An SGBM-XVA demonstrator: A scalable Python tool for pricing XVA

In this work, we developed a Python demonstrator for pricing total valuation adjustment (XVA) based on the stochastic grid bundling method (SGBM). XVA is an advanced risk management concept which became relevant after the recent financial crisis. This work is a follow-up work on Chau and Oosterlee in (Int J Comput Math 96(11):2272–2301, 2019), in which we extended SGBM to numerically solving backward stochastic differential equations (BSDEs). The motivation for this work is basically two-fold. On the application side, by focusing on a particular financial application of BSDEs, we can show the potential of using SGBM on a real-world risk management problem. On the implementation side, we explore the potential of developing a simple yet highly efficient code with SGBM by incorporating CUDA Python into our program

Deep Learning algorithms for solving high dimensional nonlinear Backward Stochastic Differential Equations

We study deep learning-based schemes for solving high dimensional nonlinear backward stochastic differential equations (BSDEs). First we show how to improve the performances of the proposed scheme in [W. E and J. Han and A. Jentzen, Commun. Math. Stat., 5 (2017), pp.349-380] regarding computational time by using a single neural network architecture instead of the stacked deep neural networks. Furthermore, those schemes can be stuck in poor local minima or diverges, especially for a complex solution structure and longer terminal time. To solve this problem, we investigate to reformulate the problem by including local losses and exploit the Long Short Term Memory (LSTM) networks which are a type of recurrent neural networks (RNN). Finally, in order to study numerical convergence and thus illustrate the improved performances with the proposed methods, we provide numerical results for several 100-dimensional nonlinear BSDEs including nonlinear pricing problems in finance.Comment: 21 pages, 5 figures, 16 table

Deep xVA solver -- A neural network based counterparty credit risk management framework

In this paper, we present a novel computational framework for portfolio-wide risk management problems, where the presence of a potentially large number of risk factors makes traditional numerical techniques ineffective. The new method utilises a coupled system of BSDEs for the valuation adjustments (xVA) and solves these by a recursive application of a neural network based BSDE solver. This not only makes the computation of xVA for high-dimensional problems feasible, but also produces hedge ratios and dynamic risk measures for xVA, and allows simulations of the collateral account.Comment: 33 pages. Several experiments adde

Analysis and numerical methods for stochastic volatility models in valuation of financial derivatives

[Abstract] The main objective of this thesis concerns to the study of the SABR stochastic volatility model for the underlyings (equity or interest rates) in order to price several market derivatives. When dealing with interest rate derivatives the SABR model is joined with the LIBOR market model (LMM) which is the most popular interest rate model in our days. In order to price derivatives we take advantage not only of Monte Carlo algorithms but also of the numerical resolution of the partial di erential equations (PDEs) associated with these models. The PDEs related to SABR/LIBOR market models are high dimensional in space. In order to cope with the curse of dimensionality we will take advantage of sparse grids. Furthermore, a detailed discussion about the calibration of the parameters of these models to market prices is included. To this end the Simulated Annealing global stochastic minimization algorithm is proposed. The above mentioned algorithms involve a high computational cost. In order to price derivatives and calibrate the models as soon as possible we will make use of high performance computing (HPC) techniques (multicomputers, multiprocessors and GPUs). Finally, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of Backward Stochastic Di erential Equations (BSDEs).[Resumen] El objetivo principal de la tesis se centra en el estudio del modelo de volatilidad estocástica SABR para los subyacentes (activos o tipos de interés) con vista a la valoración de diferentes productos derivados. En el caso de los derivados de tipos de interés, el modelo SABR se combina con el modelo de mercado de tipos de interés más popular en estos momentos, el LIBOR market model (LMM). Los métodos numéricos de valoración son fundamentalmente de tipo Monte Carlo y la resolución numérica de los modelos de ecuaciones en derivadas parciales (EDPs) correspondientes. Las EDPs asociadas a modelos SABR/LIBOR tienen alta dimensión en espacio, por lo que se estudian técnicas de sparse grid para vencer la maldición de la dimensión. Además, se discute detalladamente cómo calibrar los parámetros de los modelos a las cotizaciones de mercado, para lo cual se propone el uso del algoritmo de optimización global estocástica Simulated Annealing. Los algoritmos citados tienen un alto coste computacional. Con el objetivo de que tanto las valoraciones como las calibraciones se hagan en el menor tiempo posible se emplean diferentes técnicas de computación de altas prestaciones (multicomputadores, multiprocesadores y GPUs.) Finalmente se dise~na un nuevo algoritmo basado en Least-Squares Monte Carlo (LSMC) para aproximar la solución de Backward Stochastic Differential Equations (BSDEs).[Resumo] O obxectivo principal da tese céntrase no estudo do modelo de volatilidade estocástica SABR para os subxacentes (activos ou tipos de xuro) con vista á valoración de diferentes produtos derivados. No caso dos derivados de tipos de xuro, o modelo SABR combínase co modelo de mercado de tipos de xuro máis popular nestos momentos, o LIBOR market model (LMM). Os métodos numéricos de valoración son fundamentalmente de tipo Monte Carlo e a resolución numérica dos modelos de ecuacións en derivadas parciais (EDPs) correspondentes. As EDPs asociadas aos modelos SABR/LIBOR te~nen alta dimensión en espazo, polo que se estudan técnicas de sparse grid para vencer a maldición da dimensión. Ademais, discútese detalladamente como calibrar os parámetros dos modelos ás cotizacións de mercado, para o cal se propón o emprego do algoritmo de optimización global estocástica Simulated Annealing. Os algoritmos citados te~nen un alto custo computacional. Co obxectivo de que tanto as valoracións como as calibracións se fagan no menor tempo posible empréganse diferentes técnicas de computación de altas prestacións (multicomputadores, multiprocesadores e GPUs.) Finalmente deséñase un novo algoritmo baseado en Least-Squares Monte Carlo (LSMC) para aproximar a solución de Backward Stochastic Differential Equations (BSDEs)