THESIS
2018

xi, 113 pages : illustrations ; 30 cm

**Abstract**
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...[

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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.

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