modeling credit spreads under multifactor stochastic volatility
TRANSCRIPT
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ARTICLE IN PRESSG ModelRFE-23; No. of Pages 6
The Spanish Review of Financial Economics xxx (2013) xxx–xxx
The Spanish Review of Financial Economics
www.elsev ier .es /s r fe
rticle
odeling credit spreads under multifactor stochastic volatility�
acinto Marabel Romoa,b
BBVA, Vía de los Poblados s/n, 28033 Madrid, SpainDepartment of Management Sciences, University of Alcalá (UAH), Plaza de la Victoria 2, 28802 Alcalá de Henares, Madrid, Spain
a r t i c l e i n f o
rticle history:eceived 10 April 2013ccepted 26 June 2013vailable online xxx
EL classification:12133132
a b s t r a c t
The empirical tests of traditional structural models of credit risk tend to indicate that such models havebeen unsuccessful in the modeling of credit spreads. To address these negative findings some authorsintroduce single-factor stochastic volatility specifications and/or jumps.
In the yield curve literature it is widely accepted that one-factor is not sufficient to capture the timevariation and cross-sectional variation in the term structure. This article introduces a two-factor stochas-tic volatility specification within the structural model of credit risk. One of the factors determines thecorrelation between short-term firms’ assets returns and variance, whereas the other factor determinesthe correlation between long-term returns and variance. The numerical tests reveal how the introduc-tion of two volatility factors can generate a wide range of combinations associated with short-term andlong-term patters corresponding to credit spreads. In this sense, multi-factor stochastic volatility speci-
33eywords:redit spreadsredit ratingtochastic volatility
fications provide more flexibility than single-factor models to capture a wide range of shapes associatedwith the term structure of credit spreads consistent with the empirical evidence.
© 2013 Asociación Espanola de Finanzas. Published by Elsevier España, S.L. All rights reserved.
ultifactortructural models
. Introduction
Structural models of credit risk formulate explicit assumptionsbout the dynamics of a firm’s assets, its capital structure and itsebt. In this case, the default event happens if firm’s assets areot sufficient to pay the debt and corporate liabilities can be con-idered as contingent claims on the firm’s assets. The credit riskiterature based on the structural approach begins with the studyf Merton (1974), who applies the option pricing theory developedy Black and Scholes (1973) to model credit spreads, where theredit spread is defined as the difference between the yield of aorporate bond and the associated yield on Treasury bonds withhe closest matching maturity.
The empirical tests of traditional structural models of creditisk tend to indicate that such models have been unsuccessful inhe modeling of credit spreads. In particular, Jones et al. (1984)nd Huang and Huang (2003), among others, show that predictedredit spreads are far below observed ones. To address these neg-
Please cite this article in press as: Marabel Romo, J. Modeling credit
Econ. (2013). http://dx.doi.org/10.1016/j.srfe.2013.06.002
tive findings some authors, such as Zhang et al. (2009), introduce single-factor stochastic volatility model with jumps within theerton (1974) framework. They show that their model improves
� The content of this paper represents the author’s personal opinion and does noteflect the views of BBVA.
E-mail addresses: [email protected], [email protected]
173-1268/$ – see front matter © 2013 Asociación Espanola de Finanzas. Published by Elttp://dx.doi.org/10.1016/j.srfe.2013.06.002
the match between predicted and observed credit spreads, espe-cially for investment-grade companies. Unfortunately, the resultsare less satisfactory for low investment grade and speculativegrade. In theory, jumps can help to match the observed creditspread levels for investment grade bonds and short maturities. Butempirical evidence is rather inconclusive. Collin-Dufresne et al.(2003) have found that only a small fraction of observed creditspreads of aggregate portfolios can be explained by jump risk. Onthe other hand, Cremers et al. (2008) suggest that the addition ofjumps and jump risk premia brings predicted yield spread levelsmuch closer to observed ones.
Importantly, the study of Zhang et al. (2009) considers a sin-gle stochastic volatility factor as in Heston (1993). Within theequity option valuation context, Christoffersen et al. (2009) extendthe original Heston (1993) framework to generate a two-factorstochastic volatility model built upon the square root process. Thisarticle introduces a two-factor stochastic volatility specificationwithin the structural Merton (1974) framework. The advantageis that a two-factor model provides more flexibility to model thevolatility term structure. In this sense, one of the factors deter-mines the correlation between short-term firms’ assets returnsand variance, whereas the other factor determines the correlation
spreads under multifactor stochastic volatility. Span Rev Financ
between long-term returns and variance. Hence, the two-factorspecification is able to generate more flexible credit spread termstructures to match the observed term structures associated withcredit spreads.
sevier España, S.L. All rights reserved.
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ARTICLERFE-23; No. of Pages 6
J. Marabel Romo / The Spanish Review
The rest of the paper proceeds as follows. Section 2 presents theain features of the two-factor stochastic volatility specification
nd develops semi-closed-form solutions for the price of creditpreads and default probabilities. Section 3 provides a numericalnalysis which shows that the model is able to generate creditpread term structures consistent with the empirical evidence.his section also offers a sensitivity analysis of credit spreads withespect to the model parameters. Finally, Section 4 offers conclud-ng remarks.
. A structural model of credit risk under a two-factortochastic volatility framework
Let us assume the same market framework as in Merton (1974).ithin this framework, firms issue a zero coupon bond with a
romised payment B at maturity t = T. In this case, default occursnly at maturity with debt face value as default boundary. In thevent of default, the absolute priority rule prevails.
Let{
At ∈ R t≥0}
be the price process of the firm’s assets and lets denote by Yt = ln At the log-return vector. For simplicity, I assumehat the continuously compounded risk-free rate r and asset payoutatio q are constant. Let � denote the probability measure definedn a probability space
(�, F, �
)such that asset prices expressed in
erms of the current account are martingales. We denote this prob-bility measure as the risk-neutral measure. As in Christoffersent al. (2009), I consider a two-factor specification for the variancerocess and I assume the following dynamics for the return processt under �:
Yt =[
r − q − 12
2∑i=1
vit
]dt +
2∑i=1
√vitdZit (1)
ith:
vit = �i
(�i − vit
)dt + �i
√vitdWit
here �i represents the long-term mean corresponding to thenstantaneous variance factor i (for i = 1, 2), �i denotes the speed of
ean reversion and, finally, �i represents the volatility of the vari-nce factor i. For analytical convenience, let us rewrite the previousquation as follows:
vit = (ai − bivit) dt + �i√
vitdWit (2)
here bi = �i and ai = �i�i. In Eqs. (1) and (2) Zit and Wit are Wienerrocesses such that:
ZitdWjt ={
�idt for i = j
0 for i /= j
n the other hand, Z1t and Z2t are uncorrelated. In addition, W1t and2t are also uncorrelated. The single-factor Heston (1993) model
an be obtained as a particular case of the two-factor specificationonsidering only one volatility factor. The multifactor specificationf Eqs. (1) and (2) accounts for a richer variance–covariance struc-ure. In particular, the conditional variance of the return process is:
t:= 1dt
Var(dYt) =2∑
i=1
vit
hereas, as shown by Christoffersen et al. (2009), the correlationetween the asset return and the variance process is:
Please cite this article in press as: Marabel Romo, J. Modeling credit
Econ. (2013). http://dx.doi.org/10.1016/j.srfe.2013.06.002
AtVt :=Corr (dYt, dVt) = �1�1v1t + �2�2v2t√�2
1v1t + �22v2t
√v1t + v2t
PRESSancial Economics xxx (2013) xxx–xxx
Importantly, two-factor specification, unlike the single-factorstochastic volatility models, allows for stochastic correlationbetween the asset return and the variance process. Anotheradvantage of the two-factor model with respect to single-factorspecifications is that it provides more flexibility to model thevolatility term structure. In this sense, the two-factor model is ableto generate more flexible patterns corresponding to the term struc-ture of credit spreads.
2.1. Pricing credit spreads and default probabilities
In this section I follow the methodology of Lewis (2000) and daFonseca et al. (2007) to calculate option prices efficiently in terms ofthe generalized Fourier transform associated with the payoff func-tion and with the asset return. In this sense, let us consider a genericpayoff on the terminal value of the underlying asset, at time t = T,under the risk-neutral probability measure w(YT ). From the Funda-mental Theorem of Asset Pricing we have that the time t = 0 priceof this option, denoted OP0, is given by:
OP0 = e−rT E� [w (YT )] = e−rT
∫R
w (YT ) ıT (YT ) dYT (3)
where ıT (YT ) is the risk-neutral density function of YT. The Laplacetransform of the asset return is defined as:
� (; Y0, T) :=E� [eYT ] =∫R
eYT ıT (YT ) dYT ∈ R
On the other hand, the Fourier transform corresponding to w(YT )is given by:
w (z) =∫R
eizYT w (YT ) dYT z ∈ C
with
w (YT ) = 12
∫�
e−izYT w (z) dz
where � ⊂ C is the admissible integration domain in the complexplain corresponding to the generalized Fourier transform associ-ated with the payoff function w (z) and where i2 = −1. Substitutingprevious expression in Eq. (3) yields:
OP0 = e−rT
2
∫�
� ( = −iz; Y0, T) w (z) dz (4)
where we have used the Fubini theorem. From the previous equa-tion, to obtain a semi-closed-form solution for the option price wehave to calculate the Laplace transform of the asset return, as wellas the Fourier transform associated with the payoff function.
2.1.1. The Laplace transform of the asset returnMarabel Romo (2013) shows that, under the risk-neutral mea-
sure �, the Laplace transform associated with the two-factorspecification �( ; Y0,T) is given by:
�(
; Y0,T)
= e
B(,T)+2∑
i=1
Mi(,T)vi0+Y0
(5)
where Mi (, T) for i = 1, 2, is given by:
spreads under multifactor stochastic volatility. Span Rev Financ
Mi (, T) = bi − �i�i + ˛i
�2i
[1 − eT˛i
1 − ˇieT˛i
]
IN PRESSG ModelS
of Financial Economics xxx (2013) xxx–xxx 3
w
2p
ittzcsD
D
Hawcp
P
wpa
w
Ui
w
w
�
TcalIcpp
e
D
T
C
1 1.5 2 2.5 3 3.5 4 4.5 540
60
80
100
120
140
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
Panel A: Credit spreads
Specification ISpecification II
1 1. 5 2 2. 5 3 3. 5 4 4. 5 50
2
4
6
8
10
12
Maturity
Def
ault
prob
abili
ty (
%)
Panel B: Default probabilities
Specification ISpecification II
Fig. 1. Credit spreads term structure and default probabilities generated under the
ARTICLERFE-23; No. of Pages 6
J. Marabel Romo / The Spanish Review
ith:
˛i =[(�i�i − bi)
2 − 2d1 () �2i
]1/2
ˇi = bi − �i�i + ˛i
bi − �i�i − ˛i
d1 () =
2( − 1)
On the other hand, B (, T) can be expressed as:
B (, T) = d0 () T +2∑
i=1
ai
�2i
[(bi − �i�i + ˛i) T − 2 ln
(1 − ˇieT˛i
1 − ˇi
)]d0 () = (r − q)
.1.2. The pricing formula for credit spreads and defaultrobabilities
Let St denote the time t equity price corresponding to the firm. Its well known that, within Merton (1974) framework, it is possibleo express the time t = 0 equity price as a European call option onhe firm’s asset with maturity t = T equal to the expiration of theero coupon bond with promised payment B, where this paymentoincides with the option strike. From the put-call parity, it is pos-ible to express the time t = 0 debt value corresponding the firm,0, as:
0 = Be−rT − e−rT E�
[(B − AT )+]
= Be−rT − P (A0, T, B)
ence, the debt value can be expressed as the difference between zero coupon bond and a European put option on the firm’s assetsith strike equal to the debt payment at maturity P(A0, T, B). In this
ase, we can use the generic payoff Eq. (4) to express the time t = 0rice of a European put with strike B and maturity t = T as follows:
(A0, T, B) = e−rT
2
∫�P
� ( = −iz; Y0, T) wP (z) dz (6)
here �P ⊂ C is the admissible integration domain in the complexlain corresponding to the generalized Fourier transform associ-ted with the payoff function
P (YT ) = (B − AT )+ = (eln B − eYT )+
nder the assumption that Im (z) < 0, where Im (c) denotes themaginary part of c ∈ C, the Fourier transform is given by:
P (z) = eln B(1+iz)
iz (1 + iz)(7)
ith the corresponding integrability domain:
P ={
z ∈ C : Im (z) < 0}
herefore, we can combine Eqs. (5), (6) and (7) to obtain a semi-losed form solution for the price of a European put on the firm’sssets. Note that the integral of Eq. (6) is an integral along a straightine in the complex plane. This line can lie anywhere in the regionm (z) < 0. Since we know explicitly the marginal Laplace transformorresponding to the firms’ assets, it is also possible to obtain therice of European options using the fast Fourier transform methodroposed by Carr and Madan (1999).
Taking into account the previous expressions, it is possible toxpress the time t = 0 debt value:
0 = e−rT
[B − 1
2
∫�P
� (� = −iz; Y0, T) wP (z) dz
](8)
Please cite this article in press as: Marabel Romo, J. Modeling credit
Econ. (2013). http://dx.doi.org/10.1016/j.srfe.2013.06.002
herefore, the time t = 0 credit spread, CS0, is given by:
S0 = − 1T
ln[
D0
B
]− r (9)
parametric specifications of Table 1. The figure has been generated using the follow-ing parameter values: r = 0.05, q = 0.02, A0 = 1 and finally, B = 0.43 for specification Iand B = 0.48 under specification II.
where D0 is given by Eq. (8).Within this framework, it is assumed that the firm defaults, at
maturity t = T, if the firm’s asset value AT is not enough to pay backthe face value of the debt B to bondholders. As a consequence, theprobability, at time t = 0, of the firm defaulting at T, PdT
0, is given by:
PdT0:=E�
[1(YT <ln B)
]where 1(.) is the Heaviside step function or unit step function. Inthis case, we can express the default probability associated withthe firm as follows:
PdT0 = 1
2
∫�d
� (� = −iz; Y0, T) wd (z) dz (10)
where �d ⊂ C is the admissible integration domain in the complexplain corresponding to the generalized Fourier transform associ-ated with the payoff function wd (z), where the payoff functionis:
wd (YT ) = 1(YT <ln B)
and, under the assumption that the assumption that Im (z) < 0, theFourier transform associated with the previous payoff function isgiven by:
wd (z) = eiz ln B
iz(11)
with the corresponding integrability domain:
�d ={
z ∈ C : Im (z) < 0}
(12)
Hence, we can combine Eqs. (10), (11) and (12) to obtain a semi-closed-form solution corresponding to the default probability.
3. Numerical illustration
spreads under multifactor stochastic volatility. Span Rev Financ
This section offers a numerical analysis of the ability of the two-factor specification to generate term structures corresponding tocredit spreads consistent with empirical evidence. It also provides
ARTICLE IN PRESSG ModelSRFE-23; No. of Pages 6
4 J. Marabel Romo / The Spanish Review of Financial Economics xxx (2013) xxx–xxx
Table 1Parameters specifications in the two-factor stochastic volatility model for firm’s assets returns.
Parameter Specification I Specification II Parameter Specification I Specification II
�1 1.2017 1.5141 �2 0.3605 0.4542�1 0.0524 0.0660 �2 0.0157 0.0198�1 0.8968 1.1300 �2 0.2690 0.3390�1 −0.5590 −0.7043 �2 −0.1677 −0.2113v10 0.0581 0.0732 v20 0.0174 0.0220
1 2 3 4 540
50
60
70
80
90
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
Panel A: Effect of κ1 Panel B: Effect of κ2
Panel C: Effect of θ1
Panel E: Effect of σ1 Panel E: Effect of σ2
Panel D: Effect of θ2
Initial
Change
1 2 3 4 540
50
60
70
80
90
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
Initial
Change
1 2 3 4 540
50
60
70
80
90
100
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
Initial
Change
1 2 3 4 540
50
60
70
80
90
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
Initial
Change
40
50
60
70
80
90
Cre
dit s
prea
d (b
asis
poi
nts)
Initial
Change
40
50
60
70
80
90
Cre
dit s
prea
d (b
asis
poi
nts)
Initial
Change
ion I o
ap
3
sccrtsatoc
abilities. The credit spreads are calculated1 using the analyticalexpression of Eq. (9), while the default probabilities are calculatedusing Eq. (10).
1 I have performed several numerical simulation tests to verify that the optionprices obtained from the semi-closed-form solution presented in this article match
1 2 3 4 5
Maturity
Fig. 2. Sensitivity of credit spreads generated by specificat
study of the credit spread sensitivities with respect to the modelarameters.
.1. The term structure of credit spreads
To analyze the flexibility of the two-factor stochastic volatilitypecification to generate different patterns corresponding to theredit spread term structure, I consider the two parametric specifi-ations of Table 1. Regarding the asset payout ratio q, the risk-freeate r, the initial firm’s asset value A0 and the debt face value B, Iake these parameters from Zhang et al. (2009). In this sense, I con-ider r = 0.05, q = 0.02 and A0 = 1. With regard to the debt face value
Please cite this article in press as: Marabel Romo, J. Modeling credit
Econ. (2013). http://dx.doi.org/10.1016/j.srfe.2013.06.002
ssociated with the parametric specification I of Table 1, I considerhe debt face value corresponding to rating category A in the studyf Zhang et al. (2009), whereas for the parametric specification II, Ionsider the debt face value corresponding to rating category BBB
1 2 3 4 5
Maturity
f Table 1 together with r = 0.05, q = 0.02, A0 = 1 and B = 0.43.
of Zhang et al. (2009). Hence, I set B = 0.43 for specification I andB = 0.48 for specification II.
Panel A of Fig. 1 shows the generated credit spreads correspond-ing to each rating category for different maturities (expressed inyears), whereas panel B displays the corresponding default prob-
spreads under multifactor stochastic volatility. Span Rev Financ
perfectly with those produced with Monte Carlo simulations. Regarding the MonteCarlo specification, I consider daily time steps and 50,000 trials and I implement thequadratic exponential scheme as described in Andersen (2008). Moreover, I haveimplemented the pricing equations through the fast Fourier transform proposed byCarr and Madan (1999) obtaining the same results.
ARTICLE IN PRESSG ModelSRFE-23; No. of Pages 6
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1 1.5 2 2.5 3 3.5 4 4.5 545
50
55
60
65
70
75
80
85
90
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
Panel A: Effect of ρ1 Panel B: Effect of ρ2
Panel C: Effect of v10 Panel D: Effect of v20
InitialChange
1 1.5 2 2.5 3 3.5 4 4.5 545
50
55
60
65
70
75
80
85
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
InitialChange
1 1.5 2 2.5 3 3.5 4 4.5 545
50
55
60
65
70
75
80
85
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
InitialChange
1 1.5 2 2.5 3 3.5 4 4.5 545
50
55
60
65
70
75
80
85
Maturity
Cre
dit s
prea
d (b
asis
poi
nts)
InitialChange
ion I o
masfisatcZpitcttaatttlwtits
Fig. 3. Sensitivity of credit spreads generated by specificat
The credit spread term structure generated by the two-factorodel for specification I is of the same order of magnitude
nd it has a similar shape to the average credit spread termtructure displayed by Zhang et al. (2009) in their sample ofrms associated with the A rating category. Similarly, the termtructure of credit spreads generated by the two-factor modelssociated with specification II, displayed in Fig. 1, is similaro the credit spread term structure corresponding to firms ofredit rating BBB in the sample of companies considered byhang et al. (2009). Note that the difference between the defaultrobabilities associated with both parametric specifications is
ncreasing with the maturity. This effect is also consistent withhe evidence presented by Zhang et al. (2009) using histori-al data on default probabilities. The examples of Fig. 1 showhat the two-factor stochastic model is able to generate pat-erns corresponding to the credit spread term structure, as wells default probabilities consistent with empirical evidence. Thedvantage of this model with respect to single-factor specifica-ions, such as the one used by Zhang et al. (2009), is that awo-factor model provides more flexibility to capture the volatilityerm structure since one of the factors determines the corre-ation between short-term firms’ assets returns and variance,
hereas the other factor determines the correlation between long-
Please cite this article in press as: Marabel Romo, J. Modeling credit
Econ. (2013). http://dx.doi.org/10.1016/j.srfe.2013.06.002
erm returns and variance. Hence, the two-factor specifications able to generate more flexible credit spread term structureso match the observed term structures associated with creditpreads.
f Table 1 together with r = 0.05, q = 0.02, A0 = 1 and B = 0.43.
3.2. Sensitivity analysis
As said previously, the two-factor stochastic volatility modelprovides more flexibility than single-factor specifications tomodel the volatility term structure and, hence, the credit spreadterm structure. This section considers the effects of the modelparameters on the credit spread term structure associated withspecification I of Table 1. In this sense, Fig. 2 provides the creditspread sensitivity with respect to the mean reversion parameters�1 and �2, the long-term variance factors �1 and �2, and, finally, thevolatility of variance for each factor �1 and �2. On the other hand,Fig. 3 displays the sensitivities associated with the correlation coef-ficients �1 and �2, as well as the initial levels corresponding to eachvariance factor v10 and v20. The line initial shows the credit spreadterm structure associated with parametric specification I of Table 1,whereas the line change represents the credit spread term struc-ture obtained when we multiply the corresponding parameter bythe factor 1.25.
We can see from Fig. 2 that an increase in the mean reversionparameters reduces credit spreads. But in the case of �1 the decreaseof credit spreads is almost parallel for all maturities, whereas thechange of �2 only affects long-term spreads. Regarding the meanreversion parameters, both �1 and �2 increase the credit spread,
spreads under multifactor stochastic volatility. Span Rev Financ
especially in the long-term, but the effect of �1 is more pronounced.With regard to the volatilities of the variance factors, �1 increasesabove all the short-term credit spreads, while �2 has impact espe-cially in the long-term spreads. Higher correlation coefficients (in
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ARTICLERFE-23; No. of Pages 6
J. Marabel Romo / The Spanish Review
bsolute value) and higher initial volatility levels for the varianceactors imply higher credit spreads. But, as in the case of the meaneversion levels, the effect of the parameters associated with factor
on the term structure of credit spreads is almost parallel, whereashe second factor affects especially the long-term.
These numerical examples reveal how the introduction of twoolatility factors can generate a wide range of combinations associ-ted with short-term and long-term patters corresponding to creditpreads. In this sense, multifactor stochastic volatility specifica-ions provide more flexibility than single-factor models to capture
wide range of shapes associated with the term structure of creditpreads.
. Conclusion
The empirical tests of traditional structural models of creditisk tend to indicate that such models have been unsuccessful inhe modeling of credit spreads. To address these negative findingsome authors, such as Zhang et al. (2009), introduce a single-factortochastic volatility specification within the Merton (1974) frame-ork. They show that their model improves the match betweenredicted and observed credit spreads, especially for investment-rade companies.
In the yield curve literature the use of multifactor models ofhe short rate is widespread. In fact, it is widely accepted in theiterature that one-factor is not sufficient to capture the time vari-tion and cross-sectional variation in the term structure. In thisense, it is perhaps somewhat surprising that multifactor modelsave not yet become more popular in the credit default litera-ure. To fill this gap, this article introduces a two-factor stochasticolatility specification within the structural Merton (1974) model.he advantage is that a two-factor model provides more flexi-ility to model the volatility term structure. In this sense, onef the factors determines the correlation between short-term
Please cite this article in press as: Marabel Romo, J. Modeling credit
Econ. (2013). http://dx.doi.org/10.1016/j.srfe.2013.06.002
rms’ assets returns and variance, whereas the other factor deter-ines the correlation between long-term returns and variance.
he numerical tests reveal how the introduction of two volatil-ty factors can generate a wide range of combinations associated
PRESSancial Economics xxx (2013) xxx–xxx
with short-term and long-term patters corresponding to creditspreads. In this sense, multifactor stochastic volatility specifica-tions provide more flexibility than single-factor models to capturea wide range of shapes associated with the term structure ofcredit spreads consistent with the empirical evidence. Future workwill be devoted to the calibration of this model to market creditspreads.
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