ESR1 worked on the modeling of wrong way risk (WWR), a cutting-edge topic was the joint modelling of the probability of default (PD) and the loss given default (LGD). ESR1 modeled the dependence between PD and LGD, and took into consideration the discrepancies between risk-neutral and physical measures. ESR1’s work was novel in the sense that reinforcement urn processes were being considered for the modeling. In many applications, data is only available in an incomplete form. ESR1 proposed an extension of the Expectation-Maximization (EM) algorithm for RUPs, both in the univariate and the bivariate case. Furthermore, a new methodology combining EM and the prior elicitation mechanism of RUPs was developed: the Expectation-Reinforcement algorithm.
ESR2 has worked on Credit Valuation Adjustment (CVA), the difference between the risk-free portfolio value and the risky portfolio value, i.e. the portfolio value when taking default risk of the counterparty into account. Beforehand, it was not completely clear how the risky portfolio value should be computed, whether the exercise policy should be adjusted for the fact that the counterparty may default. One consequence was that, in the presence of a netting agreement, the exercise decisions could no longer be made individually, whereas almost all existing algorithms relied on exercise decisions made in isolation. In practice, this meant that the counterparty could be paying a CVA which was based on a sub-optimal exercise strategy. ESR2 proposed a neural network-based method for approximating the expected exposures and potential future exposures.
ESR3 worked on counterparty risk, the risk of default by the counterparty in a financial contract.ESR3 combined PDE and Monte Carlo methods in a hybrid solution approach. From February 2022 we continued with a new researcher ESR3B whose work was also framed into the modelling, mathematical analysis and numerical methods for the pricing of XVA, with the main emphasis devoted to capital value adjustment (KVA). When a bank executes a financial derivatives trade there is an incremental impact on its regulatory capital requirement. The cost is reflected in the spread charged for the trade, known as KVA, which gives rise to a nontrivial calculation, and traditional approaches make specific assumptions regarding the adjustment to a bank’s return on equity. ESR3B worked on the numerical solution of the involved BSDEs and PDEs.
ESR4 worked on the extension of valuation adjustments from a single currency to a multi-currency setting, focussing on the pricing of European options with valuation adjustments. Moreover, stochastic intensities of default were assumed and underlying assets in different currencies were involved. ESR4 solved formulations of the XVA pricing problem by the Monte Carlo Method which was not affected by the curse of dimensionality. The numerical examples helped in the analysis of the XVA behavior and its dependence on the underlying assets and the investor’s credit spread.
ESR5 worked on collateralization, a market mechanism which efficiently reduces credit risk from over-the-counter derivatives. The optimal choice of collateral securities is known as the cheapest to deliver (CTD) collateral. Under full substitution rights, the entire collateral account could be switched from one collateral security to another at any time. ESR5 considered the case of full substitution rights, assuming an instantaneous exchange of collateral in continuous time. The focus of ESR5’s modeling approach was on a conditionally independent approximation of the involved processes at interpolation points, which led to improved analytical tractability in a second-order model, while preserving some of the correlation structures observed in the market.
ESR6 analyzed a stochastic version of the Magnus expansion for the solution of linear systems of Ito stochastic differential equations (SDEs). He proved existence and provided a representation formula for the logarithm associated to the solution of the matrix-valued SDEs. The Magnus expansion provided a novel method for solving stochastic PDEs (SPDEs). ESR6 presented tests utilizing parallelization on a GPU to accelerate the evaluation of the model. ESR6, in close collaboration with Banco Santander, has been working on a novel idea for the introduction of SPDEs in the context of so-called rating triggers, i.e. the consideration of rating changes of a counterparty and not only a direct change to default.
