A Unified Model for XVA, including Interest Rates and Rating

We start in Chapter 2 to investigate linear matrix-valued SDEs and the Itô-stochastic Magnus expansion. The Itô-stochastic Magnus expansion provides an efficient numerical scheme to solve matrix-valued SDEs. We show convergence of the expansion up to a stopping time τ and provide an asymptotic estimate of the cumulative distribution function of τ. Moreover, we show how to apply it to solve SPDEs with one and two spatial dimensions by combining it with the method of lines with high accuracy. We will see that the Magnus expansion allows us to use GPU techniques leading to major performance improvements compared to a standard Euler-Maruyama scheme. In Chapter 3, we study a short-rate model in a Cox-Ingersoll-Ross (CIR) framework for negative interest rates. We define the short rate as the difference of two independent CIR processes and add a deterministic shift to guarantee a perfect fit to the market term structure. We show how to use the Gram-Charlier expansion to efficiently calibrate the model to the market swaption surface and price Bermudan swaptions with good accuracy. We are taking two different perspectives for rating transition modelling. In Section 4.4, we study inhomogeneous continuous-time Markov chains (ICTMC) as a candidate for a rating model with deterministic rating transitions. We extend this model by taking a Lie group perspective in Section 4.5, to allow for stochastic rating transitions. In both cases, we will compare the most popular choices for a change of measure technique and show how to efficiently calibrate both models to the available historical rating data and market default probabilities. At the very end, we apply the techniques shown in this thesis to minimize the collateral-inclusive Credit/ Debit Valuation Adjustments under the constraint of small collateral postings by using a collateral account dependent on rating trigger.

Development of a hydrogel construct to prevent endoleaks occurrence following endovascular abdominal aortic aneurysm repair

Abdominal aortic aneurysm is the pathological dilation of the abdominal tract of the aorta and, if left untreated, could undergo rupture with a mortality rate of up to 90%. EVAR is the most common method for AAA treatment consisting in the internal coverage of the aorta with a metallic stent to isolate the aneurysmatic segment from the systemic circulation. Although EVAR technical success rate is high, reinterventions are common. Among the causes of reinterventions typeII endoleaks are the most frequent and consist in retrograde blood flow into the aneurysmal sac from collateral aortic branches. Continued perfusion of the aneurysm sac may lead to aneurysm rupture, therefore AAA sac embolization is performed using metallic coils. However, the presence of artifacts caused by the presence of metallic coils is a limitation because they are radiopaque and can hamper the endoleak during imaging follow-up. This study is aimed at developing a biocompatible hydrogel that could be injected into the aneurysmal sac and may allow a selective intraprocedural sac embolization to reduce post procedural typeII endoleak and eventual AAA rupture. P(BT75BSI25) was synthesized by polycondensation and its biocompatibility tested to assess whether the polymers had no toxic effects. HUVEC cell line was used to mimic the environment in which the polymer would be in contact with, PBS was used as a positive control and MTT assay was performed to evaluate cellular viability after being in contact with the hydrogel. MTT assay showed no significant difference between PBS and P(BT75BSI25), thus the polymer is biocompatible, as confirmed by the analysis of apoptosis by flow cytometry. An aromatic copolymer was obtained via polycondensation and was found to be biocompatible in contact with endothelial cells. This suggests that the hydrogel could be potentially used in the clinical setting for the treatment of type II endoleak after EVAR.