This article presents a generic model for pricing financial derivatives subject to counterparty credit risk. Both unilateral and bilateral types of credit risks are considered. Our study shows that credit risk should be modeled as American style options in most cases, which require a backward induction valuation. To correct a common mistake in the literature, we emphasize that the market value of a defaultable derivative is actually a risky value rather than a risk-free value. Credit value adjustment (CVA) is also elaborated. A practical framework is developed for pricing defaultable derivatives and calculating their CVAs at a portfolio level.
A broad range of financial instruments bear credit risk. Credit risk may be unilateral or bilateral. Some derivatives such as, debt instruments (e.g., loans, bills, notes, bonds, etc), by nature contain only unilateral credit risk because only the default risk of one party appears to be relevant. Whereas some other derivatives, such as, over the counter (OTC) derivatives and securities financing transactions (SFT), bear bilateral credit risk because both parties are susceptible to default risk.
In the market, risk-free values are quoted for most financial derivatives. In other words, credit risk is not captured. Historical experience shows that credit risk often leads to significant losses. Therefore, it is obvious to all market participants that credit risk should be taken into account when reporting the fair value of any defaultable derivative. The adjustment to the risk-free value is known as the credit value adjustment (CVA). CVA offers an opportunity for banks to dynamically price credit risk into new trades and has become a common practice in the financial industry, especially for trading books.
By definition, CVA is the difference between the risk-free value and the true (or risky or defaultable) value that takes into account the possibility of default. The risk-free value is what brokers quote or what trading systems or models normally report. The risky value, however, is a relatively less explored and less transparent area, which is the main challenge and core theme for credit risk measurement and management.
There are two primary types of models that attempt to describe default processes in the literature: structural models and reduced-form (or intensity) models. The structural models, studied by Merton (1974) and Longstaff and Schwartz (1995), regard default as an endogenous event, focusing on the capital structure of a firm. The reduced-form models introduced by Jarrow and Turnbull (1995) and Duffie and Singleton (1999) do not explain the event of default endogenously, but instead characterize it exogenously as a jump process. For pricing and hedging, reduced-form models are the preferred methodology.
Three different recovery models exist in the literature. The default payoff is either 1) a fraction of par (Madan and Unal (1998)), 2) a fraction of an equivalent default-free bond (Jarrow and Turnbull (1995)), or 3) a fraction of market value (Duffie and Singleton (1999)). The last one is most commonly used in the market. In their paper, Duffie and Singleton (1999) do not clearly state whether the market value of a defaultable derivative is a risky value or a risk-free value. Instead, the authors implicitly treat the market value as a risky value because the market price therein evolves in a defaultable manner.
Otherwise, they cannot obtain the desired result. However, most of the later papers in the literature mistakenly think that the market value of a defaultable derivative is a risk-free value. Consequently, the results are incorrect. For instance, the current most popular CVA model described by Pykhtin and Zhu (2007), and Gregory (2009) is inappropriate in theory because the authors do not distinguish risky value and risk-free value when they conduct a risky valuation (i.e. valuing a defaultable derivative). In fact, the model has never been rigorously proved.
In this paper, we present generic models for valuing defaultable financial derivatives. For completeness, our study covers various cases: unilateral and bilateral, single payment and multiple payments, positive and negative payoffs. Although a couple of simple cases have been studied before by other authors, e.g. Duffie and Singleton (1999), Duffie and Huang (1996), who only provide heuristic derivations in a non-rigorous manner; analytic work on the other cases is novel. In contrast with the current recursive integral solution (see Duffie and Huang (1996)), our theory shows that the valuation of defaultable derivatives in most situations requires a backward induction procedure.
There is an intuitive way of understanding these backward induction behaviours: We can think that any defaultable derivative with bilateral credit risk embeds two default options. In other words, when entering a defaultable financial transaction, one party grants the other party an option to default and, at the same time, also receives an option to default itself. In theory, default may occur at any time. Therefore, default options are American style options that normally require a backward induction valuation.
We explicitly indicate that, within the context of risky valuation, the market value of a defaultable derivative is actually a risky value rather than a risk-free value, because a firm may default at any future time even if it survives today. An intuitive explanation is that after charging the upfront CVA, one already converted the market value of a defaultable derivative from the risk-free value to the risky value.
From a practical perspective,
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