A Dynamic Stochastic Recovery Rate Model With Applications to Credit Derivatives Pricing
(2018) Quantitative Finance and Risk Analysis (QFRA) — Location: Mykonos (Grece) (7.June.2018)
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- The expected loss (EL), or the present value of the EL are probably the most important measures to quantify credit risk of transactions initiated with a counterparty. They feature : (i) the default probability (PD) of the counterparty, (ii) the exposure at default (EAD) and (iii) the loss given default (LGD). Most recent developments in quantitative risk management aim at improving the assessment of the first two points. Step (i) can be addressed either using historical default rates or market-implied (e.g. Bonds or CDS) information whereas (ii) depends on the (historical or risk-neutral) dynamics of the book or market value of a traded portfolio. The last factor however is the part of the exposure that will be lost (i.e. non-recovered) upon default of the counterparty risk. This factor is a scaling coefficient in $[0,1)$ and has thus a huge impact on the risk measure. Nowadays, in spite of its acknowledged importance, most of credit risk models still consider the LGD to be a constant expressed in terms of the recovery rate (R), i.e. LGD=1-R with R a known constant. This is of course a major shortcoming : R being unknown prior to the auction, it should be treated as a random variable. To circumvent this issue, one could think of R as the expectation (in the chosen measure) of the recovery rate. Whereas this is a fair point when R is assumed independent from the other risk drivers, it does not hold true when the recovery rate is linked to other risk factors, and precisely, there is a wide scientific literature arguing in that direction (e.g.~\cite{Altm05}). Several techniques have been proposed to deal with recovery rates being correlated to other factors. However, these models are essentially static. For instance, default time and recovery rate are typically linked using a Gaussian copula. In particular, the dynamics of default risk is completely disregarded. In this paper, we propose a dynamic framework to account for the dependency between the recovery rate and the default likelihood. This is achieved by adopting a reduced-form default model combined with a recovery rate process, defined as the conditional expected value of the recovery rate. The stochastic intensity is modeled as a square-root diffusion and the recovery rate process by a $\Phi$-martingale~\cite{Vrins16}. The dependency between the default intensity and the recovery rate is controlled via the correlation between the Brownian drivers. The tractability of the model is enhanced by relying on the change-of-measure approach derived in~\cite{Brigo17}. The impact of the model is discussed on various examples including digital CDS spread, recovery swaps or credit linked notes. Eventually, this model provides an appealing method to address a major drawback of reduced-form models: the additional source of uncertainty resulting from the stochastic recovery rate contributes to increasing the volatility of CDS par spread. Such a method is thus expected to generate CDS option prices and value-at-risk levels that are more in line with observations.
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Vrins, F. (2018). A Dynamic Stochastic Recovery Rate Model With Applications to Credit Derivatives Pricing. Quantitative Finance and Risk Analysis (QFRA), Mykonos (Grece). https://hdl.handle.net/2078.5/172285