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CVA Solution The earlier works on CVA are mainly focused on unilateral CVA that assumes that only one counterparty is defaultable and the other one is default-free. The unilateral treatment neglects the fact that both counterparties may default, i.e., counterparty risk can be bilateral. A trend that has become increasingly relevant and popular has been to consider the bilateral nature of counterparty credit risk. Although most institutions view bilateral considerations as important in order to agree on new transactions, Hull and White (2013) argue that bilateral CVA is more controversial than unilateral CVA as the possibility that a dealer might default is in theory a benefit to the dealer. CVA is the difference between the risk-free portfolio value and the true (or risky or defaultable) 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. In general, risky valuation can be classified into two categories: the default time approach (DTA) and the default probability approach (DPA). The DTA involves 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 the DTA is very intuitive, it has 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. The current popular CVA methodology (Pykhtin and Zhu (2006) and Gregory (2009), etc.) is first derived using DTA and then discretized over a time grid in order to yield a feasible solution. The discretization, however, is inaccurate. In fact, this model has never been rigorously proved. Since CVA is used for financial accounting and pricing, its accuracy is essential. Moreover, this current model is based on a well-known assumption, in which credit exposure and counterparty’s credit quality are independent. Obviously, it can not capture wrong/right way risk properly. We present a framework for risky valuation and CVA. In contrast to previous studies, the model relies on the DPA rather than the DTA. Our study shows that the pricing process of a defaultable contract normally has a backward recursive nature if its payoff could be positive or negative. An intuitive way of understanding these backward recursive behaviours is that we can think of that any contingent claim embeds two default options. In other words, when entering an OTC derivatives 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, the default options are American style options that normally require a backward induction valuation. 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. Credit Valuation Adjustment (CVA) and Wrong Way Risk
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