Utility valuation of multi-name credit derivatives and application to CDOs

Abstract

We study the impact of risk-aversion on the valuation of credit derivatives. Using the technology of utility-indifference pricing in intensity-based models of default risk, we analyze resulting yield spreads in multiname credit derivatives, particularly CDOs. We study first the idealized problem with constant intensities where solutions are essentially explicit. We also give the large portfolio asymptotics for this problem. We then analyze the case where the firms have stochastic default intensities driven by a common factor, which can be viewed as another extreme from the independent case. This involves the numerical solution of a system of reaction-diffusion PDEs. We observe that the nonlinearity of the utility-indifference valuation mechanism enhances the effective correlation between the times of the credit events of the various firms leading to non-trivial senior tranche spreads, as often seen from market data.

Multiscale Intensity Models and Name Grouping for Valuation of Multi-Name Credit Derivatives

“…This premium associated with the senior and super senior tranches is exogenous to the default premium dictated by pricing models. Sircar and Zariphopoulou (2006) reach the same conclusion via an alternative pricing approach using utility valuation.…”

A Multidimensional Exponential Utility Indifference Pricing Model with Applications to Counterparty Risk

“…Utility based pricing has also been utilized by Bielecki and Jeanblanc [6], Sircar and Zariphopolou [43] and recently Jiao et al [27] [28] in an intensity based setting. Several authors have applied it in modeling of defaultable bonds where the problem remains one dimensional, see in particular Leung et al [35], Jaimungal and Sigloch [25], and Liang and Jiang [36].…”

Credit derivatives and risk aversion

“…This may be done if we assume independence between default times and "effectively correlate" them through utility valuation: see [7] for small correlation expansions around the independent case with risk-neutral valuation. Another possibility is to assume a large degree of homogeneity between the names (see for example [19] with indifference pricing of CDOs under intensity models), or to adapt a homogeneous group structure to reduce dimension as in [16].…”

“…He has to liquidate holdings in the stock and deposit in the bank account, reducing his investment opportunities. (Throughout, we are neglecting other potential investment opportunities, but a more complex model might include these; in multi-name problems, such as valuation of CDOs, this is particularly important: see [19]). For simplicity, we also assume that he receives full pre-default market value on his stock holdings on liquidation, though one might extend to consider some loss at the default time.…”

“…Some examples include the introduction of jumps, [3,11,21], stochastic interest rates [13], imperfect information on the firm's asset value [4], uncertainty in the default threshold [9] and fast mean-reverting stochastic volatility [6]. In related work, utility-based valuation has been applied within the framework of intensity-based models for both single-name derivatives [1,18,20] and, in addressing the second bullet point, for multi-name products [19].…”