Market implied volatilities for defaultable bonds


Typically, implied volatilities for defaultable instruments are not available in the financial market since quotations related to options on defaultable bonds or on credit default swaps are usually not quoted by brokers. However, an estimate of their volatilities is needed for pricing purposes. In this paper, we provide a methodology to infer market implied volatilities for defaultable bonds using equity implied volatilities and CDS spreads quoted by the market in relation to a specific issuer. The theoretical framework we propose is based on the Merton’s model under stochastic interest rates where the short rate is assumed to follow the Hull–White model. A numerical analysis is provided to illustrate the calibration process to be performed starting from financial market data. The market implied volatility calibrated according to the proposed methodology could be used to evaluate options where the underlying is a risky bond, i.e. callable bond or other types of credit-risk sensitive financial instruments.

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  1. 1.

    Data related to the Debt securities statistics published by the Bank for International Settlements,

  2. 2.

    See Black and Scholes (1973) and Merton (1974).

  3. 3.

    The Merton model assumes the following: (1) there is an overly simple debt structure, (2) there are no bankruptcy costs (i.e., the liquidation value equals the firm value), and (3) the debt and equity are frictionless tradable assets.

  4. 4.

    An alternative formulation of the Hull–White model can be used where the short rate is expressed as \(r(t) = \alpha (t) + x(t)\). See Russo and Fabozzi (2016) for further details.

  5. 5.

    See Crosbie and Bohn (1997) for further details.

  6. 6.

    For all quantities observed in the market we use the suffix M.

  7. 7.

    See Lando (1998).

  8. 8.

    An alternative approach could be adopted using, directly, prices of zero-coupon defaultable bonds (if they exist for the relevant issuer) in place of CDS quotations. Our decision to use CDS in place of bonds because of the fact that CDS are quoted in a standardized and liquid market while defaultable bond quotations are not always available for the desired maturity and sometimes they are not liquid.

  9. 9.

    Market data were obtained from Datastream.

  10. 10.

    Statistics we calculate are the following: average (Mean), standard deviation (Std), skewness (Skew), Kurtosis (Kurt), minimum (Min), and maximum (Max).

  11. 11.

    We have referred to RiskMetrics technical documentation RiskMetrics (1996) for theoretical framework and used a smoothing parameter equal to 0.94.


  1. Bao, J., & Pan, J. (2013). Bond illiquidity and excess volatility. The Review of Financial Studies, 26(12), 30683103.

    Article  Google Scholar 

  2. Bayraktar, E. (2008). Pricing options on defaultable stocks. Applied Mathematical Finance, 15, 277–304.

    Article  Google Scholar 

  3. Bayraktar, E., & Yang, B. (2011). A unified framework for pricing credit and equity derivatives. Mathematical Finance, 21, 493–517.

    Article  Google Scholar 

  4. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–54.

    Article  Google Scholar 

  5. Cao, C., Yu, F., & Zhong, Z. (2010). The information content of option-implied volatility for credit default swap valuation. Journal of Financial Markets, 13, 321–343.

    Article  Google Scholar 

  6. Crosbie, P. J., & Bohn, J. R. (1997). Modeling default risk. San Francisco, CA: KMV Corporation.

    Google Scholar 

  7. Hull, J., & White, A. (1990). Pricing interest rate derivative securities. Review of Financial Studies, 3, 573–592.

    Article  Google Scholar 

  8. Kelly, B. T., Manzo, G., & Palhares, D. (2017). Credit-implied volatility, Chicago Booth Research Paper, No. 17-12.

  9. Lando, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Research, 2, 99–120.

    Google Scholar 

  10. Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, 449–70.

    Google Scholar 

  11. RiskMetrics. (1996). Technical Document - Technical report J.P. Morgan.

  12. Russo, V., & Fabozzi, F. J. (2016). Pricing coupon bond options and swaptions under the two-factor Hull-White model. Journal of Fixed Income, 25, 76–82.

    Article  Google Scholar 

  13. Zheng, C. K. (1999). Default implied volatility for credit spread.

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Correspondence to Rosella Giacometti.

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Russo, V., Giacometti, R. & Fabozzi, F.J. Market implied volatilities for defaultable bonds. Ann Oper Res 275, 669–683 (2019).

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  • Defaultable bonds’ implied volatility
  • Credit default swap (CDS)
  • Merton model
  • Hull and White model