Abstract
The ISDA CDS pricer is the market-standard model to value credit default swaps (CDS). Since the Big Bang protocol moreover, it became a central quotation tool: just like options prices are quoted as implied vols with the help of the Black-Scholes formula, CDSs are quoted as running (conventional) spreads. The ISDA model sets the procedure to convert the latter to an upfront amount that compensates for the fact that the actual premia are now based on a standardized coupon rate. Finally, it naturally offers an easy way to extract a risk-neutral default probability measure from market quotes. However, this model relies on unrealistic assumptions, in particular about the deterministic nature of the recovery rate. In this paper, we compare the default probability curve implied by the ISDA model to that obtained from a simple variant accounting for stochastic recovery rate. We show that the former typically leads to underestimating the reference entity’s credit risk compared to the latter. We illustrate our views by assessing the gap in terms of implied default probabilities as well as on credit value adjustments (CVA) figures and pricing mismatches of financial products like deep in-/out-of-the-money standard CDSs and digital CDSs (main building block of credit linked notes, CLNs).
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Notes
- 1.
On the European contracts, a standard coupon rate of 25 bps is also considered for a couple of entities having a very high creditworthiness.
- 2.
This is particularly true in the banking sector, even though there are famous counter-examples.
- 3.
Note that the period between the last payment date (or inception) \(t_0\) and the valuation date does not enter the picture as it deals with past protection, and there is no reason to pay for it. The corresponding amount \(kN\Delta (t_0,t)\) is called the accrued and explains the difference between clean and dirty prices as for Bonds.
- 4.
This notation is needed to deal with the term \(i=1\), to make sure that \(t_{i-1}^+=t_0^+=t\) and not \(t_0\).
- 5.
In this paper, we assume that \(\tau \) admits a density. In particular, \(\mathbb {Q}(\tau =s)=0\) for all \(s\in \mathbb {R}^+\).
- 6.
In particular, we adopt a discretization scheme in line with the payment schedule (i.e. quarterly), and assume that all payments impacted by the occurence of the reference entity’s default take place at the first payment date following the default event.
- 7.
Usually, the term-structure of recovery rate is flat, i.e. \(x_1=\cdots =x_n=x\).
- 8.
The reason why a negative relationship between recovery rate and default intensity corresponds to a positive \(\rho \) stems from the fact that the default-time is negatively correlated with the default probability: the higher the default intensity, the sooner the default time, on average.
- 9.
Note that, in the log-likelihood function we have to subtract a machine-precision quantity from full recoveries (i.e. when the recovery rate is equal to 100%).
- 10.
Uniform distribution is a particular case of the Beta distribution when both the shape parameters \(\alpha \) and \(\beta \) are equal to one (indeed our estimation of this conditional distribution yields \(\alpha =0.9\) and \(\beta =1\)). The flat distribution for Senior Secured bonds is in accordance with the findings of Schuermann (2004).
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Acknowledgements
This work was supported by the Fonds de la Recherche Scientifique - FNRS under Grant J.0037.18.
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Gambetti, P., Gauthier, G., Vrins, F. (2018). Stochastic Recovery Rate: Impact of Pricing Measure’s Choice and Financial Consequences on Single-Name Products. In: Mili, M., Samaniego Medina, R., di Pietro, F. (eds) New Methods in Fixed Income Modeling. Contributions to Management Science. Springer, Cham. https://doi.org/10.1007/978-3-319-95285-7_11
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