 = 𝑆𝑡 exp {A (𝑡, 𝑇) + Δ + ∫ 𝑀 (𝑏, V, 𝑇) 𝜎2V 𝑑𝑊2V
𝑡
𝑛
𝑇
− ∫𝑡 ]𝑠 𝑑𝑠
𝑒
𝑇
(∫𝑡 ]𝑠 𝑑𝑠)
∞
(42)
=∑
𝐸 (𝑁 (𝑑3 )) .
𝑛!
𝑛=0
)
𝑛
𝑇
𝑒− ∫𝑡
]𝑠 𝑑𝑠 𝑆
𝐸 (𝑁 (𝑑4 )) ,
(47)
Finally, let
𝑇
𝐴 (𝑡, 𝑇) := 𝐸 (𝑒− ∫𝑡
𝑟𝑠 𝑑𝑠
where 𝐸𝑆 (⋅) is the expectation of 𝑌𝑗 under 𝑄𝑆 ,
𝑆𝑇 | F𝑡 )
𝑇
1 2
= 𝑆𝑡 exp {− ∫ (𝑘]𝑠 + 𝜎1𝑠
) 𝑑𝑠
2
𝑡
𝑇
𝑑4 =
(43)
𝑇
1
2
𝑑𝑠 + 𝑘 ∫ ]𝑠 𝑑𝑠} .
+ ∫ 𝜎1𝑠
2 𝑡
𝑡
ln (𝑆𝑡 /𝐾) + A (𝑡, 𝑇) + Δ + ∑𝑛𝑗=1 𝑌𝑗
√Δ
(48)
= 𝑑3 + √Δ.
From (42) and (33), we can conclude the following lemma.
We define
𝑇
Lemma 2. Consider the following:
𝑒− ∫𝑡 𝑟𝑠 𝑑𝑠 𝑆𝑇
𝑑𝑄𝑆
=
𝑇
𝑑𝑃
𝐸 [𝑒− ∫𝑡 𝑟𝑠 𝑑𝑠 𝑆𝑇 | F𝑡 ]
𝑇
= exp {∫ 𝜎1V 𝑑𝑊1V −
𝑡
𝑇
1 𝑇 2
∫ 𝜎 𝑑V
2 𝑡 2V
𝑇
𝐸 [𝑒− ∫𝑡
(44)
𝑇
=∑
𝑛=0
+ ∫ 𝑌𝑠− 𝑑𝑁 (𝑠) − 𝑘 ∫ ]𝑠 𝑑𝑠} ,
𝑡
+
(𝑆𝑇 − 𝐾) | F𝑡 ]
(∫𝑡 ]𝑠 𝑑𝑠)
∞
𝑇
𝑡
𝑟𝑠 𝑑𝑠
𝑛!
𝑇
× 𝑒− ∫𝑡
𝑆
and by Girsanov’s theorem, under 𝑄 ,
𝑛
]𝑠 𝑑𝑠
[𝐴 (𝑡, 𝑇) 𝐸𝑆 (𝑁 (𝑑4 ))−𝐾𝑃 (𝑡, 𝑇) 𝐸 (𝑁 (𝑑3 ))] .
(49)
𝑆
𝑑𝑊1𝑡𝑄 = 𝑑𝑊1𝑡 − 𝜎1𝑡 𝑑𝑡
𝑆
𝑑𝑊2𝑡𝑄
= 𝑑𝑊2𝑡 − 𝜌12 𝜎1𝑡 𝑑𝑡,
(45)
Combined with the previous lemmas, the price of European call
option at time 𝑡 is given by the following theorem.
Journal of Applied Mathematics
7
28
Table 1: The parameter values.
26
Value in state 𝑒1 Value in state 𝑒2
Volatility of 𝑆
Jump intensity
Speed of reversion of 𝑟
Long-term average of 𝑟
Volatility of 𝑟
Speed of reversion of 𝜆
Long-term average of 𝜆
Volatility of 𝜆
Initial stock price
𝜎11 = 0.2
]1 = 15
𝑏1 = 2
𝑎1 /𝑏1 = 0.04
𝜎21 = 0.15
𝛽1 = 1.5
𝛼1 /𝛽1 = 0.01
𝜎31 = 0.25
𝑆0 = 100
Initial interest rate
𝑟0 = 0.04
Initial default intensity
𝜆 0 = 0.5
𝜎12 = 0.4
]2 = 30
𝑏2 = 1
𝑎2 /𝑏2 = 0.02
𝜎22 = 0.3
𝛽2 = 2
𝛼2 𝛽2 = 0.02
𝜎32 = 0.45
24
22
Option price
Parameter name
16
14
12
10
0.8
0.9
0.95
𝜔 = 0.4
𝜔 = 0.6
𝜎𝐽 = 0.1
1
1.05
1.1
1.15
1.2
1.25
𝜔 = 0.8
𝜔=1
Figure 1: The option price with different recovery rate for 𝑇 = 1,
𝜌12 = 0.7, 𝜌13 = 0.5, and 𝜌23 = 0.6.
𝐶 (𝑡, 𝑇)
30
𝑇
(∫𝑡 ]𝑠 𝑑𝑠)
𝑛
25
𝑛!
𝑇
× 𝑒− ∫𝑡
]𝑠 𝑑𝑠
[𝑤 (𝐴 (𝑡, 𝑇) 𝐸𝑆 (𝑁 (𝑑4 ))
− 𝐾𝑃 (𝑡, 𝑇) 𝐸 (𝑁 (𝑑3 )))
(50)
+ (1 − 𝑤) 1(𝜏>𝑡) 𝑍 (𝑡, 𝑇)
Option price
𝑛=0
0.85
𝑆0 /𝐾
Theorem 3. The price of European call option with credit risk
at time 𝑡 is
∞
18
𝜇𝐽 = 0
Mean jump size
Standard deviation of jump size
=∑
20
20
15
10
𝑅
× (L (𝑡, 𝑇) 𝐸 (𝑁 (𝑑2 ))
5
0.8
− 𝐾𝐸 (𝑁 (𝑑1 )))] .
4. Numerical Analysis
In this section, we will employ Monte Carlo simulation and
perform a numerical analysis for European-style call option
with credit risk under the Markov-modulated jump-diffusion
model. We consider that the Markov chain 𝜉 has two states,
which means that macroeconomic shifts between the two
states: 𝑒1 (“boom” or “good”) and 𝑒2 (“recession” or “bad”).
We assume that the current economy is in boom and that the
transition probability matrix of the two-state Markov chain 𝜉
is given by
(
0.7 0.3
𝑝11 𝑝12
)=(
).
𝑝21 𝑝22
0.2 0.8
(51)
Assume that 𝑌𝑡 is normally distributed with mean 𝜇𝐽 and
standard deviation 𝜎𝐽 . We will adopt the specimen values
for the model parameters as Table 1. We consider a range of
spot-to-strike ratios 𝑆0 /𝐾 from 0.8 to 1.2 and assume that one
year has 252 trading days. We perform 10 000 simulations for
computing the option price.
0.85
0.9
0.95
1
1.05
𝑆0 /𝐾
1.1
1.15
1.2
1.25
𝑇 = 0.5
𝑇=1
𝑇 = 1.5
Figure 2: The option price with different maturity 𝑇 for 𝜔 = 0.4,
𝜌12 = 0.7, 𝜌13 = 0.5, and 𝜌23 = 0.6.
For each 𝜔 = 0.4, 0.6, 0.8, 1, we consider the impact of
the spot-to-strike ratio on the option price. From Figure 1, we
observe that the option price increases as the spot-to-strike
ratio increases. We can also see that the greater the 𝜔, the
greater the option price. When 𝜔 = 1, it follows that there
is not default risk. It is a result of the fact that the payoff
at the maturity will increase as the recovery rate increases.
Figure 2 depicts the plot of the option price against the spotto-strike ratio for each maturity 𝑇 = 0.5, 1, 1.5. From these
plots, we find that the longer the maturities, the greater the
option price.
In the following, we compare the option price with
different correlation coefficients for fixed maturity 𝑇 = 1
8
Journal of Applied Mathematics
24
25
22
20
18
Option price
Option price
20
16
14
15
12
10
8
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
10
0.8
1.25
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
𝑆0 /𝐾
𝑆0 /𝐾
1 = 10, 2 = 20
1 = 15, 2 = 30
1 = 20, 2 = 40
𝜌12 = 0.7
𝜌12 = −0.7
𝜌12 = −0.9
Figure 3: The option price with different correlation coefficients 𝜌12
for 𝑇 = 1, 𝜔 = 0.4, 𝜌13 = 0.5, and 𝜌23 = 0.6.
Figure 5: The option price with different ] for 𝜔 = 0.4, 𝜌12 = 0.7,
𝜌13 = 0.5, and 𝜌23 = 0.6.
35
25
30
25
Option price
Option price
20
20
15
15
10
10
0.8
0.85
0.9
0.95
1
1.05
𝑆0 /𝐾
1.1
1.15
1.2
1.25
𝜌13 = 1
𝜌13 = 0.5
𝜌13 = −0.5
Figure 4: The option price with different correlation coefficients 𝜌13
for 𝑇 = 1, 𝜔 = 0.4, 𝜌12 = 0.7, and 𝜌23 = 0.6.
and recovery rate 𝜔 = 0.4. Figure 3 illustrates that option
price increases as correlation coefficient 𝜌12 increases. From
Figure 4, we can also see that option price decreases as
correlation coefficient 𝜌13 increases.
In Figure 5, we present how the option prices vary with
the changes of the annual jump intensity ]. Figure 5 displays
a large change in the option price due to the variation of the
jump intensity. The option price increases as jump intensity
increases.
5
0.8
0.85
0.9
0.95
1
1.05
𝑆0 /𝐾
1.1
1.15
1.2
1.25
Markov-modulated model
Non-Markov-modulated model for a good economy
Non-Markov-modulated model for a bad economy
Figure 6: The option price in Markov-modulated model and nonMarkov-modulated models.
Finally, we investigate the difference of the option price
in Markov-modulated model and non-Markov-modulated
models. From Figure 6, we can observe that the call option
price in good economy is lower than that in bad economy. The
reason is that when economy is bad, the volatility of the risky
asset is great so that the option price is higher. In our model,
we assume that the current economy is good, so Markovmodulated model is close to the model for a good economy,
while the two plots and the model for a bad economy are
relatively far apart. In Markov-modulated model, we take into
Journal of Applied Mathematics
consideration the changes of the state of the economy, so
Markov-modulated model is more in accord with the reality
for the pricing of the defaultable options.
5. Conclusion
The pricing of European option with credit risk in a reduced
form model was studied, while the stock price was driven by
Markov-modulated jump-diffusion models. The interest rate
and the default intensity followed the Vasicek models, and the
parameters were also controlled by the same Markov chain.
Compared with most of the credit risk models, the main
advantage is that we incorporated Markov-modulated rates
into the models. We applied Girsanov’s theorem to obtain the
equivalent martingale measure and derived the closed form
formula for the valuation of the European option. Finally,
from the numerical illustrations, we obtain the effects of the
recovery rate, the maturity, the correlation coefficients, and
the jump intensity of stock on the option price.
Acknowledgments
The authors would like to thank the referee for many helpful
and valuable comments and suggestions. Jinzhi Li would like
to acknowledge the National Commission of Ethnic Affair
(no. 12ZYZ003) and the Ministry of Education of Humanities
and Social Sciences Project (no. 11YJC790102).
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