In this thesis, we will redefine the notions of CVA, DVA and FVA and derive the formulae for the rest of xVA as the expected present values of excessive cash flows due to funding spreads under the risk-neutral pricing measure. We uncover the corresponding replication strategies against market risk and counterparty default risk. The cost of replication is understood as no-default value and bilateral CVAs. The additional costs caused by funding spreads defines FVA, out of which we further identify FCA, MVA, ColVA and KVA as components.

In the second chapter, we extend the credit model to stochastic hazard rates, and focus on the modelling of CVA and wrong way risk. The model of wrong way risk has two forms. The weak form of wrong way risk is modelled as instantaneous diffusion c...[ Read more ]

In this thesis, we will redefine the notions of CVA, DVA and FVA and derive the formulae for the rest of xVA as the expected present values of excessive cash flows due to funding spreads under the risk-neutral pricing measure. We uncover the corresponding replication strategies against market risk and counterparty default risk. The cost of replication is understood as no-default value and bilateral CVAs. The additional costs caused by funding spreads defines FVA, out of which we further identify FCA, MVA, ColVA and KVA as components.

In the second chapter, we extend the credit model to stochastic hazard rates, and focus on the modelling of CVA and wrong way risk. The model of wrong way risk has two forms. The weak form of wrong way risk is modelled as instantaneous diffusion correlation between the market risk factors and the hazard rate. The strong form of wrong way risk is modelled as jump on default in the market risk factors. We develop a two-level variance reduction method, in which we apply important sampling and conditional sampling. In the example we use, simulation variance is reduced on average by a factor of 20 compared with the case of no variance reduction.

In the third chapter, we explore new method to compute dynamic initial margin. This technique can be further implemented in the xVA engine to compute MVA. Initial margin is defined as value at risk of P&L within a certain period of time. We choose the stochastic kriging method, which is essentially an interpolation method, to substitute nest monte carlo method to compute initial margin. We sequentially sample at certain states (equity prices, libor rates) where we can gain most information, and used this information to build a metamodel, which can help to evaluate the mean and variance of portfolio value for risk states within a certain stressed period. The mean and variance information helps to identify the state that correspond to quantile at a certain confidence level, and also instruct on how additional simulation budget shall be allocated. Compared to previous paper [68] and [32], our method can automatically sample at tail scenarios, and allocate simulation budget more effectively.