Introduction
Following the global financial crisis, different adjustments have been introduced to the valuation of derivative contracts to account for counterparty and funding risk. Credit Valuation Adjustment (CVA) is an adjustment to the valuation of a portfolio in order to explicitly account for the credit worthiness of our counterparties. The CVA of an OTC derivatives portfolio with a given counterparty is the market value of the credit risk due to any failure to perform on agreements with that counterparty. This adjustment can be either positive or negative, depending on which of the two counterparties bears the larger burden to the other of exposure and of counterparty default likelihood.
CVA allows institutions not only to quantify counterparty risk as a single measurable P&L number, but also to dynamically manage, price and hedge counterparty risk. The benefits of CVA are widely acknowledged. Many banks have set up internal credit risk trading desks to manage counterparty risk on derivatives.
CVA, by definition, is the difference between the risk-free portfolio value and the risky portfolio value that takes into account the possibility of a counterparty’s default. The risk-free portfolio value is what brokers quote or what trading systems or models normally report. The risky portfolio value, however, is a relatively less explored and less transparent area, which is the main challenge and core theme for CVA.
Previous CVA models involve the default time explicitly. Most CVA models in the literature (Brigo and Capponi (2008), Lipton and Sepp (2009), Pykhtin and Zhu (2006) and Gregory (2009), etc.) are based on this approach.
Although those models are very intuitive, they have the disadvantage that it explicitly involves the default time. We are very unlikely to have complete information about a firm’s default point, which is often inaccessible (see Duffie and Huang (1996), Jarrow and Protter (2004), etc.). Usually, valuation under the DTA is performed via Monte Carlo simulation. On the other hand, however, the DPA relies on the probability distribution of the default time rather than the default time itself. Sometimes the DPA yields simple closed form solutions.
Joshi and Kwon (2016) propose a nested Monte Carlo simulation or least square technique to compute CVA. Graaf employ a finite-difference method and Monte Carlo simulation to solve a partial differential equation (PDE) and to estimate the mean exposure respectively. Borovykh et al. (2018) use a fast Fourier transform approach to calculate CVA.
This paper presents a framework for risky valuation and CVA. In contrast to previous studies, the model relies on the probability distribution of the default time rather than the default time itself.
After the credit crisis, a simplified assumption of independent default no longer holds especially when CVA is calculated against other financial institutions. The current framework should be further enhanced by embedding default correlations.
Wrong way risk occurs when exposure to a counterparty is adversely correlated with the credit quality of that counterparty, while right way risk occurs when exposure to a counterparty is positively correlated with the credit quality of that counterparty. For example, in wrong way risk exposure tends to increase when counterparty credit quality worsens, while in right way risk exposure tends to decrease when counterparty credit quality declines. Wrong/right way risk, as an additional source of risk, is rightly of concern to banks and regulators. Since this new model allows us to incorporate correlated and potentially simultaneous defaults into risky valuation, it can naturally capture wrong/right way risk.
The rest of this paper is organized as follows: Section 2 discusses risky valuation and CVA. Section 2 presents numerical results. The conclusions are given in Section 3. All proofs and a practical framework that embraces netting agreements, margining agreements and wrong/right way risk are contained in the appendices.