Huang Haohan, Huang Huaxiong, Wang Eugene, Zhu Hongmei
1RBC Financial Group, 222 Bay St, Toronto, M5K 1G8 ON Canada.
2Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3 Canada.
Math Ind Case Stud. 2017;8(1):6. doi: 10.1186/s40929-017-0015-x. Epub 2017 Oct 3.
Credit value adjustment (CVA) is an adjustment to an existing trading price based on the counterparty-risk premium. Currently, CVA is computed with an implicit assumption that the replacement contract is default-free after the original counterparty defaults, with the assumption that those trades will not re-assigned. In the actual counterparty default settlement, it is the norm that trades will be re-assigned, especially on the buy side. Since the counterparty of the replacement contract could also default within the lifetime of an existing contract, ignoring the possibility of counterparty defaults of replacement contracts will either under or over estimate the cost of the risk. An important practical question is, therefore, how to estimate under/over pricing of CVA under current practice. In this paper, we considered the pricing of credit contingent interest rate swap (CCIRS) or credit contingent default swap (CCDS), which is considered the CVA hedge for interest rate swaps (IRS). We derived partial differential Eqs. (PDEs) satisfied by the approximated CVA with the assumption that the replacement contracts do not default. For comparison purposes, we also derived the PDEs for the cost of CVA by relaxing the assumption of default-free replacement contracts with a finite number of counterparty defaults. It shows that the no-default and two default cases can be derived within the same analytical solution framework, similar to the Funding Valuation Adjustment (FVA) problem where continuous funding is a reasonable assumption. The finite number of default case is non-trivial. The PDE for the two default case is derived in this paper. We calibrate our model based on market data and carry out extensive computations for the purpose of comparing these three CVAs. Our basic finding is that the values of the two CVAs are close for top rated counterparties. On the other hand, for counterparties with lower credit ratings, the difference among the two CVAs can be significant.
信用价值调整（CVA）是基于交易对手风险溢价对现有交易价格进行的调整。目前，计算CVA时隐含假设在原交易对手违约后，替代合约无违约风险，且假设这些交易不会重新分配。在实际的交易对手违约结算中，交易重新分配是常态，尤其是在买方。由于替代合约的交易对手在现有合约存续期内也可能违约，忽略替代合约交易对手违约的可能性会低估或高估风险成本。因此，一个重要的实际问题是，如何估计当前实践下CVA的定价偏差。在本文中，我们考虑了信用或有利率互换（CCIRS）或信用或有违约互换（CCDS）的定价，它们被视为利率互换（IRS）的CVA对冲工具。我们在假设替代合约不违约的情况下，推导了近似CVA所满足的偏微分方程（PDEs）。为作比较，我们还通过放宽替代合约无违约的假设并考虑有限数量的交易对手违约，推导了CVA成本的PDEs。结果表明，无违约和双违约情况可在同一解析解框架内得出，这类似于融资估值调整（FVA）问题，在该问题中连续融资是一个合理假设。有限数量违约情况较为复杂。本文推导了双违约情况的PDE。我们基于市场数据校准模型，并为比较这三种CVA进行了大量计算。我们的基本发现是，对于信用评级高的交易对手，两种CVA的值相近。另一方面，对于信用评级较低的交易对手，两种CVA之间的差异可能很大。
