Conference Agenda

This paper presents a study of a XVA framework in which the underlying market is incomplete due to the absence of credit default swaps for the bank and the counterparty. Therefore, neither the BSDE-based replication nor the equivalent discounting approach for XVAs can be applied in this case. We address this issue by employing the local risk-minimisation approach for hedging the bank's position. As a result, we are able to describe the resulting strategy in a multi-curve setting via BSDEs.

Dynamic recalibration of risk-neutral parameters in stochastic models to align with observed prices of financial derivatives is a widely used industry practice that, however, often lacks a tractable underlying mathematical framework. We address this gap by proposing a novel methodology wherein recalibrated parameters are treated as unobservable components embedded within a larger-scale affine or polynomial model. The estimation of the dynamics of this background model then boils down to a two-step procedure, in which unobservable states are first calibrated to observed option prices using classic least-squares optimization techniques, and risk-neutral and physical model parameters are then jointly estimated to fit the trajectories of observed components alongside with the recalibrated states. We embed this joint estimation of both measures into the general framework of estimating functions and establish weak consistency and asymptotic normality of the resulting estimators. Moreover, we derive explicitly computable expressions of the asymptotic estimator covariance matrix.

Local stochastic volatility refers to a popular model class in applied mathematical finance that allows for "calibration-on-the-fly", typically via a particle method, derived from a formal McKean-Vlasov equation. Well-posedness of this limit is a well-known problem in the field with the general case still being largely open, despite recent progress in Markovian situations. Our approach is to start with a well-defined Euler approximation to the formal McKean-Vlasov equation, followed by a newly established "half-step"-scheme, allowing for good approximations of conditional expectations.

We show that this scheme converges with weak rate one regarding the step-size, plus error terms that account for the said approximation. Furthermore, the case of particle approximation is discussed in detail and the weak error rate, in dependence of all parameters used, is derived.

We discuss the limit of risk-based prices of European contingent claims in discrete-time financial markets under volatility uncertainty when the number of intermediate trading periods goes to infinity. The limiting dynamics are obtained using recently developed results for the construction of strongly continuous convex monotone semigroups. We connect the resulting dynamics to the semigroups associated to G-Brownian motion, showing in particular that the worst-case bounds always give rise to a larger bid-ask spread than the risk-based bounds. Moreover, the worst-case bounds are achieved as limit of the risk-based bounds as the agent’s risk aversion tends to infinity. The talk is based on joint work with Jonas Blessing and Alessandro Sgarabottolo.