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Pricing Recovery Risk via a Partial Information Transform

In his seminal paper, Wang ( S. S. Wang, "A Universal Framework for Pricing Financial and Insurance Risks", ASTIN Bulletin, Volume 32, Issue 02 November 2002, pp 213-234) proposes a transform method to price both liabilities and contingent claims, whether traded or not. One application of the Wang Transform is in option pricing, which for example can be used in the structural modeling of defaultable bonds. The classical Merton model (R.C. Merton, "On the pricing of corporate debt: The risk structure of interest rates", The Journal of Finance 29.2, 449-470, 1974.) assumes that default and post-default recoverable values are driven by a common stochastic factor, that of the firm's asset value. However, a single factor for pricing such bonds assumes that bond and equity returns are perfectly correlated, which need not be true. New methods, (A. Cohen and N. Costanzino, "Bond and CDS Pricing with Recovery Risk I and II", SSRN preprints) address the entanglement of recovery risk with default risk in the pricing of defaultable bonds. The authors show that this mixture can lead to errors in estimating the total risk that investors in these bonds have undertaken. The approach taken also considers related instruments such as CDS's and equity as well as solving the related boundary value problems for bond price. The time value of money is another important aspect of bond pricing, and in the same issue that Wang's paper appeared, Buhlmann (H. Buhlmann, "New Math for Life Actuaries", ASTIN Bulletin, Volume 32, Issue 02 November 2002, pp 209-211.) proposed a paradigm shift in thinking about this technology. In his editorial, Buhlmann advocated switching to a "numeraire based approach" for teaching financial risk to actuaries. In our short note, we present a transform for a multifactor bond-pricing model that represents the partial information available to firm managers about recoverable value. In the spirit of Buhlmann's proposal, this approach is dependent on the choice of numeraire as well as the conditions for default. We will recover a version of the Wang Transform in the classical Merton model, and will calculate the transformed default probability in two other related structural models. The effect of this extra uncertainty about post-default recovery, we will see, can be observed through the distortion of the original default risk. Three examples are used to illuminate the approach, including a model that incorporates short-term credit spreads into asset and recovery value evolution.