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A first-passage-time model under regime-switching market environment
Journal of Banking & Finance 32 (2008) 2617–2627
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A first-passage-time model under regime-switching market environment Mi Ae Kim a, Bong-Gyu Jang b,*, Ho-Seok Lee c a b c
Risk Management Department, Woori Bank, Seoul, Republic of Korea Department of Industrial and Management Engineering, POSTECH, Pohang, Republic of Korea Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea
a r t i c l e
i n f o
Article history: Received 11 February 2008 Accepted 28 May 2008 Available online 6 June 2008 JEL classification: G12 G13 G32 Keywords: First-passage-time model Regime-switching model Default probability Default correlation Credit default swap
a b s t r a c t In this paper, we suggest a first-passage-time model which can explain default probability and default correlation dynamics under stochastic market environment. We add a Markov regime-switching market condition to the first-passage-time model of Zhou [Zhou, C., 2001. An analysis of default correlations and multiple defaults. Review of Financial Studies 14, 555–576]. Using this model, we try to explain various relationship between default probability, default correlation, and market condition. We also suggest a valuation method for credit default swap (CDS) with (or without) counterparty default risk (CDR) and basket default swap under this model. Our numerical results provide us with several meaningful implications. First, default swap spread is higher in economic recession than in economic expansion across default swap maturity. Second, as the difference of asset return volatility between under bear market and under bull market increases, CDS spread increases regardless of maturity. Third, the bigger the intensity shifting from bull market to bear market, the higher the spread for both CDS without CDR and basket default swap. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Recently, correlation products such as basket default swap (BDS) and synthetic collateralized debt obligation (CDO) are actively traded and hence modelling default risk and default correlation has become an important issue in pricing correlation products and managing credit portfolio risk. Until now, two approaches of modelling default risk have been popular: the reduced-form approach and the structural approach. The reduced-form approach is a method of modelling default risk by specifying directly default process as a totally inaccessible stopping time, while the structural approach is a method of deriving default probability implicitly by calculating the probability that the specified firm value dynamic drops below the liability level of the firm. The most popular structural model is based on Merton’s (1974) framework, but he assumed that default can occur only at the maturity of a security. Due to this restrictive assumption, many researchers have developed structural models different from Merton’s. One of the most widely used structural model is first-passage-time model. In particular, Zhou (2001) developed a first-passage-time model and exhibited some advantages of his model over Merton’s
* Corresponding author. Tel.: +82 54 279 2372; fax: +82 54 279 2870. E-mail addresses: [email protected] (M.A. Kim), [email protected] (B.-G. Jang), [email protected] (H.-S. Lee). 0378-4266/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2008.05.013
model. Moreover, he provided an analytical formula for calculating default correlations and joint default probabilities. He also suggested a readily implementable approach to default correlations that makes use of various firm-specific information, and gave some theoretical explanations for various empirical results. Recently, first-passage-time models were extended to excursion models permitting firms to reorganize operations after a default. Gauthier (2002) and Giesecke (2004) gave some details for excursion models. In his paper, Zhou (2001) assumed that the volatilities of asset returns are constant regardless of business cycle. However in real world, the volatilities of asset returns vary according to market environments, e.g. business cycle, as pointed out by Ang and Bekaert (2002). Thus this fact should be considered in determining firm’s default probability. Furthermore, empirical literatures, such as Das et al. (2006) and Lucas and Klaassen (2006), showed that default correlations as well as default probability are higher conditional on the economy starting in a recession than an expansion. All these results highly recommend that it is necessary to reflect the effects of such market environments in modelling default risk.1 Adopting a Markov regime-switching model is an easy way to capture the cyclical features of the drift and volatility of asset return depending on market environments. For a long time, lots of empirical evidences have supported the existence of regime-switching 1 New Basel Accord also suggests that banks reflect business cycle when they group the counterparties into several groups.
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property in financial markets. For example, Campbell (1991) and Lewellen (2004) found that expected returns on equities change over time. Schwert (1989) and Campbell and Hentschel (1992) concluded that the volatilities of stock returns also vary substantially over time. Related researches to regime-switching model have been fulfilled actively. So et al. (1998) generalized stochastic volatility model to encompass a regime-switching process. Bollen et al. (2000) investigated the ability of regime-switching models to capture the dynamics of foreign exchange rates. In the literature related with correlated default risk, Das and Geng (2004) suggested that regime-switching model provides a better representation of the properties of correlated default than jump model. Their approach was based on hazard rate process and they assumed that hazard rate level changes over regime. In this paper, we extend Zhou’s (2001) model to a regimeswitching first-passage-time model. Our model assumes that market condition has two states, bull market (or economic expansion) and bear market (or economic recession), and the volatilities of asset returns as well as risk-free interest rate change according to such market condition. Under this set-up, we try to suggest a method calculating individual and joint default probabilities. In fact, we provide a numerical method for calculating individual and joint default probabilities. We also show that our model can explain various relationship between default probability, default correlation, and market condition. For example, firms having large differences in volatilities between regimes have high default correlation. In addition, both default probability and default correlation increase as the intensity shifting from bull market to bear market increases. As an application to our model we calculate the spread of credit default swap (CDS) with counterparty default risk (CDR) and that of BDS. Also we obtain affluent implications from numerical results: First, CDS spread is higher in economic recession than in economic boom regardless of the swap maturity. Second, as the volatility fluctuation according to market condition increases, CDS spread increases regardless of maturity. Third, the higher the tendency shifting from bull market to bear market, the higher the spread for both CDS without CDR and basket default swap. However, the spread of CDS with CDR may decline against the intensity when the default risk of counterparty is higher than that of reference entity under stochastic market condition. The rest of the paper is organized as follows. In Section 2, we present assumptions underlying our regime-switching first-passagetime model and default correlation structure, and introduce the basic framework of the regime-switching model. We discuss individual and joint default probabilities under the model in Section 3. Section 4 shows numerical results and some implications for default correlation dynamics. Section 5 applies our model to the valuation of CDS with (or without) CDR and BDS, and Section 6 concludes.
W 1t ; W 2t and two Poisson processes uB and ub , which are independent each other under a risk-neutral measure P. Also we assume each Poisson process is independent of the Brownian motions. Market condition changes from regime i to regime jð–iÞ at the next jump time of Poisson process ui for i; j 2 fB; bg. Under the risk-neutral measure P, the intensity of the Poisson process ui is assumed to be constant ki > 0 in our model. Then Markov chain Y t satisfies
PðY tþdt ¼ ei jY t ¼ ej Þ ¼ kj dt þ oðdtÞ; PðY tþdt ¼ ej jY t ¼ ej Þ ¼ 1 kj dt þ oðdtÞ: This implies that, during the infinitesimal-time interval ðt; t þ dtÞ, the risk-neutral probability which market condition changes from regime j to regime i is kj dt, and the probability which market condition stays in regime j is ð1 kj dtÞ. Assumption 1. The asset value X i of firm i 2 f1; 2g under riskneutral probability measure P is given by
dX i ðtÞ ¼ r s ðtÞX i ðtÞdt þ rsi ðtÞX i ðtÞdW i ðtÞ where r s ðtÞ is a risk-free rate, rsi ðtÞ is the volatility of asset return, s is an indicator describing market condition, and W i ðtÞ is a standard Brownian motion with dW i ðtÞdW j ðtÞ ¼ qij dt. According to this assumption, unlike earlier structural models,2 our model admits the stochastic changes of risk-free rate and asset return volatility, which are geared to market condition (or business cycle). Thus our model can reflect firm characteristics varying with economic state more precisely. We assume that market condition switches across two regimes of bull market B and bear market b, therefore, the indicator s can take two symbols: B and b. Viewed in this light, r B (r b ) and rBi (rbi ) represent the interest rate and the asset volatility under bull (bear) market, respectively. Assumption 2. A firm defaults when its asset value first hits a default boundary. Let sit be the default time of firm i at time t, defined by
sit ¼ inffs P t; X is 6 Bis g; where Bis is an exogenously pre-specified default boundary at time s for firm i. In addition, Bis ¼ K i egi s is a time-dependent default boundary.3 The default boundary can be defined in various forms. For example, Black and Cox (1976) took an exponential form in time t because the expected debt value usually takes this form. Default boundary can also be modelled stochastically, moving in random with firm value and business cycle.4 2.2. Default correlation Suppose there is a random variable Di ðTÞ that describes the default status of firm i over a given horizon T:
2. The model
Di ðTÞ ¼ 2.1. Assumptions We extend the first-passage-time model by Zhou (2001) into a model incorporating regime-switching market condition, where the expectation and the volatility of asset return change across the regimes. Firstly, we assume that there are two regimes of market condition, i.e. bull regime (regime B) and bear regime (regime b). These regimes are determined by a two-state Markov chain Y t evolving in continuous time, and we represent the set of corresponding states of Y t as the orthogonal basis feB ; eb g, where eB ¼ ð1; 0Þ0 , eb ¼ ð0; 1Þ0 2 R2 . For the probability space ðX; F; PÞ, we assume that the filtration fFt g is generated by two standard Brownian motions
1 if firm i defaults by T; i 2 f1; 2g; 0
otherwise:
Then default correlation DCðTÞ over given horizon T is defined as follows:
DCðTÞ ¼
E½D1 ðTÞD2 ðTÞ E½D1 ðTÞE½D2 ðTÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Var½D1 ðTÞVar½D2 ðTÞ
2 Earlier structural models have been developed by Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995) and Shimko et al. (1993). 3 Hereafter, we denote X i ðsÞ as X is . 4 Default boundary may fluctuate with business cycle and this can be an extension of our approach. We put aside this kind of model for future research.
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Because Di ðtÞ is a Bernoulli binomial random variable, we have
E½Di ðTÞ ¼ PðDi ðTÞ ¼ 1Þ; Var½Di ðTÞ ¼ PðDi ðTÞ ¼ 1Þ½1 PðDi ðTÞ ¼ 1Þ; where PðAÞ is the probability that an event A occurs. In addition, the probability that at least one firm defaults is presented as
X 2t ; Y t Þ and using Feynmann-Kac formula, a PDE with respect to uðt; X 1t ; X 2t ; Y t Þ is derived as follows:
ou ou ou 1 o2 u 1 o2 u þ rX 2 þ ðr1 X 1 Þ2 2 þ ðr2 X 2 Þ2 2 þ rX 1 ot oX 1 oX 2 2 2 oX 1 oX 2 þ qr1 r2 X 1 X 2
PðD1 ðTÞ ¼ 1 or D2 ðTÞ ¼ 1Þ ¼ E½D1 ðTÞ þ E½D2 ðTÞ E½D1 ðTÞ D2 ðTÞ: where A ¼ Since the default correlation also switches across the regimes, we denote those in bull and bear markets by DC B and DC b , respectively. 3. Individual and joint default probability In this section, we suggest the methods for obtaining the individual and joint survival probability under the regime-switching first-passage-time model. More specifically, we derive a system of partial differential equations (PDEs) for each individual and joint default probability. We also adopt a finite difference method (FDM) as the numerical scheme for solving such PDEs. The relationship between the probability of default and survival probability is presented as
PðDi ðTÞ ¼ 1Þ ¼ 1 PðSi ðTÞ ¼ 1Þ; where Si ðtÞ is 1 if firm i 2 f1; 2g survives over a given horizon t, and 0 if firm i 2 f1; 2g defaults. And the probability that at least one firm defaults is the same as one minus joint survival probability as follows:
PðD1 ðTÞ ¼ 1 or D2 ðTÞ ¼ 1Þ ¼ 1 PðS1 ðTÞ ¼ 1 and S2 ðTÞ ¼ 1Þ: The survival probability and joint survival probability are defined by, for i 2 f1; 2g and y ¼ feB ; eb g,
kB kB
kb . By letting uj ðt; X 1t ; X 2t Þ ¼ uðt; X 1t ; X 2t ; ej Þ kb
for j 2 fB; bg, each component uj of U satisfies the equation
ouj ouj ouj 1 j o2 uj 1 o2 uj þ rj X 2 þ ðr1 X 1 Þ2 2 þ ðrj2 X 2 Þ2 2 þ rj X 1 ot oX 1 oX 2 2 oX 1 2 oX 2 þ qrj1 rj2 X 1 X 2
o2 uj þ hU; Aej i ¼ 0: oX 1 oX 2
The boundary and terminal conditions are given by
uj ðt; x1 ; x2 Þ ¼ 0; j
u ðT; x1 ; x2 Þ ¼ 1;
9j 2 fB; bg;
xi 6 Bit ;
t 6 T;
xi > BiT ;
for all i 2 f1; 2g. Consequently, we have a system of second-order PDEs as follows:
ouB ouB ouB 1 B o2 uB 1 B o2 u B þ rB X 2 þ ðr1 X 1 Þ2 þ ðr2 X 2 Þ2 þ rB X 1 2 ot oX 1 oX 2 2 oX 1 2 oX 22 o2 uB þ kB ðub uB Þ ¼ 0; oX 1 oX 2 oub oub oub 1 b o2 u b 1 b o2 u b þ rb X 2 þ ðr X 1 Þ2 þ ðr2 X 2 Þ2 þ rb X 1 2 2 ot oX 1 oX 2 2 1 oX 1 oX 22 þ qrB1 rB2 X 1 X 2
þ qrb1 rb2 X 1 X 2
PðSi ðTÞ ¼ 1Þ ¼ Pfsit > TjX it ¼ xi ; Y t ¼ yg; PðS1 ðTÞ ¼ 1 and S2 ðTÞ ¼ 1Þ ¼ Pfs1t > T;
o2 u þ hU; AY t i ¼ 0; oX 1 oX 2
o2 ub þ kb ðuB ub Þ ¼ 0: oX 1 oX 2
ð1Þ
Using the argument similar to above, we can derive the system of PDEs for individual survival probability v as follows:
s2t > TjX 1t ¼ x1 ; X 2t ¼ x2 ; Y t ¼ yg: For convenience sake, we adopt the functions v and w as the individual survival probabilities such as
vðt; x1 ; yÞ ¼ Pfs1t > TjX 1t ¼ x1 ; Y t ¼ yg; wðt; x2 ; yÞ ¼ Pfs2t > TjX 2t ¼ x2 ; Y t ¼ yg: Also we let the function u be the joint survival probability, namely,
ovB ovB 1 B o2 vB þ ðr1 X 1 Þ2 þ kB ðvb vB Þ ¼ 0; þ rB X 1 ot oX 1 2 oX 21 ovb ovb 1 b o2 vb þ ðr X 1 Þ2 þ kb ðvB vb Þ ¼ 0: þ rb X 1 ot oX 1 2 1 oX 21
ð2Þ
Let uj ¼ uð; ; ; ej Þ; vj ¼ vð; ; ; ej Þ and wj ¼ wð; ; ; ej Þ for j 2 fB; bg, then the default correlations are determined by the following relationship:
The equations for individual survival probability w are of the same form with these. All these coupled PDEs can be solved by the FDM described in Appendix or other numerical methods for calculating the solution of PDEs.
uB vB wB DC B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; vB wB ð1 vB Þð1 wB Þ
4. Default correlation dynamics
uðt; x1 ; x2 ; yÞ ¼ Pfs1t > T; s2t > TjX 1t ¼ x1 ; X 2t ¼ x2 ; Y t ¼ yg:
ub vb wb DC b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : vb wb ð1 vb Þð1 wb Þ
Under the set-up, we derive firstly the system of PDEs for joint survival probability
uðt; x1 ; x2 ; yÞ ¼ Pfs1t > T; s2t > TjX 1t ¼ x1 ; X 2t ¼ x2 ; Y t ¼ yg with terminal and boundary conditions
uðt; x1 ; x2 ; yÞ ¼ 0;
9i 2 f1; 2g;
uðT; x1 ; x2 ; yÞ ¼ 1;
xi > BiT ;
xi 6 Bit ;
t 6 T;
for all i:
Let us define Uðt; x1 ; x2 ; Y t Þ as the matrix form of
Uðt; x1 ; x2 ; Y t Þ ¼ ðuðt; x1 ; x2 ; eB Þ; uðt; x1 ; x2 ; eb ÞÞ0 so that uðt; X 1t ; X 2t ; Y t Þ ¼ hUðt; X 1t ; X 2t ; Y t Þ; Y t i. Here, h; i represents Euclidean inner product. By applying Ito’s lemma to uðt; X 1t ;
4.1. Asset correlation Fig. 1 presents the term structure of default correlation with various asset correlations. Default correlation increases as asset correlation increases and this is consistent with the results in earlier literatures such as Zhou (2001) and Kim and Kim (2005). Positive asset correlation means that the increase (resp. decrease) of one asset return causes the increase (resp. decrease) of the other asset return. If two firms have the strong interdependence of asset return, default events of two firms are also strongly interdependent. In fact, firms in the same industry or region usually have strong asset correlation and so they have high default correlation. Default correlation increases firstly and then gradually decreases as maturity increases, whenever asset correlation is positive.
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4.2. Asset-to-liability ratio Firms with high asset-to-liability ratio have low default correlation as shown in Fig. 2. High asset-to-liability ratio implies that current asset value is large enough to cover liability and consequently individual default probability is low. Fig. 2 plots the term structure of default correlation with various asset-to-liability ratios and we assumed that the change of volatility according to market condition is the same irrespective of asset-to-liability ratio. However we can observe different aspect from Fig. 3. From Jang et al. (2007), we define the average volatility ri (in steady state) of firm i across both regimes as
ri
Default Correlation
0.25
– :
X1 K1
=
X2 K2
= 4,
X1 K1
=
X2 K2
= 5,
:
X1
=
K1
X2 K2
=6
0.2
0.15
0.1
0.05
kB k ¼ rb þ b rB : kB þ kb i kB þ kb i
5
Then, Fig. 3 says that, when the volatility change is higher for a firm with higher asset-to-liability ratio, default correlation do not need to have a negative relation with asset-to-liability ratio in spite of the same average volatility. Default correlation between firms with higher asset-to-liability ratio and higher volatility change can be lower for short maturity but higher for long maturity than that with lower asset-to-liability and lower volatility change.
—:
10 Maturity (year)
15
20
Fig. 2. The relation between default correlation and maturity for various asset-toliability ratios. Here, q ¼ 0:4, rB1 ¼ 0:13, rb1 ¼ 0:26, rB2 ¼ 0:16, rb2 ¼ 0:33, kB ¼ 0:23, kb ¼ 1:73, r B ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bullish.
4.3. Market condition
Default Correlation
0.25
As already examined by lots of researchers such as Ang and Bekaert (2002), firms under bull market have lower volatility than firms under bear market. So the default correlation as well as default probability may be lower in economic expansion than in economic recession. Fig. 4 explains this intuition. We can say that the diversification effect in a credit portfolio with positive correlation is higher in economic expansion than in economic recession. Of course, the regime-switching approach can also explain the empirical evidence that individual default probability is higher in economic recession than in economic expansion.
0.2 —:
0.15 :
0.1
5
In our regime-switching model, r (resp. r represents the asset return volatility of firm i when market is bullish (resp. bearish). Therefore, we can think of the difference of jrbi rBi j as the sensitivity of the volatility for firm i to market condition. Fig. 5 plots default correlation for various volatilities rB1 and rB2 where r1 ¼ 0:145 and r2 ¼ 0:18. For this case, the larger rBi , the smaller rbi for i ¼ 1; 2. The smaller magnitude of the difference
— : ρ = 0.4, — : ρ = 0.2,
b i)
: ρ = – 0.2,
0.25
Default Correlation
0.2 0.15 0.1
=
X2 K2 X2 K2
= 4, σ Bi = 0.223, σ bi = 0.327, i =1,2 = 6, σ Bi = 0.2, σ bi σ = 0.5, i =1,2
10 Maturity (year)
15
20
Fig. 3. The relation between default correlation and maturity for various asset-toliability ratios (with different volatility change). Here, q ¼ 0:4, kB ¼ 0:23, kb ¼ 1:73, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bullish and we use the average volatilities of r1 ¼ r2 ¼ 0:2352.
Default Correlation
0.25
X1 K1
=
0.05
4.4. Asset return volatility B i
X1 K1
— : Bullish market,
: Bearish market
0.2 0.15 0.1
0.05 0.05 5
10
15
20 5
0.05 Maturity (year) Fig. 1. The relation between default correlation and maturity for given asset return correlations. Here, q represents the asset return correlation between firms where X 1 =K 1 ¼ X 2 =K 2 ¼ 4, rB1 ¼ 0:13, rb1 ¼ 0:26, rB2 ¼ 0:16, rb2 ¼ 0:33, kB ¼ 0:23, kb ¼ 1:73, r B ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bullish.
10 Maturity (year)
15
20
Fig. 4. The relation between default correlation and maturity in two regimes: bull and bear. Here, X 1 =K 1 ¼ X 2 =K 2 ¼ 4 and other parameters are q ¼ 0:4, rB1 ¼ 0:13, rb1 ¼ 0:26, rB2 ¼ 0:16, rb2 ¼ 0:33, kB ¼ 0:23, kb ¼ 1:73, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05.
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0.22
0.2 — : Bullish market, : Bearish market
— : Bullish market, : Bearish market
0.195 Default correlation
Default correlation
0.21
0.2
0.19
0.19
0.185
0.18
0.18
0.175 0.17 0.115
0.12
0.125
0.13
σ B1
0.135
0.14
0.15
(a) σB1 (σB2 = 0.16, σb2 = 0.33)
0.155
0.16
0.165
σ B2
0.17
0.175
(b) σB2 (σB1 = 0.13, σb1 = 0.26)
Fig. 5. The relation between default correlation and volatility for different market conditions. For both cases, the average volatilities are assumed to be r2 ¼ 0:18. Other parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 4, q ¼ 0:4, kB ¼ 0:23, kb ¼ 1:73, T ¼ 20, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05.
r1 ¼ 0:145 and
jrbi rBi j implies the lower sensitivity of asset value to the change of market condition. A firm is considered to be stable if it has small difference in volatilities between two regimes. As shown in Fig. 5, more stable firms have lower default correlation. Hence, it is advantageous to choose stable firms in selecting a credit portfolio in order to increase diversification effect.
ing empirical fact (see Das et al. (2006) and Das and Geng (2004)) that default correlation dynamically changes depending on the intensity shifting between two regimes.
4.5. Regime-switching intensity
In this section, we introduce two applications of our regimeswitching first-passage-time model. The first is the valuation of CDS with CDR and the second is that of BDS referencing two entities.
Another key factor in determining default correlation is the regime-switching intensity ki . kB denotes the transition intensity from bull regime to bear regime and kb vice versa. For a fixed kb , the increase of kB implies that bull regime will last for shorter time than before. Fig. 6 shows default correlation for various intensities kB and kb . As kB increases market stays for relatively more time in bear regime, thus default correlation increases. On the contrary, as kb increases market stays for more time in bull regime, thus default correlation decreases. Therefore, our model justifies the exist-
5. Applications to credit derivatives
5.1. Credit default swap with counterparty default risk A CDS is an agreement between two parties that allows one party to have a long position on a third-party credit risk, and the other party to have a short position on the credit risk. In other words, one party (called as the protection seller) is selling the insurance
0.24
0.24
0.23
Default correlation
Default correlation
0.235
— : Bullish market, : Bearish market
0.225
— : Bullish market, : Bearish market
0.22
0.2
0.22
0.18 0.215 0
0.5
1
1.5
2
2.5
(a) λB (λb = 1.73)
3
3.5
λB
0.5
1
1.5
2
2.5
3
3.5
λB
(b) λb (λB = 0.23)
Fig. 6. The relation between default correlation and intensity for different market conditions. Here, the parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 4, q ¼ 0:4, rB2 ¼ 0:16, rb2 ¼ 0:33, T ¼ 20, r B ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05.
rB1 ¼ 0:13, rb1 ¼ 0:26,
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a
1200
Single CDS without CDR —: Bullish market, : Bearish market
Spread (bp)
1000 800 600 400 200
2
b
6 Maturity (year)
8
10
8
10
8
10
1200 Single CDS without CDR — : Bullish market, : Bearish market
1000 Spread (bp)
4
800 600 400 200
2
4
6 Maturity (year)
c 1200
Basket Default Swap — : Bullish market,
Spread (bp)
1000
: Bearish market
800 600 400 200
2
4
6 Maturity (year)
Fig. 7. The relation between default swap rates and maturity for different market conditions. Here, the default parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 3, q ¼ 0:4, R ¼ 0:4, rB2 ¼ 0:3, rb2 ¼ 0:4, kB ¼ 0:23, kb ¼ 1:73, r B ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Furthermore, we use rB1 ¼ 0:2, rb1 ¼ 0:5 for (b) and (c).
against the default of the third-party and the other party (called as the protection buyer) is buying it. The third-party, which is called as a reference entity, can be a specific firm or its corporate bond. In CDS with no CDR, the protection buyer pays a fixed payment of cT dollars per unit time, called as CDS spread with maturity T, to the protection seller unless the reference entity defaults. However, in the credit default swap with CDR, the protection buyer pays the CDS spread to the protection seller unless either the reference entity or the protection seller default.5 In order to calculate the spread of CDS with CDR under our regime-switching first-passage-time model, we use the following discrete-time formula. Here, the recovery rate R of the notional amount is assumed to be constant. Then,
5
See Kim and Kim (2003) for the pricing of CDS with CDR.
cT ¼
ð1 RÞ
PN
n¼1 ½Q R ðt n Þ Q R;C ðt n Þ ðQ R ðt n1 Þ Q R;C ðt n1 ÞÞBð0; t n ÞDt ; PN n¼1 ½1 Q R ðt n Þ Q C ðt n Þ þ Q R;C ðt n ÞBð0; t n ÞDt
where Dt ¼ T=N, tn ¼ nDt, Q R ðtÞ (resp. Q C ðtÞ) is the default probability of the reference entity (resp. the counterparty) over a given horizon t under the risk-neutral probability measure P. Q R;C ðtÞ is the joint default probability of the reference entity and the counterparty over a given horizon t under P. Bð0; tÞ is a default-free discount bond price at time 0 with maturity t and principal amount 1. 5.2. Basket default swap In a BDS (sometimes called as a first-to-default swap), there are a number of different reference entities and reference obligations. The default correlation is the fundamental driver of BDS premium,
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M.A. Kim et al. / Journal of Banking & Finance 32 (2008) 2617–2627
a
1200
Spread (bp)
1000
Single CDS without CDR
– : σ b2 − σ B2
=
0.2,
— : σ
b 2−
σ B2
=
: σ
0.1,
b 2−
σ B2
=
0
800 600 400 200 2
b
6 Maturity (year)
8
10
500 400
Spread (bp)
4
Single CDS without CDR X2 : = 3 & Unstable K2
X2 = 4 & Unstable K2
— :
300 200 100
2
c
1750 1500
4
6 Maturity (year)
8
10
8
10
Xi = 3 for all cases, i =1,2 Ki
Basket Default Swap, — : Bullish market & Stable,
: Bullish market & Unstable : Bearish market & Unstable
– : Bearish market & Stable,
Spread (bp)
1250 1000 750 500 250 2
4
6 Maturity (year)
Fig. 8. The relation between default swap rate and maturity for various stabilities. (a) X 1 =K 1 ¼ X 2 =K 2 ¼ 3, q ¼ 0:4, R ¼ 0:4, rB2 ¼ 0:2, rb2 ¼ 0:5, kB ¼ 0:23, kb ¼ 1:73, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. (b) q ¼ 0:4, R ¼ 0:4, kB ¼ 0:23, kb ¼ 1:73, rB ¼ rb ¼ 0:05, g1 ¼ g2 ¼ 0:05, and rB2 ¼ 0:223, rb2 ¼ 0:327 for the stable case, rB2 ¼ 0:2, rb2 ¼ 0:5 for the unstable case. (c) The same parameters with (b). For the cases of (a) and (b), current market is bullish.
and hence joint default probabilities are necessary to obtain it. When the correlation between reference entities is non-zero, the joint default probability of two entities can be easily obtained under our model.6 The probability of the first default happening between times t n1 and t n ðn ¼ 1; 2; . . . ; N), when the default correlation between reference entities is considered, is
e 1;2;...;I ðtn Þ; e 1;2;...;I ðtn1 Þ Q Q
e 1;2;...;I ðtÞ is the probability of no earlier default by any of refwhere Q erence entities for the time interval ð0; tÞ and I is the number of reference entities. Similar to the case with a plain CDS, the following discrete equation can be used as an approximate formula for the spread of a BDS.
cT ¼
ð1 RÞ
PN
e
e
n¼1 ½ Q 1;2;...;I ðt n1 Þ Q 1;2;...;I ðt n ÞBð0; t n ÞDt PN e n¼1 Q 1;2;...;I ðt n ÞBð0; t n ÞDt
5.3. Implications 6 The joint default probability of multiple entities are needed, in order to determine the spread of BDS. However, the calculation of such probability for more than three entities can be executed well only after resolving some numerical problems, for example, the inaccuracy from cumulative numerical errors. The problems remain for future research.
We assume that there are two firms in this section. In determining the spread of CDS with CDR, reference entity (resp. counterparty) is assumed to be firm 2 (resp. firm 1). In addition, we assume that basket is composed of firm 1 and firm 2. The intensity parameters
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M.A. Kim et al. / Journal of Banking & Finance 32 (2008) 2617–2627
a
1200
Single CDS without CDR – : λB = 1.5, — : λB = 1,
: λB = 0.23
Spread (bp)
1000 800 600 400 200 2
4
6
8
10
8
10
8
10
Maturity (year)
b 1200
Single CDS with CDR – : λB = 1.5, —: λB = 1,
Spread (bp)
1000
: λB = 0.23
800 600 400 200 2
4
6 Maturity (year)
c
Basket Default Swap – : λB = 1.5, —: λB = 1,
1200
: λB = 0.23
Spread (bp)
1000 800 600 400 200 2
4
6 Maturity (year)
Fig. 9. The relation between default swap rate and maturity for various regime-switching intensities kB . Here, the parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 3, q ¼ 0:4, R ¼ 0:4, rB1 ¼ 0:2, rb1 ¼ 0:5, rB2 ¼ 0:3, rb2 ¼ 0:4, kb ¼ 1:73, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bullish.
were empirically estimated by Ang and Bekaert (2002) and also used in Jang et al. (2007). The volatility of asset return for each firm has two values depending on market condition. In particular, we assume that the volatility of firm 1 is the lower than that of firm 2 in bull market, but the volatility of firm 1 is higher than that of firm 2 in bear market. In addition, we assume that the intensity kB switching from bull market to bear market is lower than the intensity kb switching from bear market to bull market, as already shown in Ang and Bekaert (2002). 5.3.1. Market condition Fig. 7 shows the term structures of CDS spreads depending on market condition. Three types of CDS are considered: single CDS without CDR, single CDS with CDR, and BDS with two reference entities. First, Fig. 7 says that default swap spread is higher in bearish market than in bullish market for both CDS and BDS cases. This result is consistent with the empirical result that, when market goes
down, the productivity of a firm declines and, therefore, the credit quality of the firm decreases. Second, relative spread change7 is higher for BDS than for single CDS. In addition, relative spread change is higher for single CDS without CDR than for single CDS with CDR. This result is due to CDR which may cause an early termination of the CDS contract. 5.3.2. Stability When market shifts from one regime to the other regime, a firm is said to be stable if it has the small difference of volatility between two regimes. Fig. 8 shows the relationship between single CDS (with firm 2 as the reference entity) spread and maturity. From Fig. 8a, we obtain that as the difference between rb2 and rB2 increases, CDS spread increases regardless of maturity. On the other hand, it is well known that, when firm F 1 has higher asset-to-liability 7 Relative spread change is defined as ðCDSb CDSB Þ=CDSB , where CDSb (resp. CDSB ) is spread under bear (resp. bull) market.
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a
Single CDS without CDR – : λb = 0.5, — : λb = 1,
1750
: λb = 1.73
Spread (bp)
1500 1250 1000 750 500 250 2
4
6
8
10
Maturity (year)
b
Single CDS with CDR – : λb = 0.5, — : λb = 1,
1750
Spread (bp)
1500
: λb = 1.73
1250 1000 750 500 250 2
4
6
8
10
8
10
Maturity (year)
c
Basket Default Swap – : λb = 0.5, — : λb = 1,
1750
: λb = 1.73
Spread (bp)
1500 1250 1000 750 500 250
2
4
6 Maturity (year)
Fig. 10. The relation between default swap rate and maturity for various regime-switching intensities kb . Here, the parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 3, q ¼ 0:4, R ¼ 0:4, rB1 ¼ 0:2, rb1 ¼ 0:5, rB2 ¼ 0:3, rb2 ¼ 0:4, kB ¼ 0:23, r B ¼ r b ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bearish.
ratio than that of firm F 2 , the spread of default swap referencing firm F 1 is lower than that referencing firm F 2 under earlier existing firstpassage-time models. However, as seen in Fig. 8b, we find the fact that the gap between the spreads for two firms diminishes with maturity. The result analogues to this is also depicted in Fig. 8c. In this figure we find that the spreads in bear market are more sensitive to maturity than those in bull market. 5.3.3. Regime-switching intensity Fig. 9 shows the term structures of CDS spread for single CDS without CDR, single CDS with CDR, and BDS where kB changes. The higher the intensity kB shifting from bull market to bear market, the higher the spreads for both CDS without CDR and BDS. The effect of higher intensity on the spread increases firstly, but diminishes gradually with longer maturity. Also, single CDS spread increases with kB across maturity. However, for sufficiently long maturity,
the spread of CDS with CDR declines against kB as shown in Fig. 9b. This result stems from the fact that CDR increases as kB increases. Fig. 10 shows the term structure of CDS spread for single CDS without CDR, single CDS with CDR, and BDS where the intensity kb changes. Higher intensity shifting from bear market to bull market declines both average volatility of firm and default correlation. Single CDS spread declines against kb , since the lower volatility of asset return reduces the default probability of reference entity. Also, the spread of BDS declines against kb since the first-to-default probability declines. Two factors determine the first-to-default probability: One is individual default probability and the other is the default correlation determining joint default probability. When the intensity shifting from economic recession to economic expansion increases, the lower individual default probability decreases BDS spread and, at the same time, the lower default correlation increases BDS spread. But, because the former is more dominant
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M.A. Kim et al. / Journal of Banking & Finance 32 (2008) 2617–2627
than the latter in determining default swap spread, the spread of BDS declines against kb .
6. Conclusion We suggest a first-passage-time model that can explain default probability and default correlation dynamics under stochastic market condition. We add a Markov regime-switching market condition to a first-passage-time model, and consequently extend Zhou’s (2001) default correlation model to the model incorporating regime-switching market condition. Using this regime-switching first-passage-time model, we can show various relationship between default probability, default correlation, and market condition. First, firms having larger difference in volatilities between two regimes have higher default correlation. This result suggests that, in order to increase diversification effect, it is better to choose firms less sensitive to business cycle in investing a credit portfolio. Second, as the intensity shifting from bull market to bear market increases, both default probability and default correlation increase and vice versa. In fact, high possibility for market to go down may increase implied default probability and implied correlation, and this phenomenon must be reflected in pricing credit derivatives and managing credit portfolio risk. This relationship has not been involved in earlier structural models for multiple firms but our approach can solve this problem. We also suggest a valuation method for single CDS with(or without) CDR and BDS under the set-up. The numerical results show several implications: First, default swap spread is higher in economic recession than in economic expansion across maturity. Second, as the difference in volatilities between bear market and bull market increases, CDS spread increases regardless of maturity. Third, under earlier first-passage-time models, the spread of default swap referencing firm with higher asset-to-liability ratio is always lower. But, if the firm has larger difference in volatilities according to market condition, the spread of default swap may not be always lower in our regime-switching first-passage-time model. Fourth, the higher the intensity shifting from bull market to bear market, the higher the spread for both CDS without CDR and BDS. However, the spread of CDS with CDR may decline against the intensity when the default risk of counterparty is higher than that of reference entity under stochastic market condition.
We thank an anonymous referee for a helpful comment.
Appendix. Implementation of the system of the partial differential equations in Section 3 by finite difference method Eq. (1), which is the PDE for joint survival probability, can be rewritten by using the symbols p ¼ ln X 1 and q ¼ ln X 2 as follows:
B ouB 1 ou 1 2 ouB 1 B 2 o2 uB þ rB ðrB1 Þ2 þ r B rB2 þ ðr Þ 2 2 ot op oq 2 1 op2 ð3Þ
oU 1 oU 1 oU 1 2 o2 U 1 2 o2 U þ r þ r r21 þ r r22 þ r ot 2 op 2 oq 2 1 op2 2 2 oq2 o2 U ~ UÞ ¼ 0: þ kðU opoq
þ
iþ1;j i;j i1;j i;jþ1 i;j i;j1 1 2 U kþ1 2U kþ1 þ U kþ1 1 2 U kþ1 2U kþ1 þ U kþ1 r1 þ r2 2 2 2 2 ðDpÞ ðDqÞ
þ qr1 r2
i1;j1 U iþ1;jþ1 U i1;jþ1 U iþ1;j1 þ U kþ1 kþ1 kþ1 kþ1 4DpDq
~ i;j U i;j Þ ¼ 0: þ kðU kþ1 kþ1 In other words,
"
U i;j k
ðDpÞ2
2 1
r
Dt
#
Dt iþ1;jþ1 qr1 r2 U kþ1 4DpDq " # Dt Dt 1 2 Dt 1 2 iþ1;j qr1 r2 U iþ1;j1 þ r r r þ 1 U kþ1 kþ1 4DpDq 2Dp 2 1 ðDpÞ2 2 " # Dt 1 Dt 1 2 i;jþ1 Dt 1 2 U þ r r r r22 þ r 2 kþ1 2 Dq 2 2 Dq 2 2 ðDqÞ2 2 # " # Dt 1 2 i;j1 Dt 1 Dt 1 2 i1;j r2 U kþ1 r1 U kþ1 r r22 2Dp 2 ðDqÞ2 2 ðDpÞ2 2
¼ 1
Dt
ðDqÞ2
r Dtk U i;jkþ1 þ 2 2
Dt Dt i1;j1 ~ i;j : qr1 r2 U i1;jþ1 þ qr1 r2 U kþ1 þ DtkU kþ1 kþ1 4DpDq 4DpDq ð5Þ
In case of individual default probability, by using the variable change p ¼ ln X 1 we can write a generalized version of the equations of (2) as
oV 1 oV 1 2 o2 V ~ VÞ ¼ 0: þ kðV þ r r2 þ r ot 2 op 2 op2
ð6Þ
Since the discrete version of Eq. (6) is
iþ1 i1 iþ1 i1 V ikþ1 V ik V kþ1 V kþ1 2V ikþ1 þ V kþ1 1 1 V þ r r2 þ r2 kþ1 2 2 2 Dt 2Dp ðDpÞ ~ i V i Þ ¼ 0; þ kðV kþ1 kþ1
ð7Þ
Eq. (7) can be rearranged as
"
# Dt 1 2 1 Dt iþ1 2 ¼ r V kþ1 r r þ 2 Dp 2 2 ðDpÞ2 " # Dt 2 þ 1 r k D t V ikþ1 ðDpÞ2 " # Dt 1 2 1 Dt 2 ~i þ r V i1 r r þ kþ1 þ kDt V kþ1 : 2Dp 2 2 ðDpÞ2
ð8Þ
We can calculate the joint and individual survival probabilities by using the difference Eqs. (5) and (8). References
Eq. (3) can be generalized as follows by using representative func~ and ignoring the regime-state superscripts B and b: tions U; U
þ qr1 r2
iþ1;j i;jþ1 i;j i1;j U i;j U kþ1 U kþ1 U kþ1 U i;j1 1 1 kþ1 U k kþ1 þ r r21 þ r r22 2 2 Dt 2 Dp 2Dq
V ik
Acknowledgement
1 o2 uB o2 uB þ ðrB2 Þ2 2 þ qrB1 rB2 þ kB ðub uB Þ ¼ 0: 2 oq opoq
The discrete version of Eq. (4) is
ð4Þ
Ang, A., Bekaert, G., 2002. International asset allocation with regime shifts. Review of Financial Studies 15, 1137–1187. Black, F., Cox, J., 1976. Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance 31, 351–367. Bollen, N.P.B., Gray, S.F., Whaley, R.E., 2000. Regime switching in foreign exchange rates: Evidence from currency option prices. Journal of Econometrics 94, 239– 276. Campbell, J., 1991. A variance decomposition for stock returns. Economic Journal 101, 157–179. Campbell, J., Hentschel, L., 1992. No news is good news: An asymmetric model of changing volatility in stock returns. Journal of Financial Economics 31, 281–318. Das, S.R., Geng, G., 2004. Correlated default processes: A criterion-based copula approach. Working paper, Santa Clara University.
M.A. Kim et al. / Journal of Banking & Finance 32 (2008) 2617–2627 Das, S.R., Freed, L., Geng, G., Kapadia, N., 2006. Correlated default risk. Journal of Fixed Income 14, 7–32. Gauthier, L., 2002. Excursions height- and length-related stopping times, and application to finance. Advances in Applied Probability 34, 846–868. Giesecke, K., 2004. Credit risk modeling and valuation: An introduction. Working paper, Cornell University. Jang, B.-G., Koo, H.K., Liu, H., Loewenstein, M., 2007. Liquidity premia and transaction costs. Journal of Finance 62, 2329–2366. Kim, M.A., Kim, T.S., 2003. Credit default swap valuation with counterparty default risk and market risk. Journal of Risk 6, 49–80. Kim, M.A., Kim, T.S., 2005. Default correlation dynamics with business cycle and credit quality changes. Journal of Derivatives (Fall), 8–27. Lewellen, J., 2004. Predicting returns with financial ratios. Journal of Financial Economics 74, 209–235.
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Longstaff, F., Schwartz, E., 1995. A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 50, 789–819. Lucas, A., Klaassen, P., 2006. Discrete versus continuous state switching models for portfolio credit risk. Journal of Banking and Finance 30, 23–35. Merton, R.C., 1974. On the pricing of corporate debt: The risk structure of interest rates. Journal of finance 29, 449–470. Schwert, W., 1989. Why does stock market volatility change over time? Journal of Finance 44, 1115–1153. Shimko, D., Tejima, N., van Deventer, D., 1993. The pricing of risky debt when interest rates are stochastic. Journal of Fixed Income 3, 58–65. So, M.K.P., Lam, K., Li, W.K., 1998. A stochastic volatility model with Markov switching. Journal of Business and Economic Statistics 16, 244–253. Zhou, C., 2001. An analysis of default correlations and multiple defaults. Review of Financial Studies 14, 555–576.
Contents lists available at ScienceDirect
Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf
A first-passage-time model under regime-switching market environment Mi Ae Kim a, Bong-Gyu Jang b,*, Ho-Seok Lee c a b c
Risk Management Department, Woori Bank, Seoul, Republic of Korea Department of Industrial and Management Engineering, POSTECH, Pohang, Republic of Korea Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea
a r t i c l e
i n f o
Article history: Received 11 February 2008 Accepted 28 May 2008 Available online 6 June 2008 JEL classification: G12 G13 G32 Keywords: First-passage-time model Regime-switching model Default probability Default correlation Credit default swap
a b s t r a c t In this paper, we suggest a first-passage-time model which can explain default probability and default correlation dynamics under stochastic market environment. We add a Markov regime-switching market condition to the first-passage-time model of Zhou [Zhou, C., 2001. An analysis of default correlations and multiple defaults. Review of Financial Studies 14, 555–576]. Using this model, we try to explain various relationship between default probability, default correlation, and market condition. We also suggest a valuation method for credit default swap (CDS) with (or without) counterparty default risk (CDR) and basket default swap under this model. Our numerical results provide us with several meaningful implications. First, default swap spread is higher in economic recession than in economic expansion across default swap maturity. Second, as the difference of asset return volatility between under bear market and under bull market increases, CDS spread increases regardless of maturity. Third, the bigger the intensity shifting from bull market to bear market, the higher the spread for both CDS without CDR and basket default swap. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Recently, correlation products such as basket default swap (BDS) and synthetic collateralized debt obligation (CDO) are actively traded and hence modelling default risk and default correlation has become an important issue in pricing correlation products and managing credit portfolio risk. Until now, two approaches of modelling default risk have been popular: the reduced-form approach and the structural approach. The reduced-form approach is a method of modelling default risk by specifying directly default process as a totally inaccessible stopping time, while the structural approach is a method of deriving default probability implicitly by calculating the probability that the specified firm value dynamic drops below the liability level of the firm. The most popular structural model is based on Merton’s (1974) framework, but he assumed that default can occur only at the maturity of a security. Due to this restrictive assumption, many researchers have developed structural models different from Merton’s. One of the most widely used structural model is first-passage-time model. In particular, Zhou (2001) developed a first-passage-time model and exhibited some advantages of his model over Merton’s
* Corresponding author. Tel.: +82 54 279 2372; fax: +82 54 279 2870. E-mail addresses: [email protected] (M.A. Kim), [email protected] (B.-G. Jang), [email protected] (H.-S. Lee). 0378-4266/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2008.05.013
model. Moreover, he provided an analytical formula for calculating default correlations and joint default probabilities. He also suggested a readily implementable approach to default correlations that makes use of various firm-specific information, and gave some theoretical explanations for various empirical results. Recently, first-passage-time models were extended to excursion models permitting firms to reorganize operations after a default. Gauthier (2002) and Giesecke (2004) gave some details for excursion models. In his paper, Zhou (2001) assumed that the volatilities of asset returns are constant regardless of business cycle. However in real world, the volatilities of asset returns vary according to market environments, e.g. business cycle, as pointed out by Ang and Bekaert (2002). Thus this fact should be considered in determining firm’s default probability. Furthermore, empirical literatures, such as Das et al. (2006) and Lucas and Klaassen (2006), showed that default correlations as well as default probability are higher conditional on the economy starting in a recession than an expansion. All these results highly recommend that it is necessary to reflect the effects of such market environments in modelling default risk.1 Adopting a Markov regime-switching model is an easy way to capture the cyclical features of the drift and volatility of asset return depending on market environments. For a long time, lots of empirical evidences have supported the existence of regime-switching 1 New Basel Accord also suggests that banks reflect business cycle when they group the counterparties into several groups.
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property in financial markets. For example, Campbell (1991) and Lewellen (2004) found that expected returns on equities change over time. Schwert (1989) and Campbell and Hentschel (1992) concluded that the volatilities of stock returns also vary substantially over time. Related researches to regime-switching model have been fulfilled actively. So et al. (1998) generalized stochastic volatility model to encompass a regime-switching process. Bollen et al. (2000) investigated the ability of regime-switching models to capture the dynamics of foreign exchange rates. In the literature related with correlated default risk, Das and Geng (2004) suggested that regime-switching model provides a better representation of the properties of correlated default than jump model. Their approach was based on hazard rate process and they assumed that hazard rate level changes over regime. In this paper, we extend Zhou’s (2001) model to a regimeswitching first-passage-time model. Our model assumes that market condition has two states, bull market (or economic expansion) and bear market (or economic recession), and the volatilities of asset returns as well as risk-free interest rate change according to such market condition. Under this set-up, we try to suggest a method calculating individual and joint default probabilities. In fact, we provide a numerical method for calculating individual and joint default probabilities. We also show that our model can explain various relationship between default probability, default correlation, and market condition. For example, firms having large differences in volatilities between regimes have high default correlation. In addition, both default probability and default correlation increase as the intensity shifting from bull market to bear market increases. As an application to our model we calculate the spread of credit default swap (CDS) with counterparty default risk (CDR) and that of BDS. Also we obtain affluent implications from numerical results: First, CDS spread is higher in economic recession than in economic boom regardless of the swap maturity. Second, as the volatility fluctuation according to market condition increases, CDS spread increases regardless of maturity. Third, the higher the tendency shifting from bull market to bear market, the higher the spread for both CDS without CDR and basket default swap. However, the spread of CDS with CDR may decline against the intensity when the default risk of counterparty is higher than that of reference entity under stochastic market condition. The rest of the paper is organized as follows. In Section 2, we present assumptions underlying our regime-switching first-passagetime model and default correlation structure, and introduce the basic framework of the regime-switching model. We discuss individual and joint default probabilities under the model in Section 3. Section 4 shows numerical results and some implications for default correlation dynamics. Section 5 applies our model to the valuation of CDS with (or without) CDR and BDS, and Section 6 concludes.
W 1t ; W 2t and two Poisson processes uB and ub , which are independent each other under a risk-neutral measure P. Also we assume each Poisson process is independent of the Brownian motions. Market condition changes from regime i to regime jð–iÞ at the next jump time of Poisson process ui for i; j 2 fB; bg. Under the risk-neutral measure P, the intensity of the Poisson process ui is assumed to be constant ki > 0 in our model. Then Markov chain Y t satisfies
PðY tþdt ¼ ei jY t ¼ ej Þ ¼ kj dt þ oðdtÞ; PðY tþdt ¼ ej jY t ¼ ej Þ ¼ 1 kj dt þ oðdtÞ: This implies that, during the infinitesimal-time interval ðt; t þ dtÞ, the risk-neutral probability which market condition changes from regime j to regime i is kj dt, and the probability which market condition stays in regime j is ð1 kj dtÞ. Assumption 1. The asset value X i of firm i 2 f1; 2g under riskneutral probability measure P is given by
dX i ðtÞ ¼ r s ðtÞX i ðtÞdt þ rsi ðtÞX i ðtÞdW i ðtÞ where r s ðtÞ is a risk-free rate, rsi ðtÞ is the volatility of asset return, s is an indicator describing market condition, and W i ðtÞ is a standard Brownian motion with dW i ðtÞdW j ðtÞ ¼ qij dt. According to this assumption, unlike earlier structural models,2 our model admits the stochastic changes of risk-free rate and asset return volatility, which are geared to market condition (or business cycle). Thus our model can reflect firm characteristics varying with economic state more precisely. We assume that market condition switches across two regimes of bull market B and bear market b, therefore, the indicator s can take two symbols: B and b. Viewed in this light, r B (r b ) and rBi (rbi ) represent the interest rate and the asset volatility under bull (bear) market, respectively. Assumption 2. A firm defaults when its asset value first hits a default boundary. Let sit be the default time of firm i at time t, defined by
sit ¼ inffs P t; X is 6 Bis g; where Bis is an exogenously pre-specified default boundary at time s for firm i. In addition, Bis ¼ K i egi s is a time-dependent default boundary.3 The default boundary can be defined in various forms. For example, Black and Cox (1976) took an exponential form in time t because the expected debt value usually takes this form. Default boundary can also be modelled stochastically, moving in random with firm value and business cycle.4 2.2. Default correlation Suppose there is a random variable Di ðTÞ that describes the default status of firm i over a given horizon T:
2. The model
Di ðTÞ ¼ 2.1. Assumptions We extend the first-passage-time model by Zhou (2001) into a model incorporating regime-switching market condition, where the expectation and the volatility of asset return change across the regimes. Firstly, we assume that there are two regimes of market condition, i.e. bull regime (regime B) and bear regime (regime b). These regimes are determined by a two-state Markov chain Y t evolving in continuous time, and we represent the set of corresponding states of Y t as the orthogonal basis feB ; eb g, where eB ¼ ð1; 0Þ0 , eb ¼ ð0; 1Þ0 2 R2 . For the probability space ðX; F; PÞ, we assume that the filtration fFt g is generated by two standard Brownian motions
1 if firm i defaults by T; i 2 f1; 2g; 0
otherwise:
Then default correlation DCðTÞ over given horizon T is defined as follows:
DCðTÞ ¼
E½D1 ðTÞD2 ðTÞ E½D1 ðTÞE½D2 ðTÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Var½D1 ðTÞVar½D2 ðTÞ
2 Earlier structural models have been developed by Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995) and Shimko et al. (1993). 3 Hereafter, we denote X i ðsÞ as X is . 4 Default boundary may fluctuate with business cycle and this can be an extension of our approach. We put aside this kind of model for future research.
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Because Di ðtÞ is a Bernoulli binomial random variable, we have
E½Di ðTÞ ¼ PðDi ðTÞ ¼ 1Þ; Var½Di ðTÞ ¼ PðDi ðTÞ ¼ 1Þ½1 PðDi ðTÞ ¼ 1Þ; where PðAÞ is the probability that an event A occurs. In addition, the probability that at least one firm defaults is presented as
X 2t ; Y t Þ and using Feynmann-Kac formula, a PDE with respect to uðt; X 1t ; X 2t ; Y t Þ is derived as follows:
ou ou ou 1 o2 u 1 o2 u þ rX 2 þ ðr1 X 1 Þ2 2 þ ðr2 X 2 Þ2 2 þ rX 1 ot oX 1 oX 2 2 2 oX 1 oX 2 þ qr1 r2 X 1 X 2
PðD1 ðTÞ ¼ 1 or D2 ðTÞ ¼ 1Þ ¼ E½D1 ðTÞ þ E½D2 ðTÞ E½D1 ðTÞ D2 ðTÞ: where A ¼ Since the default correlation also switches across the regimes, we denote those in bull and bear markets by DC B and DC b , respectively. 3. Individual and joint default probability In this section, we suggest the methods for obtaining the individual and joint survival probability under the regime-switching first-passage-time model. More specifically, we derive a system of partial differential equations (PDEs) for each individual and joint default probability. We also adopt a finite difference method (FDM) as the numerical scheme for solving such PDEs. The relationship between the probability of default and survival probability is presented as
PðDi ðTÞ ¼ 1Þ ¼ 1 PðSi ðTÞ ¼ 1Þ; where Si ðtÞ is 1 if firm i 2 f1; 2g survives over a given horizon t, and 0 if firm i 2 f1; 2g defaults. And the probability that at least one firm defaults is the same as one minus joint survival probability as follows:
PðD1 ðTÞ ¼ 1 or D2 ðTÞ ¼ 1Þ ¼ 1 PðS1 ðTÞ ¼ 1 and S2 ðTÞ ¼ 1Þ: The survival probability and joint survival probability are defined by, for i 2 f1; 2g and y ¼ feB ; eb g,
kB kB
kb . By letting uj ðt; X 1t ; X 2t Þ ¼ uðt; X 1t ; X 2t ; ej Þ kb
for j 2 fB; bg, each component uj of U satisfies the equation
ouj ouj ouj 1 j o2 uj 1 o2 uj þ rj X 2 þ ðr1 X 1 Þ2 2 þ ðrj2 X 2 Þ2 2 þ rj X 1 ot oX 1 oX 2 2 oX 1 2 oX 2 þ qrj1 rj2 X 1 X 2
o2 uj þ hU; Aej i ¼ 0: oX 1 oX 2
The boundary and terminal conditions are given by
uj ðt; x1 ; x2 Þ ¼ 0; j
u ðT; x1 ; x2 Þ ¼ 1;
9j 2 fB; bg;
xi 6 Bit ;
t 6 T;
xi > BiT ;
for all i 2 f1; 2g. Consequently, we have a system of second-order PDEs as follows:
ouB ouB ouB 1 B o2 uB 1 B o2 u B þ rB X 2 þ ðr1 X 1 Þ2 þ ðr2 X 2 Þ2 þ rB X 1 2 ot oX 1 oX 2 2 oX 1 2 oX 22 o2 uB þ kB ðub uB Þ ¼ 0; oX 1 oX 2 oub oub oub 1 b o2 u b 1 b o2 u b þ rb X 2 þ ðr X 1 Þ2 þ ðr2 X 2 Þ2 þ rb X 1 2 2 ot oX 1 oX 2 2 1 oX 1 oX 22 þ qrB1 rB2 X 1 X 2
þ qrb1 rb2 X 1 X 2
PðSi ðTÞ ¼ 1Þ ¼ Pfsit > TjX it ¼ xi ; Y t ¼ yg; PðS1 ðTÞ ¼ 1 and S2 ðTÞ ¼ 1Þ ¼ Pfs1t > T;
o2 u þ hU; AY t i ¼ 0; oX 1 oX 2
o2 ub þ kb ðuB ub Þ ¼ 0: oX 1 oX 2
ð1Þ
Using the argument similar to above, we can derive the system of PDEs for individual survival probability v as follows:
s2t > TjX 1t ¼ x1 ; X 2t ¼ x2 ; Y t ¼ yg: For convenience sake, we adopt the functions v and w as the individual survival probabilities such as
vðt; x1 ; yÞ ¼ Pfs1t > TjX 1t ¼ x1 ; Y t ¼ yg; wðt; x2 ; yÞ ¼ Pfs2t > TjX 2t ¼ x2 ; Y t ¼ yg: Also we let the function u be the joint survival probability, namely,
ovB ovB 1 B o2 vB þ ðr1 X 1 Þ2 þ kB ðvb vB Þ ¼ 0; þ rB X 1 ot oX 1 2 oX 21 ovb ovb 1 b o2 vb þ ðr X 1 Þ2 þ kb ðvB vb Þ ¼ 0: þ rb X 1 ot oX 1 2 1 oX 21
ð2Þ
Let uj ¼ uð; ; ; ej Þ; vj ¼ vð; ; ; ej Þ and wj ¼ wð; ; ; ej Þ for j 2 fB; bg, then the default correlations are determined by the following relationship:
The equations for individual survival probability w are of the same form with these. All these coupled PDEs can be solved by the FDM described in Appendix or other numerical methods for calculating the solution of PDEs.
uB vB wB DC B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; vB wB ð1 vB Þð1 wB Þ
4. Default correlation dynamics
uðt; x1 ; x2 ; yÞ ¼ Pfs1t > T; s2t > TjX 1t ¼ x1 ; X 2t ¼ x2 ; Y t ¼ yg:
ub vb wb DC b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : vb wb ð1 vb Þð1 wb Þ
Under the set-up, we derive firstly the system of PDEs for joint survival probability
uðt; x1 ; x2 ; yÞ ¼ Pfs1t > T; s2t > TjX 1t ¼ x1 ; X 2t ¼ x2 ; Y t ¼ yg with terminal and boundary conditions
uðt; x1 ; x2 ; yÞ ¼ 0;
9i 2 f1; 2g;
uðT; x1 ; x2 ; yÞ ¼ 1;
xi > BiT ;
xi 6 Bit ;
t 6 T;
for all i:
Let us define Uðt; x1 ; x2 ; Y t Þ as the matrix form of
Uðt; x1 ; x2 ; Y t Þ ¼ ðuðt; x1 ; x2 ; eB Þ; uðt; x1 ; x2 ; eb ÞÞ0 so that uðt; X 1t ; X 2t ; Y t Þ ¼ hUðt; X 1t ; X 2t ; Y t Þ; Y t i. Here, h; i represents Euclidean inner product. By applying Ito’s lemma to uðt; X 1t ;
4.1. Asset correlation Fig. 1 presents the term structure of default correlation with various asset correlations. Default correlation increases as asset correlation increases and this is consistent with the results in earlier literatures such as Zhou (2001) and Kim and Kim (2005). Positive asset correlation means that the increase (resp. decrease) of one asset return causes the increase (resp. decrease) of the other asset return. If two firms have the strong interdependence of asset return, default events of two firms are also strongly interdependent. In fact, firms in the same industry or region usually have strong asset correlation and so they have high default correlation. Default correlation increases firstly and then gradually decreases as maturity increases, whenever asset correlation is positive.
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4.2. Asset-to-liability ratio Firms with high asset-to-liability ratio have low default correlation as shown in Fig. 2. High asset-to-liability ratio implies that current asset value is large enough to cover liability and consequently individual default probability is low. Fig. 2 plots the term structure of default correlation with various asset-to-liability ratios and we assumed that the change of volatility according to market condition is the same irrespective of asset-to-liability ratio. However we can observe different aspect from Fig. 3. From Jang et al. (2007), we define the average volatility ri (in steady state) of firm i across both regimes as
ri
Default Correlation
0.25
– :
X1 K1
=
X2 K2
= 4,
X1 K1
=
X2 K2
= 5,
:
X1
=
K1
X2 K2
=6
0.2
0.15
0.1
0.05
kB k ¼ rb þ b rB : kB þ kb i kB þ kb i
5
Then, Fig. 3 says that, when the volatility change is higher for a firm with higher asset-to-liability ratio, default correlation do not need to have a negative relation with asset-to-liability ratio in spite of the same average volatility. Default correlation between firms with higher asset-to-liability ratio and higher volatility change can be lower for short maturity but higher for long maturity than that with lower asset-to-liability and lower volatility change.
—:
10 Maturity (year)
15
20
Fig. 2. The relation between default correlation and maturity for various asset-toliability ratios. Here, q ¼ 0:4, rB1 ¼ 0:13, rb1 ¼ 0:26, rB2 ¼ 0:16, rb2 ¼ 0:33, kB ¼ 0:23, kb ¼ 1:73, r B ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bullish.
4.3. Market condition
Default Correlation
0.25
As already examined by lots of researchers such as Ang and Bekaert (2002), firms under bull market have lower volatility than firms under bear market. So the default correlation as well as default probability may be lower in economic expansion than in economic recession. Fig. 4 explains this intuition. We can say that the diversification effect in a credit portfolio with positive correlation is higher in economic expansion than in economic recession. Of course, the regime-switching approach can also explain the empirical evidence that individual default probability is higher in economic recession than in economic expansion.
0.2 —:
0.15 :
0.1
5
In our regime-switching model, r (resp. r represents the asset return volatility of firm i when market is bullish (resp. bearish). Therefore, we can think of the difference of jrbi rBi j as the sensitivity of the volatility for firm i to market condition. Fig. 5 plots default correlation for various volatilities rB1 and rB2 where r1 ¼ 0:145 and r2 ¼ 0:18. For this case, the larger rBi , the smaller rbi for i ¼ 1; 2. The smaller magnitude of the difference
— : ρ = 0.4, — : ρ = 0.2,
b i)
: ρ = – 0.2,
0.25
Default Correlation
0.2 0.15 0.1
=
X2 K2 X2 K2
= 4, σ Bi = 0.223, σ bi = 0.327, i =1,2 = 6, σ Bi = 0.2, σ bi σ = 0.5, i =1,2
10 Maturity (year)
15
20
Fig. 3. The relation between default correlation and maturity for various asset-toliability ratios (with different volatility change). Here, q ¼ 0:4, kB ¼ 0:23, kb ¼ 1:73, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bullish and we use the average volatilities of r1 ¼ r2 ¼ 0:2352.
Default Correlation
0.25
X1 K1
=
0.05
4.4. Asset return volatility B i
X1 K1
— : Bullish market,
: Bearish market
0.2 0.15 0.1
0.05 0.05 5
10
15
20 5
0.05 Maturity (year) Fig. 1. The relation between default correlation and maturity for given asset return correlations. Here, q represents the asset return correlation between firms where X 1 =K 1 ¼ X 2 =K 2 ¼ 4, rB1 ¼ 0:13, rb1 ¼ 0:26, rB2 ¼ 0:16, rb2 ¼ 0:33, kB ¼ 0:23, kb ¼ 1:73, r B ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bullish.
10 Maturity (year)
15
20
Fig. 4. The relation between default correlation and maturity in two regimes: bull and bear. Here, X 1 =K 1 ¼ X 2 =K 2 ¼ 4 and other parameters are q ¼ 0:4, rB1 ¼ 0:13, rb1 ¼ 0:26, rB2 ¼ 0:16, rb2 ¼ 0:33, kB ¼ 0:23, kb ¼ 1:73, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05.
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0.22
0.2 — : Bullish market, : Bearish market
— : Bullish market, : Bearish market
0.195 Default correlation
Default correlation
0.21
0.2
0.19
0.19
0.185
0.18
0.18
0.175 0.17 0.115
0.12
0.125
0.13
σ B1
0.135
0.14
0.15
(a) σB1 (σB2 = 0.16, σb2 = 0.33)
0.155
0.16
0.165
σ B2
0.17
0.175
(b) σB2 (σB1 = 0.13, σb1 = 0.26)
Fig. 5. The relation between default correlation and volatility for different market conditions. For both cases, the average volatilities are assumed to be r2 ¼ 0:18. Other parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 4, q ¼ 0:4, kB ¼ 0:23, kb ¼ 1:73, T ¼ 20, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05.
r1 ¼ 0:145 and
jrbi rBi j implies the lower sensitivity of asset value to the change of market condition. A firm is considered to be stable if it has small difference in volatilities between two regimes. As shown in Fig. 5, more stable firms have lower default correlation. Hence, it is advantageous to choose stable firms in selecting a credit portfolio in order to increase diversification effect.
ing empirical fact (see Das et al. (2006) and Das and Geng (2004)) that default correlation dynamically changes depending on the intensity shifting between two regimes.
4.5. Regime-switching intensity
In this section, we introduce two applications of our regimeswitching first-passage-time model. The first is the valuation of CDS with CDR and the second is that of BDS referencing two entities.
Another key factor in determining default correlation is the regime-switching intensity ki . kB denotes the transition intensity from bull regime to bear regime and kb vice versa. For a fixed kb , the increase of kB implies that bull regime will last for shorter time than before. Fig. 6 shows default correlation for various intensities kB and kb . As kB increases market stays for relatively more time in bear regime, thus default correlation increases. On the contrary, as kb increases market stays for more time in bull regime, thus default correlation decreases. Therefore, our model justifies the exist-
5. Applications to credit derivatives
5.1. Credit default swap with counterparty default risk A CDS is an agreement between two parties that allows one party to have a long position on a third-party credit risk, and the other party to have a short position on the credit risk. In other words, one party (called as the protection seller) is selling the insurance
0.24
0.24
0.23
Default correlation
Default correlation
0.235
— : Bullish market, : Bearish market
0.225
— : Bullish market, : Bearish market
0.22
0.2
0.22
0.18 0.215 0
0.5
1
1.5
2
2.5
(a) λB (λb = 1.73)
3
3.5
λB
0.5
1
1.5
2
2.5
3
3.5
λB
(b) λb (λB = 0.23)
Fig. 6. The relation between default correlation and intensity for different market conditions. Here, the parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 4, q ¼ 0:4, rB2 ¼ 0:16, rb2 ¼ 0:33, T ¼ 20, r B ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05.
rB1 ¼ 0:13, rb1 ¼ 0:26,
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a
1200
Single CDS without CDR —: Bullish market, : Bearish market
Spread (bp)
1000 800 600 400 200
2
b
6 Maturity (year)
8
10
8
10
8
10
1200 Single CDS without CDR — : Bullish market, : Bearish market
1000 Spread (bp)
4
800 600 400 200
2
4
6 Maturity (year)
c 1200
Basket Default Swap — : Bullish market,
Spread (bp)
1000
: Bearish market
800 600 400 200
2
4
6 Maturity (year)
Fig. 7. The relation between default swap rates and maturity for different market conditions. Here, the default parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 3, q ¼ 0:4, R ¼ 0:4, rB2 ¼ 0:3, rb2 ¼ 0:4, kB ¼ 0:23, kb ¼ 1:73, r B ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Furthermore, we use rB1 ¼ 0:2, rb1 ¼ 0:5 for (b) and (c).
against the default of the third-party and the other party (called as the protection buyer) is buying it. The third-party, which is called as a reference entity, can be a specific firm or its corporate bond. In CDS with no CDR, the protection buyer pays a fixed payment of cT dollars per unit time, called as CDS spread with maturity T, to the protection seller unless the reference entity defaults. However, in the credit default swap with CDR, the protection buyer pays the CDS spread to the protection seller unless either the reference entity or the protection seller default.5 In order to calculate the spread of CDS with CDR under our regime-switching first-passage-time model, we use the following discrete-time formula. Here, the recovery rate R of the notional amount is assumed to be constant. Then,
5
See Kim and Kim (2003) for the pricing of CDS with CDR.
cT ¼
ð1 RÞ
PN
n¼1 ½Q R ðt n Þ Q R;C ðt n Þ ðQ R ðt n1 Þ Q R;C ðt n1 ÞÞBð0; t n ÞDt ; PN n¼1 ½1 Q R ðt n Þ Q C ðt n Þ þ Q R;C ðt n ÞBð0; t n ÞDt
where Dt ¼ T=N, tn ¼ nDt, Q R ðtÞ (resp. Q C ðtÞ) is the default probability of the reference entity (resp. the counterparty) over a given horizon t under the risk-neutral probability measure P. Q R;C ðtÞ is the joint default probability of the reference entity and the counterparty over a given horizon t under P. Bð0; tÞ is a default-free discount bond price at time 0 with maturity t and principal amount 1. 5.2. Basket default swap In a BDS (sometimes called as a first-to-default swap), there are a number of different reference entities and reference obligations. The default correlation is the fundamental driver of BDS premium,
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a
1200
Spread (bp)
1000
Single CDS without CDR
– : σ b2 − σ B2
=
0.2,
— : σ
b 2−
σ B2
=
: σ
0.1,
b 2−
σ B2
=
0
800 600 400 200 2
b
6 Maturity (year)
8
10
500 400
Spread (bp)
4
Single CDS without CDR X2 : = 3 & Unstable K2
X2 = 4 & Unstable K2
— :
300 200 100
2
c
1750 1500
4
6 Maturity (year)
8
10
8
10
Xi = 3 for all cases, i =1,2 Ki
Basket Default Swap, — : Bullish market & Stable,
: Bullish market & Unstable : Bearish market & Unstable
– : Bearish market & Stable,
Spread (bp)
1250 1000 750 500 250 2
4
6 Maturity (year)
Fig. 8. The relation between default swap rate and maturity for various stabilities. (a) X 1 =K 1 ¼ X 2 =K 2 ¼ 3, q ¼ 0:4, R ¼ 0:4, rB2 ¼ 0:2, rb2 ¼ 0:5, kB ¼ 0:23, kb ¼ 1:73, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. (b) q ¼ 0:4, R ¼ 0:4, kB ¼ 0:23, kb ¼ 1:73, rB ¼ rb ¼ 0:05, g1 ¼ g2 ¼ 0:05, and rB2 ¼ 0:223, rb2 ¼ 0:327 for the stable case, rB2 ¼ 0:2, rb2 ¼ 0:5 for the unstable case. (c) The same parameters with (b). For the cases of (a) and (b), current market is bullish.
and hence joint default probabilities are necessary to obtain it. When the correlation between reference entities is non-zero, the joint default probability of two entities can be easily obtained under our model.6 The probability of the first default happening between times t n1 and t n ðn ¼ 1; 2; . . . ; N), when the default correlation between reference entities is considered, is
e 1;2;...;I ðtn Þ; e 1;2;...;I ðtn1 Þ Q Q
e 1;2;...;I ðtÞ is the probability of no earlier default by any of refwhere Q erence entities for the time interval ð0; tÞ and I is the number of reference entities. Similar to the case with a plain CDS, the following discrete equation can be used as an approximate formula for the spread of a BDS.
cT ¼
ð1 RÞ
PN
e
e
n¼1 ½ Q 1;2;...;I ðt n1 Þ Q 1;2;...;I ðt n ÞBð0; t n ÞDt PN e n¼1 Q 1;2;...;I ðt n ÞBð0; t n ÞDt
5.3. Implications 6 The joint default probability of multiple entities are needed, in order to determine the spread of BDS. However, the calculation of such probability for more than three entities can be executed well only after resolving some numerical problems, for example, the inaccuracy from cumulative numerical errors. The problems remain for future research.
We assume that there are two firms in this section. In determining the spread of CDS with CDR, reference entity (resp. counterparty) is assumed to be firm 2 (resp. firm 1). In addition, we assume that basket is composed of firm 1 and firm 2. The intensity parameters
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a
1200
Single CDS without CDR – : λB = 1.5, — : λB = 1,
: λB = 0.23
Spread (bp)
1000 800 600 400 200 2
4
6
8
10
8
10
8
10
Maturity (year)
b 1200
Single CDS with CDR – : λB = 1.5, —: λB = 1,
Spread (bp)
1000
: λB = 0.23
800 600 400 200 2
4
6 Maturity (year)
c
Basket Default Swap – : λB = 1.5, —: λB = 1,
1200
: λB = 0.23
Spread (bp)
1000 800 600 400 200 2
4
6 Maturity (year)
Fig. 9. The relation between default swap rate and maturity for various regime-switching intensities kB . Here, the parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 3, q ¼ 0:4, R ¼ 0:4, rB1 ¼ 0:2, rb1 ¼ 0:5, rB2 ¼ 0:3, rb2 ¼ 0:4, kb ¼ 1:73, rB ¼ rb ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bullish.
were empirically estimated by Ang and Bekaert (2002) and also used in Jang et al. (2007). The volatility of asset return for each firm has two values depending on market condition. In particular, we assume that the volatility of firm 1 is the lower than that of firm 2 in bull market, but the volatility of firm 1 is higher than that of firm 2 in bear market. In addition, we assume that the intensity kB switching from bull market to bear market is lower than the intensity kb switching from bear market to bull market, as already shown in Ang and Bekaert (2002). 5.3.1. Market condition Fig. 7 shows the term structures of CDS spreads depending on market condition. Three types of CDS are considered: single CDS without CDR, single CDS with CDR, and BDS with two reference entities. First, Fig. 7 says that default swap spread is higher in bearish market than in bullish market for both CDS and BDS cases. This result is consistent with the empirical result that, when market goes
down, the productivity of a firm declines and, therefore, the credit quality of the firm decreases. Second, relative spread change7 is higher for BDS than for single CDS. In addition, relative spread change is higher for single CDS without CDR than for single CDS with CDR. This result is due to CDR which may cause an early termination of the CDS contract. 5.3.2. Stability When market shifts from one regime to the other regime, a firm is said to be stable if it has the small difference of volatility between two regimes. Fig. 8 shows the relationship between single CDS (with firm 2 as the reference entity) spread and maturity. From Fig. 8a, we obtain that as the difference between rb2 and rB2 increases, CDS spread increases regardless of maturity. On the other hand, it is well known that, when firm F 1 has higher asset-to-liability 7 Relative spread change is defined as ðCDSb CDSB Þ=CDSB , where CDSb (resp. CDSB ) is spread under bear (resp. bull) market.
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a
Single CDS without CDR – : λb = 0.5, — : λb = 1,
1750
: λb = 1.73
Spread (bp)
1500 1250 1000 750 500 250 2
4
6
8
10
Maturity (year)
b
Single CDS with CDR – : λb = 0.5, — : λb = 1,
1750
Spread (bp)
1500
: λb = 1.73
1250 1000 750 500 250 2
4
6
8
10
8
10
Maturity (year)
c
Basket Default Swap – : λb = 0.5, — : λb = 1,
1750
: λb = 1.73
Spread (bp)
1500 1250 1000 750 500 250
2
4
6 Maturity (year)
Fig. 10. The relation between default swap rate and maturity for various regime-switching intensities kb . Here, the parameters are X 1 =K 1 ¼ X 2 =K 2 ¼ 3, q ¼ 0:4, R ¼ 0:4, rB1 ¼ 0:2, rb1 ¼ 0:5, rB2 ¼ 0:3, rb2 ¼ 0:4, kB ¼ 0:23, r B ¼ r b ¼ 0:05, and g1 ¼ g2 ¼ 0:05. Current market is bearish.
ratio than that of firm F 2 , the spread of default swap referencing firm F 1 is lower than that referencing firm F 2 under earlier existing firstpassage-time models. However, as seen in Fig. 8b, we find the fact that the gap between the spreads for two firms diminishes with maturity. The result analogues to this is also depicted in Fig. 8c. In this figure we find that the spreads in bear market are more sensitive to maturity than those in bull market. 5.3.3. Regime-switching intensity Fig. 9 shows the term structures of CDS spread for single CDS without CDR, single CDS with CDR, and BDS where kB changes. The higher the intensity kB shifting from bull market to bear market, the higher the spreads for both CDS without CDR and BDS. The effect of higher intensity on the spread increases firstly, but diminishes gradually with longer maturity. Also, single CDS spread increases with kB across maturity. However, for sufficiently long maturity,
the spread of CDS with CDR declines against kB as shown in Fig. 9b. This result stems from the fact that CDR increases as kB increases. Fig. 10 shows the term structure of CDS spread for single CDS without CDR, single CDS with CDR, and BDS where the intensity kb changes. Higher intensity shifting from bear market to bull market declines both average volatility of firm and default correlation. Single CDS spread declines against kb , since the lower volatility of asset return reduces the default probability of reference entity. Also, the spread of BDS declines against kb since the first-to-default probability declines. Two factors determine the first-to-default probability: One is individual default probability and the other is the default correlation determining joint default probability. When the intensity shifting from economic recession to economic expansion increases, the lower individual default probability decreases BDS spread and, at the same time, the lower default correlation increases BDS spread. But, because the former is more dominant
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M.A. Kim et al. / Journal of Banking & Finance 32 (2008) 2617–2627
than the latter in determining default swap spread, the spread of BDS declines against kb .
6. Conclusion We suggest a first-passage-time model that can explain default probability and default correlation dynamics under stochastic market condition. We add a Markov regime-switching market condition to a first-passage-time model, and consequently extend Zhou’s (2001) default correlation model to the model incorporating regime-switching market condition. Using this regime-switching first-passage-time model, we can show various relationship between default probability, default correlation, and market condition. First, firms having larger difference in volatilities between two regimes have higher default correlation. This result suggests that, in order to increase diversification effect, it is better to choose firms less sensitive to business cycle in investing a credit portfolio. Second, as the intensity shifting from bull market to bear market increases, both default probability and default correlation increase and vice versa. In fact, high possibility for market to go down may increase implied default probability and implied correlation, and this phenomenon must be reflected in pricing credit derivatives and managing credit portfolio risk. This relationship has not been involved in earlier structural models for multiple firms but our approach can solve this problem. We also suggest a valuation method for single CDS with(or without) CDR and BDS under the set-up. The numerical results show several implications: First, default swap spread is higher in economic recession than in economic expansion across maturity. Second, as the difference in volatilities between bear market and bull market increases, CDS spread increases regardless of maturity. Third, under earlier first-passage-time models, the spread of default swap referencing firm with higher asset-to-liability ratio is always lower. But, if the firm has larger difference in volatilities according to market condition, the spread of default swap may not be always lower in our regime-switching first-passage-time model. Fourth, the higher the intensity shifting from bull market to bear market, the higher the spread for both CDS without CDR and BDS. However, the spread of CDS with CDR may decline against the intensity when the default risk of counterparty is higher than that of reference entity under stochastic market condition.
We thank an anonymous referee for a helpful comment.
Appendix. Implementation of the system of the partial differential equations in Section 3 by finite difference method Eq. (1), which is the PDE for joint survival probability, can be rewritten by using the symbols p ¼ ln X 1 and q ¼ ln X 2 as follows:
B ouB 1 ou 1 2 ouB 1 B 2 o2 uB þ rB ðrB1 Þ2 þ r B rB2 þ ðr Þ 2 2 ot op oq 2 1 op2 ð3Þ
oU 1 oU 1 oU 1 2 o2 U 1 2 o2 U þ r þ r r21 þ r r22 þ r ot 2 op 2 oq 2 1 op2 2 2 oq2 o2 U ~ UÞ ¼ 0: þ kðU opoq
þ
iþ1;j i;j i1;j i;jþ1 i;j i;j1 1 2 U kþ1 2U kþ1 þ U kþ1 1 2 U kþ1 2U kþ1 þ U kþ1 r1 þ r2 2 2 2 2 ðDpÞ ðDqÞ
þ qr1 r2
i1;j1 U iþ1;jþ1 U i1;jþ1 U iþ1;j1 þ U kþ1 kþ1 kþ1 kþ1 4DpDq
~ i;j U i;j Þ ¼ 0: þ kðU kþ1 kþ1 In other words,
"
U i;j k
ðDpÞ2
2 1
r
Dt
#
Dt iþ1;jþ1 qr1 r2 U kþ1 4DpDq " # Dt Dt 1 2 Dt 1 2 iþ1;j qr1 r2 U iþ1;j1 þ r r r þ 1 U kþ1 kþ1 4DpDq 2Dp 2 1 ðDpÞ2 2 " # Dt 1 Dt 1 2 i;jþ1 Dt 1 2 U þ r r r r22 þ r 2 kþ1 2 Dq 2 2 Dq 2 2 ðDqÞ2 2 # " # Dt 1 2 i;j1 Dt 1 Dt 1 2 i1;j r2 U kþ1 r1 U kþ1 r r22 2Dp 2 ðDqÞ2 2 ðDpÞ2 2
¼ 1
Dt
ðDqÞ2
r Dtk U i;jkþ1 þ 2 2
Dt Dt i1;j1 ~ i;j : qr1 r2 U i1;jþ1 þ qr1 r2 U kþ1 þ DtkU kþ1 kþ1 4DpDq 4DpDq ð5Þ
In case of individual default probability, by using the variable change p ¼ ln X 1 we can write a generalized version of the equations of (2) as
oV 1 oV 1 2 o2 V ~ VÞ ¼ 0: þ kðV þ r r2 þ r ot 2 op 2 op2
ð6Þ
Since the discrete version of Eq. (6) is
iþ1 i1 iþ1 i1 V ikþ1 V ik V kþ1 V kþ1 2V ikþ1 þ V kþ1 1 1 V þ r r2 þ r2 kþ1 2 2 2 Dt 2Dp ðDpÞ ~ i V i Þ ¼ 0; þ kðV kþ1 kþ1
ð7Þ
Eq. (7) can be rearranged as
"
# Dt 1 2 1 Dt iþ1 2 ¼ r V kþ1 r r þ 2 Dp 2 2 ðDpÞ2 " # Dt 2 þ 1 r k D t V ikþ1 ðDpÞ2 " # Dt 1 2 1 Dt 2 ~i þ r V i1 r r þ kþ1 þ kDt V kþ1 : 2Dp 2 2 ðDpÞ2
ð8Þ
We can calculate the joint and individual survival probabilities by using the difference Eqs. (5) and (8). References
Eq. (3) can be generalized as follows by using representative func~ and ignoring the regime-state superscripts B and b: tions U; U
þ qr1 r2
iþ1;j i;jþ1 i;j i1;j U i;j U kþ1 U kþ1 U kþ1 U i;j1 1 1 kþ1 U k kþ1 þ r r21 þ r r22 2 2 Dt 2 Dp 2Dq
V ik
Acknowledgement
1 o2 uB o2 uB þ ðrB2 Þ2 2 þ qrB1 rB2 þ kB ðub uB Þ ¼ 0: 2 oq opoq
The discrete version of Eq. (4) is
ð4Þ
Ang, A., Bekaert, G., 2002. International asset allocation with regime shifts. Review of Financial Studies 15, 1137–1187. Black, F., Cox, J., 1976. Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance 31, 351–367. Bollen, N.P.B., Gray, S.F., Whaley, R.E., 2000. Regime switching in foreign exchange rates: Evidence from currency option prices. Journal of Econometrics 94, 239– 276. Campbell, J., 1991. A variance decomposition for stock returns. Economic Journal 101, 157–179. Campbell, J., Hentschel, L., 1992. No news is good news: An asymmetric model of changing volatility in stock returns. Journal of Financial Economics 31, 281–318. Das, S.R., Geng, G., 2004. Correlated default processes: A criterion-based copula approach. Working paper, Santa Clara University.
M.A. Kim et al. / Journal of Banking & Finance 32 (2008) 2617–2627 Das, S.R., Freed, L., Geng, G., Kapadia, N., 2006. Correlated default risk. Journal of Fixed Income 14, 7–32. Gauthier, L., 2002. Excursions height- and length-related stopping times, and application to finance. Advances in Applied Probability 34, 846–868. Giesecke, K., 2004. Credit risk modeling and valuation: An introduction. Working paper, Cornell University. Jang, B.-G., Koo, H.K., Liu, H., Loewenstein, M., 2007. Liquidity premia and transaction costs. Journal of Finance 62, 2329–2366. Kim, M.A., Kim, T.S., 2003. Credit default swap valuation with counterparty default risk and market risk. Journal of Risk 6, 49–80. Kim, M.A., Kim, T.S., 2005. Default correlation dynamics with business cycle and credit quality changes. Journal of Derivatives (Fall), 8–27. Lewellen, J., 2004. Predicting returns with financial ratios. Journal of Financial Economics 74, 209–235.
2627
Longstaff, F., Schwartz, E., 1995. A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 50, 789–819. Lucas, A., Klaassen, P., 2006. Discrete versus continuous state switching models for portfolio credit risk. Journal of Banking and Finance 30, 23–35. Merton, R.C., 1974. On the pricing of corporate debt: The risk structure of interest rates. Journal of finance 29, 449–470. Schwert, W., 1989. Why does stock market volatility change over time? Journal of Finance 44, 1115–1153. Shimko, D., Tejima, N., van Deventer, D., 1993. The pricing of risky debt when interest rates are stochastic. Journal of Fixed Income 3, 58–65. So, M.K.P., Lam, K., Li, W.K., 1998. A stochastic volatility model with Markov switching. Journal of Business and Economic Statistics 16, 244–253. Zhou, C., 2001. An analysis of default correlations and multiple defaults. Review of Financial Studies 14, 555–576.