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Price discovery in the markets for credit risk: a Markov switching approach

  • Thomas Dimpfl EMAIL logo and Franziska J. Peter


We examine price discovery in the Credit Default Swap and corporate bond market. Using a Markov switching framework enables us to analyze the dynamic behavior of the information shares during tranquil and crisis periods. The results show that price discovery takes place mostly on the CDS market. The importance of the CDS market even increases during the more volatile crisis periods. According to a cross sectional analysis liquidity is the main determinant of a market’s contribution to price discovery. During the crisis period, however, we also find a positive link between leverage and CDS market information shares. Overall the results indicate that price discovery measures and their determinants change during tranquil and crisis periods, which emphasizes the importance of more flexible frameworks, such as Markov switching models.

JEL: C14; G15

Corresponding author: Thomas Dimpfl, University of Tübingen, Department of Statistics, Econometrics, and Empirical Economics, Mohlstr. 36, 72074 Tübingen, Germany, Phone: +49 7071 29 76417, Fax: +49 7071 29 5546, e-mail:


We are indebted to Irina Dyshko who did an excellent job when implementing a first version of the code during her master thesis. We also thank Joachim Grammig, the editor Bruce Mizrach, and an anonymous referee for helpful comments. This research project is funded by the German Research Foundation (DFG) through grant GR 2288-3, data are provided through the DFG SFB 649 “Economic Risk”.


A Construction of the likelihood function

Denote the vector of price series by yt . Conditional on the unobserved state variable st , the density function of yt is known. As the innovations are assumed to be conditionally normally distributed, the conditional density of yt is given by

(A.1)f(yt|st=j,Yt1)=(2π)n2det(j)12exp(12utj1ut) (A.1)

for j=1, 2 and Yt–1={y1, …, yt–1}. Multiplying the conditional density in Equation (A.1) with the marginal density of st yields the joint density of yt and st

(A.2)f(yt,st=j|Yt1)=Pr(st=j|Yt1)f(yt|st=j,Yt1) (A.2)

for j=1, 2. Summing over all regimes allows to integrate out st , which results in the unconditional density of yt :

(A.3)f(yt|Yt1)=j=12f(yt,st=j|Yt1). (A.3)

This density corresponds to the average of the conditional densities weighted by the probability of the particular regime. To compute the weighting factors, we have to account for the Markov structure of st . In time t, the probability of regime j can be calculated as Pr(st =j|Yt–1)=Pr(st =j|st–1=i)×Pr(st–1=j|Yt–1), where the probabilities in the first term on the right hand side are summarized in the transition matrix P=[p111p221p11p22] with pij =Pr(st =j|st–1=i) ∀i, j∈1, 2.

Using Bayes’ theorem, the probability Pr(st =j|Yt ) can be filtered out, which together with the transition matrix P delivers Pr(st+1=j|Yt ). Iterating these steps until the last observation generates f(yt |Yt–1) for each point in time. Finally, the log-likelihood function is given by

(A.4)logLT=t=1Tlogf(yt|Yt1). (A.4)

It is maximised with respect to W, Ψ, p11, and p22.

The main difficulty of this optimization approach concerns the search of appropriate initial values. Due to the complexity of the log-likelihood function, finding the global peak proves to be challenging. In order to verify the maximization results and to find a more reliable method in terms of starting values, we recommend to apply the EM-algorithm.

B The EM-algorithm

In this study, the approach described by Herwartz and Lütkepohl (2014) is implemented. It represents the EM-algorithm enhanced by a Baum-Lindgren-Hamilton-Kim (BLHK) filter, which is explained by Krolzig (1997). Contrary to the approach of Herwartz and Lütkepohl (2014), we do not compute the residual series within the EM-algorithm, but estimate the VECM model in Equation (1) and subsequently use the VECM residuals in the optimization of the log-likelihood.

Let P denote the transition matrix, ι is a 2×1 vector of ones, and


⊙ stands for element-wise multiplication,

⊘ stands for element-wise division and

⊗ for the Kronecker product.

  1. Initialization of the starting values:

    P(0),ξ0|0,W0 and Ψ(0).

  2. Expectation step:

    The expectation step of the EM-Algorithm begins with the filtering or a “forward recursion

    ξt|t=ηtPξt1|t1ι(ηtPξt1|t1) for t=1,,T,




    f(yt|st=m,Yt1)=(2π)K2det(m)12exp{12utm1ut}  for m=1,2.

    Then, the filtered probabilities are smoothed by means of “backward recursion”:

    ξt|T=(P(ξt+1|TPξt|t))ξt|t   for t=T1,,0.

  3. Maximization step:

    Along the maximization step, the transition matrix is estimated with the Hidden Markov Chain formula (Krolzig 1997)



    ξt|T(2)=vec(P)[(ξt+1|TPξt|t)ξt|t]   for t=0,,T1.

    Finally, the values of W and Ψ, which maximize the expected log-likelihood function, have to be found. The expected log-likelihood is given by


    Writing out the regimes yields


The likelihood function is then maximized subject to constraints on the W and Ψ matrices discussed in Section 2. In addition, a lower bound of 0.001 is imposed on the determinant of each covariance matrix. The estimates become the initial values for the next iteration:

P^=P(0),ξ0|T=ξ0|0,W^=W0, andΨ^=Ψ(0).

Steps 2 and 3 are iterated until convergence is achieved, e.g. until


Unfortunately, using numerical optimization does not guarantee monotonic convergence of the expected log-likelihood Schaefer (1997). This is the case in our study as shown by Figure B.1. It shows that using numerical optimization leads to non-monotonic convergence in our empirical application. Consequently we verify our results by applying both approaches, direct maximization of the loglikelihood function an the EM-algorithm outlined above.

Figure B.1: Convergence of the log-likelihood function.The graphic illustrates the convergence of the log-likelihood function for the estimation of Allianz.
Figure B.1:

Convergence of the log-likelihood function.

The graphic illustrates the convergence of the log-likelihood function for the estimation of Allianz.

Standard errors of the estimates are obtained by means of a parametric bootstrap. This bootstrap is a slightly adapted version of the procedure described in Grammig and Peter (2013). It proves to be convenient, bearing in mind the two-step estimation structure.

Estimation is done in Gauss and all program codes are available upon request.


Alexander, C., and A. Kaeck. 2008. “Regime Dependent Determinants of Credit Default Swap Spreads.” Journal of Banking & Finance 32: 1008–1021.10.1016/j.jbankfin.2007.08.002Search in Google Scholar

Blanco, R., S. Brennan, and I. W. Marsh. 2005. “An Empirical Analysis of the Dynamic Relation Between Investment-grade Nonds and Credit Default Swaps.” Journal of Finance 60: 2255–2281.10.1111/j.1540-6261.2005.00798.xSearch in Google Scholar

Booth, G., J.-C. Lin, T. Martikainen, and Y. Tse. 2002. “Trading and Pricing in Upstairs and Downstairs Stock Markets.” Review of Financial Studies 15: 1111–1135.10.1093/rfs/15.4.1111Search in Google Scholar

Chen, H., P. M. S. Choi, and Y. Hong. 2013. “How Smooth is Price Discovery? Evidence from Cross-listed Stock Trading.” Journal of International Money and Finance 32: 668–699.10.1016/j.jimonfin.2012.06.005Search in Google Scholar

Coro, F., A. Dufour, and S. Varotto. 2013. “Credit and Liquidity Components of Corporate CDS Spreads.” Journal of Banking & Finance 37: 5511–5525.10.1016/j.jbankfin.2013.07.010Search in Google Scholar

Davidson, R., and J. G. MacKinnon. 2000. Econometric Theory and Methods. Oxford: Oxford University Press.Search in Google Scholar

Dimpfl, T., and F. J. Peter. 2013. “Using Transfer Entropy to Measure Information Flows Between Financial Markets.” Studies in Nonlinear Dynamics and Econometrics 17: 85–102.10.1515/snde-2012-0044Search in Google Scholar

Doetz, N. 2007. “Time-varying Contributions by the Corporate Bond and CDS Markets to Credit Risk Price Discovery,” Discussion Paper Series 2: Banking and Financial Studies.10.2139/ssrn.2793993Search in Google Scholar

Grammig, J., and F. J. Peter. 2013. “Telltale Tails: A New Approach to Estimate Unique Market Information Shares.” Journal of Financial and Quantitative Analysis 48: 459–488.10.1017/S0022109013000215Search in Google Scholar

Hamilton, J. 1994. Time Series Analysis. Princeton, New Jersey: Princeton University Press.10.1515/9780691218632Search in Google Scholar

Hasbrouck, J. 1995. “One Security, Many Markets: Determining the Contributions to Price Discovery.” Journal of Finance 50: 1175–1199.10.1111/j.1540-6261.1995.tb04054.xSearch in Google Scholar

Herwartz, H., and H. Lütkepohl. 2014. “Structural Vector Autoregressions with Markov Switching: Combining Conventional with Statistical Identification of Shocks.” Journal of Econometrics 183: 104–116.10.1016/j.jeconom.2014.06.012Search in Google Scholar

Hupperets, E. C., and A. J. Menkveld. 2002. “Intraday Analysis of Market Integration: Dutch Blue Chips Traded in Amsterdam and New York.” Journal of Financial Markets 5: 57–82.10.1016/S1386-4181(01)00019-2Search in Google Scholar

Johansen, S. 1995. Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press.10.1093/0198774508.001.0001Search in Google Scholar

Kehrle, K., and F. J. Peter. 2013. “Who Moves First? An Intensity-based Measure for Information Flows Between Stock Exchanges.” Journal of Banking & Finance 37: 1629–1642.10.1016/j.jbankfin.2012.12.011Search in Google Scholar

Krolzig, H.-M. 1997. Markov-switching Vector Autoregressions: Modelling, Statistical Inference, and Application to Business Cycle Analysis. Berlin: Springer-Verlag.10.1007/978-3-642-51684-9Search in Google Scholar

Lanne, M., and H. Lütkepohl. 2010. “Structural Vector Autoregressions with Nonnormal Residuals.” Journal of Business & Economic Statistics 28: 159–168.10.1198/jbes.2009.06003Search in Google Scholar

Lanne, M., H. Lütkepohl, and K. Maciejowska. 2010. “Structural Vector Autoregressions with Markov Switching.” Journal of Economic Dynamcis and Control 34: 121–131.10.1016/j.jedc.2009.08.002Search in Google Scholar

Leppin, J. S., and S. Reitz. 2014. “The Role of a Changing Market Environment for Credit Default Swap Pricing.” FinMaP-Working Papers.Search in Google Scholar

Longin, F., and B. Solnik. 2001. “Extreme Correlation of International Equity Markets.” Journal of Finance 56: 649–676.10.1111/0022-1082.00340Search in Google Scholar

Rigobon, R. 2003. “Identification Through Heteroskedasticity.” Review of Economics and Statistics 85: 777–792.10.1162/003465303772815727Search in Google Scholar

Schaefer, J. 1997. Analysis of Incomplete Multivariate Data. New York: Chapman and Hall.10.1201/9781439821862Search in Google Scholar

Yan, B., and E. Zivot. 2010. “A Structural Analysis of Price Discovery Measures.” Journal of Financial Markets 13: 1–19.10.1016/j.finmar.2009.09.003Search in Google Scholar

Supplemental Material:

The online version of this article (DOI: 10.1515/snde-2015-0032) offers supplementary material, available to authorized users.

Published Online: 2015-12-17
Published in Print: 2016-6-1

©2016 by De Gruyter

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