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.

Radiocarbon and stable isotopes in the shell organic matrix: a new approach for the use of mollusk shells in the study of human evolution

Mollusk shells are often found in archeological sites, given their great preservation potential and high value as a multipurpose resource. They are often the only available material to use for radiocarbon dating, due to a lack of well-preserved bones in many archeological sites, especially for the key period of the Middle to Upper Paleolithic transition. However, radiocarbon dating on mollusk shells is often regarded as less reliable compared to bones, wood, or charcoals due to the various factors influencing their radiocarbon content (e.g., Isotope fractionation, marine reservoir effect etc.). For the development of more accurate chronologies using shells, it is fundamental to continue improving the precision of the techniques applied, as has been done for other materials (wood and bones). Thus, improving the chemical pretreatment on mollusk shells might allow researchers to obtain more reliable radiocarbon determinations allowing for the construction of new radiocarbon chronologies in archeological sites where so far it has not been possible. Furthermore, mollusk shells can provide information on the climatic and environmental variables present during their growth. Using shells for paleoclimatic reconstruction adds more evidence helpful for the interpretation of scenarios of human migration, adaptation, and behavior. Standard methods for both radiocarbon and stable isotope studies use the carbonate fraction of the shell. However, being biogenic structures, mollusk shells also consist of a minor organic fraction. The shell organic matrix has an important role in the formation of the calcium carbonate structure and is still not fully understood. This thesis explores the potential of using the shell organic matrix for radiocarbon dating and paleoenvironmental studies. The results of the work performed for this thesis represent a starting point for future research to build on, and further develop the approach and methodology proposed here.