# Bond and CDS Pricing via the Stochastic Recovery Black-Cox Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Background and Motivation

## 2. Review of Credit Risk and Pricing in the Black-Cox Model

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

#### 2.1. Bond Pricing with the Black-Cox Model

**Definition**

**1.**

**Definition**

**2.**

- Weak Covenant $(K\le N)$$$\begin{array}{c}\hfill \begin{array}{c}{d}_{0}^{w}=\frac{ln\left(\right)open="("\; close=")">\frac{{A}_{t}}{N}}{+}{\sigma}_{A}\sqrt{T-t}\hfill \end{array}{d}_{1}^{w}=\frac{ln\left(\right)open="("\; close=")">\frac{{A}_{t}}{N}}{+}{\sigma}_{A}\sqrt{T-t}\hfill & {x}_{0}^{w}=\frac{ln\left(\right)open="("\; close=")">\frac{{K}^{2}}{N{A}_{t}}}{+}{\sigma}_{A}\sqrt{T-t}\hfill \\ {x}_{1}^{w}=\frac{ln\left(\right)open="("\; close=")">\frac{{K}^{2}}{N{A}_{t}}}{+}{\sigma}_{A}\sqrt{T-t}\hfill \end{array}$$
- Strong Covenant $(K\ge N)$$$\begin{array}{c}\hfill \begin{array}{c}{d}_{0}^{s}=\frac{ln\left(\right)open="("\; close=")">\frac{{A}_{t}}{K}}{+}{\sigma}_{A}\sqrt{T-t}\hfill \end{array}{d}_{1}^{s}=\frac{ln\left(\right)open="("\; close=")">\frac{{A}_{t}}{K}}{+}{\sigma}_{A}\sqrt{T-t}\hfill & {x}_{0}^{s}=\frac{ln\left(\right)open="("\; close=")">\frac{K}{{A}_{t}}}{+}{\sigma}_{A}\sqrt{T-t}\hfill \\ {x}_{1}^{s}=\frac{ln\left(\right)open="("\; close=")">\frac{K}{{A}_{t}}}{+}{\sigma}_{A}\sqrt{T-t}\hfill \end{array}$$

**Lemma**

**1.**

**Proof.**

**Note:**The result (27) can also be found in, for instance, Appendix B of [25]. We provide another direct proof within the constructive proof of Theorem 3 below, by computing ${\int}_{K}^{\infty}{\tilde{\mathbb{P}}}_{t}\left(\right)open="["\; close="]">{A}_{T}\in da,{\tau}_{\mathrm{K}}T$ in the strong covenant case.

**Theorem**

**1.**

- I. Weak Covenant Case. If $K\le N$ the price of a zero-coupon bond is given by$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {B}_{t,T}^{\mathrm{BC}}\left(w\right)& ={e}^{-r(T-t)}N\left(\right)open="["\; close="]">\Phi \left({d}_{0}^{w}\right)-{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1\Phi \left({x}_{0}^{w}\right)\hfill \end{array}& & +{A}_{t}\left(\right)open="["\; close="]">\Phi (-{d}_{1}^{w})+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}+1\Phi \left({x}_{1}^{w}\right)\hfill \end{array}$$$$\begin{array}{c}\hfill {\tilde{\mathrm{PD}}}_{t,T}^{\mathrm{BC}}\left(w\right)=\Phi \left(\right)open="("\; close=")">-{d}_{0}^{w}+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1& \Phi \left({x}_{0}^{w}\right)\end{array}$$$$\begin{array}{c}\hfill {\tilde{\mathrm{LGD}}}_{t,T}^{BC}\left(w\right)=1-{e}^{r(T-t)}\frac{{A}_{t}}{N}\frac{\Phi \left(\right)open="("\; close=")">-{d}_{1}^{w}}{+}\Phi \left(\right)open="("\; close=")">{x}_{1}^{w}& \Phi \left(\right)open="("\; close=")">-{d}_{0}^{w}+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1\\ \Phi \left(\right)open="("\; close=")">{x}_{0}^{w}\end{array}$$
- II. Strong Covenant Case. If $K\ge N$ the price of a zero-coupon bond is given by$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {B}_{t,T}^{\mathrm{BC}}\left(s\right)& ={e}^{-r(T-t)}N\left(\right)open="["\; close="]">\Phi \left({d}_{0}^{s}\right)-{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1\Phi \left({x}_{0}^{s}\right)\hfill \end{array}& & +{A}_{t}\left(\right)open="["\; close="]">\Phi (-{d}_{1}^{s})+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}+1\Phi \left({x}_{1}^{s}\right)\hfill \end{array}$$$$\begin{array}{c}\hfill {\tilde{\mathrm{PD}}}_{t,T}^{\mathrm{BC}}\left(s\right)=\Phi \left(\right)open="("\; close=")">-{d}_{0}^{s}+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1& \Phi \left({x}_{0}^{s}\right)\end{array}$$$$\begin{array}{c}\hfill {\tilde{\mathrm{LGD}}}_{t,T}^{\mathrm{BC}}\left(s\right)=1-{e}^{r(T-t)}\frac{{A}_{t}}{N}\frac{\Phi \left(\right)open="("\; close=")">-{d}_{1}^{s}}{+}\Phi \left(\right)open="("\; close=")">{x}_{1}^{s}& \Phi \left(\right)open="("\; close=")">-{d}_{0}^{s}+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1\\ \Phi \left(\right)open="("\; close=")">{x}_{0}^{s}\end{array}$$

**Proof.**

**Remark**

**1.**

#### 2.2. CDS Pricing with the Black-Cox Model

**Theorem**

**2.**

**Note**

- The numerator in (40) follows from direct substitution of the bond price calculated in Theorem 1 into the numerator in the general CDS formula (23). This direct substitution of the risky bond price also reflects the flexibility in assigning a weak or strong covenant, and will in fact be the only change observed when stochastic recovery is included in Section 5.
- To complete the proof of Theorem 2, we will need to compute the denominator in (23). In the BC, and SRBC model forthcoming, this reduces to computing$${\tilde{\mathbb{E}}}_{t}[{e}^{-r({\tau}_{\mathrm{K}}-t)}{\mathbb{1}}_{\left(\right)}]$$$${\tilde{\mathbb{E}}}_{t}[{e}^{-r({\tau}_{\mathrm{K}}-t)}{\mathbb{1}}_{\left(\right)}].$$

**Proof.**

## 3. Modeling Recovery Risk within a Structural Framework

#### 3.1. The Correlated Asset-Recovery Model

#### 3.2. Some Preliminary Results

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

#### 3.3. Comparison of Recovery to Asset Upon Default

**Lemma**

**5.**

**Proof.**

#### 3.4. Connection between Recovery Risk and PD-LGD Correlation

## 4. The Black-Cox Model with Recovery Risk

**Assumption**

**5.**

**Assumption**

**6.**

**Assumption**

**7.**

#### 4.1. Bond Pricing with Recovery Risk

**Theorem**

**3.**

- I. Weak Covenant Case. If $K\le N$ the price of a zero-coupon bond is given by$$\begin{array}{c}\hfill \begin{array}{c}{B}_{t,T}^{\mathrm{SRBC}}\left(w\right)=N{e}^{-r(T-t)}\left(\right)open="["\; close="]">\Phi \left({d}_{0}^{w}\right)-{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1\Phi \left({x}_{0}^{w}\right)\hfill & +{R}_{t}\left(\right)open="["\; close="]">\Phi (-{d}_{\gamma}^{w})+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}+2\gamma -1\\ \Phi \left({x}_{\gamma}^{w}\right)\end{array}.\end{array}$$The risk-neutral PD and LGD in the case of a weak covenant are$$\begin{array}{c}{\tilde{\mathrm{LGD}}}_{t,T}^{\mathrm{SRBC}}\left(w\right)=1-{e}^{r(T-t)}\frac{{R}_{t}}{N}\frac{\Phi \left(\right)open="("\; close=")">-{d}_{\gamma}^{w}}{+}\Phi \left(\right)open="("\; close=")">{x}_{\gamma}^{w}\hfill & \Phi \left(\right)open="("\; close=")">-{d}_{0}^{w}+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1\\ \Phi \left(\right)open="("\; close=")">{x}_{0}^{w}\end{array}$$
- II. Strong Covenant Case. If $K\ge N$ the price of a zero-coupon bond is given by$$\begin{array}{c}\hfill \begin{array}{c}{B}_{t,T}^{\mathrm{SRBC}}\left(s\right)=N{e}^{-r(T-t)}\left(\right)open="["\; close="]">\Phi \left({d}_{0}^{s}\right)-{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}-1\Phi \left({x}_{0}^{s}\right)\hfill & +{R}_{t}\left(\right)open="["\; close="]">\Phi (-{d}_{\gamma}^{s})+{\left(\right)}^{\frac{K}{{A}_{t}}}\frac{2r}{{\sigma}_{A}^{2}}+2\gamma -1\\ \Phi \left({x}_{\gamma}^{s}\right)\end{array}.\end{array}$$

**Remark**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 4.2. Consistency and Reduction to Black-Cox Model

**Lemma**

**6.**

**Proof.**

#### 4.3. Greeks and Comparison with Standard Black-Cox model

**Lemma**

**7.**

**Recovery Greeks**are given by

#### 4.4. CDS Pricing with Recovery Risk

**Theorem**

**5.**

**Proof.**

**Lemma**

**8.**

**Proof.**

## 5. The Implied Recovery and Recovery Risk Premium

#### 5.1. Implied Recovery Rates From Observed CDS Premia

#### 5.2. The Price of Recovery Risk

**Lemma**

**9.**

## 6. The Effect of Coupons

#### 6.1. Perpetual Bonds

- Managers set a bankruptcy level K and accordingly we recall the definition (8) of ${\tau}_{\mathrm{K}}$ as the first passage time of assets A to level K.
- Coupons are paid continuously at rate C, there is a risk-free interest rate r, and recovery at bankruptcy is ${R}_{{\tau}_{\mathrm{K}}}$.

**Lemma**

**10.**

**Remark**

**4.**

**Proof.**

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Edward I. Altman, Brooks Brady, Andrea Resti, and Andrea Sironi. “The Link Between Default and Recovery Rates: Theory, Empirical Evidence and Implications.” The Journal of Business 78 (2005): 2203–28. [Google Scholar] [CrossRef]
- Jon Frye. “Depressing Recoveries.” Risk 13 (2000): 108–11. [Google Scholar]
- Guido Giese. “The Impact of PD/LGD Correlations on Credit Risk Capital.” Risk 18 (2005): 79–85. [Google Scholar]
- Amnon Levy, and Zhenya Hu. Incorporating Systematic Risk in Recovery: Theory and Evidence, Modeling Methodology. San Francisco: Moody’s KMV, 2007. [Google Scholar]
- Qiang Meng, Amnon Levy, Andrew Kaplin, Yashan Wang, and Zhenya Hu. Implications of PD-LGD Correlation in a Portfolio Setting. New York: Moody’s Analytics, 2010. [Google Scholar]
- Peter Miu, and Bogie Ozdemir. “Basel Requirement of Downturn LGD: Modeling and Estimating PD & LGD Correlations.” Journal of Credit Risk 2 (2006): 43–68. [Google Scholar]
- Michael Pykhtin. “Unexpected Recovery Risk.” Risk 16 (2003): 74–78. [Google Scholar]
- Salah Amraoui, Laurent Cousot, Sébastien Hitier, and Jean-Paul Laurent. “Pricing CDOs with state-dependent stochastic recovery rates.” Quantitative Finance 12 (2012): 1219–40. [Google Scholar] [CrossRef]
- Norddine Bennani, and Jerome Maetz. “A Spot Stochastic Recovery Extension of the Gaussian Copula.” Working Paper. Germany: University Library of Munich, 2009. Available online: http://www.defaultrisk.com/pp_cdo_82.htm (accessed on 19 April 2017).
- Damiano Brigo, and Massimo Morini. “CDS Calibration with tractable structural models under uncertain credit quality.” Risk Magazine 19 (2006): 1–13. [Google Scholar]
- Charaf Ech-Chatbi. “CDS and CDO Pricing with Stochastic Recovery.” 2008. Available online: http://dx.doi.org/10.2139/ssrn.1271823 (accessed on 19 April 2017).
- Stephan Höcht, and Rudi Zagst. “Pricing distressed CDOs with stochastic recovery.” Review of Derivatives Research 13 (2010): 219–44. [Google Scholar] [CrossRef]
- Martin Krekel. “Pricing Distressed CDOs with Base Correlation and Stochastic Recovery.” 2008. SSRN Preprint. Available online: http://dx.doi.org/10.2139/ssrn.1134022 (accessed on 19 April 2017).
- Timo Schläfer, and Marliese Uhrig-Homburg. “Is recovery risk priced? ” Journal of Banking & Finance 40 (2014): 257–70. [Google Scholar]
- Albert Cohen, and Nick Costanzino. “Bond and CDS Pricing with Recovery Risk I: The Stochastic Recovery Merton Model.” 2015. Available online: http://dx.doi.org/10.2139/ssrn.2544532 (accessed on 19 April 2017).
- Tomasz R. Bielecki, and Marek Rutkowski. Credit Risk: Modeling, Valuation and Hedging. Berlin: Springer, 2002. [Google Scholar]
- John Hull, and Alan White. “Valuing Credit Default Swaps I: No Counterparty Default Risk.” Journal of Derivatives 8 (2000): 29–40. [Google Scholar] [CrossRef]
- Dominic O’Kane. Modelling Single-Name and Multi-Name Credit Derivatives. Somerset: John Wiley & Sons, 2008. [Google Scholar]
- Robert Merton. “On the Pricing of Corporate Debt: the Risk Structure of Interest Rates.” Journal of Finance 29 (1974): 449–70. [Google Scholar] [CrossRef]
- Fischer Black, and John C. Cox. “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions.” Journal of Finance 31 (1976): 351–67. [Google Scholar] [CrossRef]
- Christopher Finger, Vladimir Finkelstein, Jean-Pierre Lardy, George Pan, Thomas Ta, and John Tierney. CreditGrades
^{TM}Technical Document. New York: Risk Metrics Group, 2002. [Google Scholar] - C. H. Hui, C. F. Lo, and H. C. Lee. “Valuation of Corporate Bonds with Stochastic Default Barriers.” Preprint. Available online: http://www.phy.cuhk.edu.hk/~cflo/Finance/papers/EuroFinRev_paper.pdf (accessed on 19 April 2017).
- I. H. Gökgöz, Ö. Ug̎ur, and Y. Yolcu Okur. “On the single name CDS price under structural modeling.” Journal of Computational and Applied Mathematics 259 (2014): 406–12. [Google Scholar] [CrossRef]
- Paul Wilmott, Jeff Dewynne, and Sam Howison. Option Pricing: Mathematical Models and Computation. Oxford: Oxford Financial Press, 1993. [Google Scholar]
- David Lando. Credit Risk Modeling. Princeton: Princeton University Press, 2004. [Google Scholar]
- Hua He, William P. Keirstead, and Joachim Rebholz. “Double Lookbacks.” Mathematical Finance 8 (1998): 201–28. [Google Scholar] [CrossRef]
- Viral V. Acharya, Sreedhar T. Bharath, and Anand Srinivasan. “Does industry-wide distress affect defaulted firms? Evidence from creditor recoveries.” Journal of Financial Economics 85 (2007): 787–821. [Google Scholar] [CrossRef]
- Martin Hillebrand. “Modeling and Estimating Dependent Loss Given Default.” Risk 19 (2005): 120–25. [Google Scholar]
- Klaus Dullmann, and Monika Trapp. “Systematic Risk in Recovery Rates of US Corporate Credit Exposures.” In Recovery Risk The Next Challenge in Credit Risk Management. Edited by Edward Altman, Andrea Resti and Andrea Sironi. London: Risk Books, 2005, pp. 235–52. [Google Scholar]
- Jon Frye. “Collateral Damage: A source of systematic credit risk.” Risk 13 (2000): 91–94. [Google Scholar]
- Jon Frye. Collateral Damage Detected. Working Paper, Emerging Issues Series; Chicago: Federal Reserve Bank of Chicago, 2000, pp. 1–14. [Google Scholar]
- William Perraudin, and Yen Ting Hu. “Dependence of Recovery Rates and Default.” SSRN Working Paper. 2006. Available online: http://dx.doi.org/10.2139/ssrn.1961142 (accessed on 19 April 2017).
- Chanatip Kitwiwattanachai. The Stochastic Recovery Rate in CDS: Empirical Test and Model. Working Paper. 2014. Available online: http://dx.doi.org/10.2139/ssrn.2136116 (accessed on 19 April 2017).
- Todd C. Pulvino. “Do Asset Fire Sales Exist? An Empirical Investigations of Commercial Aircraft Transactions.” Journal of Finance 53 (1998): 939–78. [Google Scholar] [CrossRef]
- Young Ho Eom, Jean Helwege, and Jing-Zhi Huang. “Structural Models of Corporate Bond Pricing: An Empirical Analysis.” Review of Financial Studies 17 (2004): 499–544. [Google Scholar] [CrossRef]
- Jing-Zhi Huang, and Ming Huang. “How Much of the Corporate-Treasury Yield Spread is Due to Credit Risk? ” Review of Asset Pricing Studies 2 (2012): 153–202. [Google Scholar] [CrossRef]
- E. Philip Jones, Scott P. Mason, and Eric Rosenfeld. “Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation.” Journal of Finance 39 (1984): 611–25. [Google Scholar] [CrossRef]
- Joseph P. Ogden. “Determinants of the Ratings and Yields on Corporate Bonds: Tests of the Contingent Claims Model.” Journal of Financial Research 10 (1987): 329–40. [Google Scholar] [CrossRef]
- Gordon Gemmill. “Testing Merton’s Model for Credit Spreads on Zero-Coupon Bonds.” In Paper presented at the European Financial Management Association 2002 Annual Meetings, London, UK, 26–29 June 2002. [Google Scholar]
- Ioannis Karatzas, and Steven E. Shreve. Brownian Motion and Stochastic Calculus, 2nd ed. Berlin: Springer, 2004. [Google Scholar]
- Andrei Shleifer, and Robert W. Vishny. “Liquidation values and debt capacity: A market equilibrium approach.” Journal of Finance 47 (1992): 1343–66. [Google Scholar] [CrossRef]
- Dominique C. Badoer, and Chris M. James. “The Determinants of Long Term Corporate Debt Issuances.” Journal of Finance 71 (2016): 457–92. [Google Scholar] [CrossRef]
- Thomas T. Vogel Jr. “Disney amazes investors with sale of 100-year bonds.” The Wall Street Journal, 23 July 1993. [Google Scholar]
- Hayne E. Leland. “Corporate debt value, bond covenants, and optimal capital structure.” The Journal of Finance 49 (1994): 1213–52. [Google Scholar] [CrossRef]

^{1.}This constraint can be improved upon slightly by introducing a curved (time-dependent) boundary ${\tau}_{\mathrm{K}}:=arginf\{t\in [0,T]:{A}_{t}\le K\left(t\right)\}$ for a suitably chosen $K\left(t\right)$. However, the fact remains there is still a single driver for both recovery and default.

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Cohen, A.; Costanzino, N.
Bond and CDS Pricing via the Stochastic Recovery Black-Cox Model. *Risks* **2017**, *5*, 26.
https://doi.org/10.3390/risks5020026

**AMA Style**

Cohen A, Costanzino N.
Bond and CDS Pricing via the Stochastic Recovery Black-Cox Model. *Risks*. 2017; 5(2):26.
https://doi.org/10.3390/risks5020026

**Chicago/Turabian Style**

Cohen, Albert, and Nick Costanzino.
2017. "Bond and CDS Pricing via the Stochastic Recovery Black-Cox Model" *Risks* 5, no. 2: 26.
https://doi.org/10.3390/risks5020026