Anastasia Borovykh, Andrea Pascucci, Cornelis W. Oosterlee: We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential L ́evy-type martingale. This class of models allows for a local volatility, local default intensity and a locally dependent L ́evy measure. We present a pricing method for Bermudan options based on an analytical approximation of the characteristic function combined with the COS method. Due to a special form of the obtained characteristic function the price can be computed using a fast Fourier transform-based algorithm resulting in a fast and accurate calculation. The Greeks can be computed at almost no additional computational cost. Error bounds for the approximation of the characteristic function as well as for the total option price are given.

# ESR4 Anastasia Borovykh

Anastasia Borovykh, Andrea Pascucci, Cornelis W. Oosterlee: Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FB-SDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local L ́evy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.

Anastasia Borovykh, Sander Bohte, Cornelis W. Oosterlee : We develop a modern deep convolutional neural network for conditional time series forecasting based on the recent WaveNet architecture. The proposed network contains stacks of dilated convolutions that widen the receptive field of the forecast; multiple convolutional filters are applied in parallel to separate time series and allow for the fast processing of data and the exploitation of the correlation structure between the multivariate time series. The performance of the deep convolutional neural network is analyzed on various multivariate time series including commodities data and stock indices and compared to that of the well-known autoregressive model and a fully convolutional network. We show that our network is able to effectively learn dependencies between the series without the need of long historical time series and significantly outperforms the baseline neural forecasting models.

Anastasia Borovykh, Andrea Pascucci, Stefano La Rovere: In this paper we consider a mean-field model of interacting diffusions for the monetary reserves in which the reserves are subjected to a self- and cross-exciting shock. This is motivated by the financial acceleration and fire sales observed in the market. We derive a mean-field limit using a weak convergence analysis and find an explicit measure-valued process associated with a large interbanking system. We define systemic risk indicators and derive, using the limiting process, several law of large numbers results and verify these numerically. We conclude that self-exciting shocks increase the systemic risk in the network and their presence in interbank networks should not be ignored.

Anastasia Borovykh: In this paper we cast the well-known convolutional neural network in a Gaussian process perspective. In this way we hope to gain additional insights into the performance of convolutional networks, in particular understand under what circumstances they tend to perform well and what assumptions are implicitly made in the network. While for feedforward networks the properties of convergence to Gaussian processes have been studied extensively, little is known about situations in which the output from a convolutional network approaches a multivariate normal distribution. In the convolutional net the sum is computed over variables which are not necessarily identically distributed, rendering the general central limit theorem useless. Nevertheless we can apply a Lyapunov-type bound on the distance between the Gaussian process and convolutional network output, and use this bound to study the properties under which the convolutional network behaves approximately like a Gaussian process, so that this behavior –depending on the application– can be either obtained or avoided.

Anastasia Borovykh, Stefano La Rovere: The aim of these notes is to present a short overview of the methods that can be used to compute risk in financial systems, in particular focusing on methods that can be used as adaptations of the numerical simulations used in modeling engineering systems. When modeling an engineering system one considers the different components making up the system, determines the interactions between these components and uses this to simulate quantities of interest with regard to the total system. The quantities of interest when working with reliability analysis are e.g. system failure, what leads to the failure of the system, effects of maintenance. We start with describing a general Monte Carlo (MC) algorithm which can be used in order to compute the state of the system as a whole through time in Section 2.