Price Discovery Limits in the Credit Default Swap Market in the Financial Crisis

Abstract

We derive the credit default swap (CDS) premium a financial institution requires to assume the default risk of fixed income instruments and the maximum premium a CDS buyer is willing to offer. These premiums are functions of the institution’s capital and current risk exposure. In most cases, an institution requires an increasing premium to assume additional risk. However, we show that an under-capitalized institution that already has substantial default risk exposure would engage in risk-shifting and assume more risk at lower CDS premiums. Consistent with this, prior to the 2008 financial crisis credit default swap issuance increased substantially, as did the volume of the underlying mortgage-backed securities, but the data suggests that required CDS premiums remained constant or declined.

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Notes

  1. 1.

    The ABX Index represents 20 subprime residential mortgage-backed securities and is used as a financial benchmark of the overall value and performance of the subprime residential mortgage market. See Wachter (2018) for further discussion.

  2. 2.

    See Susan M. Wachter, Credit Risk Transfer, Informed Markets, and Securitization, Economic Policy Review 24(3), 2018. In January 2006, Markit Group, in collaboration with a group of major banks, launched the ABX.HE (the ABX), linked to the pricing of twenty specific home equity RMBS deals, including some of the largest deals during this period. The overall index incorporated a basket of indexes, differentiated by credit risk rating.

  3. 3.

    The literature identifies the mispricing of the put option embedded in nonrecourse lending (Pavlov & Wachter, 2006; Gomes, Grotteria, & Wachter, 2018), deriving from bank managers and shareholders exploitation of mispriced deposit insurance. In the model we develop here, insurance is mispriced but not due to government backing but rather due to profit maximizing firm behavior. In a related paper, Gollier, Koehl, and Rochet (1997) shows that the optimal exposure to risk of firms with limited liability is larger than firms with unlimited liability.

  4. 4.

    Two solutions exist, negative and positive. When there is a perturbation of the capital stock away from the steady state, a negative h suggests a concave value function and will bring the capital stock back to the steady state, while the positive solution is unstable.

  5. 5.

    Note that the CDS premium is not risk premium. It is akin the premium charged on any insurance policy.

  6. 6.

    In our illustration, the offer rate is lower than the ask rate in all cases. This is intuitive because the issuer needs to be compensated for putting the underlying business at risk, while the buyer needs to be compensated for the credit risk of the issuer. If taken literally, our results suggest that there would be no CDS transactions. We note, however, that from a modelling point of view this can be easily overcome. For instance, the buyer may have some hedging demand for the CDS and be willing to pay a premium in excess of the one depicted in Figure 6. Or, the buyer may be risk-averse, which would also induce them to offer a higher premium than what we compute.

References

  1. Arentsen, E., Mauer, D. C., Rosenlund, B., Zhang, H. H., & Zhao, F. (2015). Subprime mortgage defaults and credit default swaps. The Journal of Finance, 70(2), 689–731.

    Article  Google Scholar 

  2. Bernal, O., Gnabo, J. Y., & Guilmin, G. (2014). Assessing the contribution of banks, insurance and other financial services to systemic risk. Journal of Banking & Finance, 47, 270–287.

    Article  Google Scholar 

  3. Bolton, P., & Oehmke, M. (2011). Credit default swaps and the empty creditor problem. The Review of Financial Studies, 24(8), 2617–2655.

    Article  Google Scholar 

  4. Dang, T. V., Gorton, G., & Holmström, B. (2012). Ignorance, debt and financial crises. Yale University and Massachusetts institute of technology, working paper, 17.

  5. Fostel, A., & Geanakoplos, J. (2012). Tranching, CDS, and asset prices: How financial innovation can cause bubbles and crashes. American Economic Journal: Macroeconomics, 4(1), 190–225.

    Google Scholar 

  6. Gollier, C., Koehl, P. F., & Rochet, J. C. (1997). Risk-taking behavior with limited liability and risk aversion. Journal of risk and insurance, 347-370.

  7. Gomes, J. F., Grotteria, M., & Wachter, J. (2018). Foreseen risks (No. w25277). National Bureau of Economic Research.

  8. Jayasuriya, D. (2019). Icarus of the 21st century: The rise and fall of Monoline/bond insurers. Bond insurers (January 18, 2019).

  9. Levitin, A. J., Lin, D., & Wachter, S. M. (forthcoming). Mortgage risk premiums during the housing bubble. The Journal of Real Estate Finance and Economics.

  10. Markose, S., Giansante, S., & Shaghaghi, A. R. (2012). “Too interconnected to fail” financial network of us cds market: Topological fragility and systemic risk. Journal of Economic Behavior & Organization, 83(3), 627–646.

    Article  Google Scholar 

  11. Merton, Robert. (1976). Option pricing when the underlying stock returns are discontinous. Journal of Financial Economics. 4, 125–144.

  12. Min, D. (2015). Understanding the failures of market discipline. Washington University Law Review, 92, 1421.

    Google Scholar 

  13. Pavlov, A., & Wachter, S. M. (2006). The inevitability of marketwide underpricing of mortgage default risk. Real Estate Economics, 34(4), 479–496.

    Article  Google Scholar 

  14. Schwarcz, S. L. (2019). Regulating financial guarantors: Abstraction Bias as a cause of excessive risk-taking. Available at SSRN 3431345.

  15. Stanton, R., & Wallace, N. (2011). The bear’s lair: Index credit default swaps and the subprime mortgage crisis. The Review of Financial Studies, 24(10), 3250–3280.

    Article  Google Scholar 

  16. Stulz, R. M. (2010). Credit default swaps and the credit crisis. The Journal of Economic Perspectives, 24(1), 73–92.

    Article  Google Scholar 

  17. Wachter, S. M. (2018). Credit risk transfer, informed markets, and securitization. Economic Policy Review, 24(3).

  18. Zingales, L. (2008). Causes and effects of the Lehman Brothers bankruptcy. Committee on Oversight and Government Reform US House of Representatives, 23–25.

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Correspondence to Andrey Pavlov.

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Appendix

Appendix

Differential equation with a general dividend policy

The risk-neutral, no-arbitrage condition for the value of the lender, V, under a general dividend policy, δ, is:

$$ rV(X) dt=\delta\;Xdt+E(dV). $$

The Ito’s Lemma for the jump-diffusion processes remains unchanged:

$$ E(dV)=\left(\left(\mu + Ck-\varOmega \right)\frac{\partial V}{\partial X}+\frac{\sigma^2}{2}\frac{\partial^2V}{\partial {X}^2}\right)\; dt-\lambda \left(V(X)-V\left(X- Cg\right)\right) dt. $$

This provides the following differential equation for the lender value, V:.

$$ rV(X)=\left\{\begin{array}{c}\delta X+\left(m+C\ k\right)\frac{\partial V}{\partial X}+\frac{\sigma^2}{2}\frac{\partial^2V}{\partial {X}^2}-\lambda V(X),\kern6.5em \xi \le X\le \xi +C\ g\\ {}\delta X+\left(m+C\ k\right)\frac{\partial V}{\partial X}+\frac{\sigma^2}{2}\frac{\partial^2V}{\partial {X}^2}-\lambda \left(V(X)-V\left(X-C\ g\right)\right),\kern0.5em \xi +C\ g\le X\ \end{array}\right. $$

This system of equations can only be solved numerically.

Valuation of the CDS Buyer’s Claim

Applying Ito’s Lemma for jump-diffusion processes while the issuer is in business, >ξ, we obtain:

$$ E(dW)=\left(\left(\mu + Ck-\Omega \right)\frac{\partial W}{\partial X}+\frac{\sigma^2}{2}\frac{\partial^2W}{\mathrm{\eth}{X}^2}\right) dt+\lambda \left(\min \left( Cg,\max \left(X-\xi, 0\right)\right)-W\right) dt $$

Note that the second part above has (-W) because the CDS buyer receives a payout but the claim expires in case of a jump. Substitute this into Equation (7) to obtain the following differential equation for the value of the CDS claim, W, while the issuer is in business, X > ξ:

$$ rW(X)=- kC+\left(\left(\mu + Ck-\Omega \right)\frac{dW}{dX}+\frac{\sigma^2}{2}\frac{\partial^2W}{\mathrm{\eth}{X}^2}\right)+\lambda \left(\min \left( Cg,\max \left(X-\xi, 0\right)\right)-W\right) $$

Simplify:

$$ \left\{\begin{array}{c}\left(r+\lambda \right)W(X)=-k\ C+\left(\left(m+ Ck\right)\frac{dW}{dX}+\frac{\sigma^2}{2}\frac{\partial^2W}{\eth {X}^2}\right)+\lambda\ g\ C,\kern0.5em if\ X> Cg+\xi \\ {}\left(r+\lambda \right)W(X)=-k\ C+\left(\left(m+ Ck\right)\frac{dW}{dX}+\frac{\sigma^2}{2}\frac{\partial^2W}{\eth {X}^2}\right)+\lambda \left(X-\xi \right),\kern1.25em if\ \xi <X< Cg+\xi \end{array}\right. $$

The solution to this equation is:

$$ W(X)==\left\{\begin{array}{c}\frac{\left( g\lambda -k\right)C}{r+\lambda }+{B}_1\exp (pX),\kern0.5em if\ X> Cg+\xi \\ {}\frac{\left(X-\xi \right){\lambda}^2+\left(\left(X-\xi \right)r+m\right)\lambda - Ckr}{{\left(r+\lambda \right)}^2}+{B}_2\exp (pX)+{B}_3\exp (qX),\kern1.25em if\ \xi <X< Cg+\xi \end{array}\right. $$

where B1 and B2 are arbitrary constants, ξ is the abandonment boundary as determined by the issuer optimization, and

$$ p=-\frac{C\ k+m+\sqrt{C^2{k}^2+2C\ k\ m+2\left(\lambda +r\right){\sigma}^2+{m}^2}}{\sigma^2}<0 $$
$$ q=-\frac{C\ k+m-\sqrt{C^2{k}^2+2C\ k\ m+2\left(\lambda +r\right)\ {\sigma}^2+{m}^2}}{\sigma^2} $$

Determine B1, B2, and B3 with value-matching and smooth-pasting at X = Cg + ξ and at value-matching to zero at the abandonment boundary, X = ξ:

$$ {B}_3=\frac{-p\ \left(\hbox{--} \frac{\left( g\lambda -k\right)C}{r+\lambda }+\frac{\left(C\ g-2\xi \right){\lambda}^2+\left(\left(C\ g-2\xi \right)r+m\right)\lambda - Ckr}{{\left(r+\lambda \right)}^2}\right)-\frac{\lambda }{r+\lambda }}{\left(p-q\right)\exp \left(-q\left( Cg-\xi \right)\right)}<0 $$
$$ {B}_2=\frac{\frac{\lambda }{r+\lambda }-q{B}_3\exp \left(- q\xi \right)}{p\exp \left(- p\xi \right)} $$
$$ {B}_1\exp \left(-p\left( Cg-\xi \right)\right)=\frac{\left(C\ g-2\xi \right){\lambda}^2+\left(\left(C\ g-2\xi \right)r+m\right)\lambda - Ckr}{{\left(r+\lambda \right)}^2}+ $$
$$ +{B}_2\exp \left(-p\left( Cg-\xi \right)\right)+{B}_3\exp \left(-q\left( Cg-\xi \right)\right)-\frac{\left( g\lambda -k\right)C}{r+\lambda } $$

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Pavlov, A., Schwartz, E. & Wachter, S. Price Discovery Limits in the Credit Default Swap Market in the Financial Crisis. J Real Estate Finan Econ 62, 165–186 (2021). https://doi.org/10.1007/s11146-020-09747-8

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Keywords

  • Credit insurance
  • Credit default swaps
  • Price discovery