Publications

ESR1 Luis Souto

Joint and Survivor Annuity Valuation with a Bivariate Reinforced Urn Process, 27 July 2020

Using a Bi-variate Reinforced Urn Process (B-RUP), a novel way of modeling the dependence of coupled lifetimes is introduced, with application to the pricing of joint and survivor annuities. In line with the machine learning paradigm, the model is able to improve its performances over time, but it also allows for the use of a priori information, like for example experts’ judgement, to complement the empirical data. Using a well-known Canadian data set, the performances of the B-RUP are studied and compared with the existing literature.

Estimation for Univariate and Bivariate Reinforced Urn Processes under Left-Truncation and Right-Censoring, 21 October 2020

Reinforced Urn Processes (RUPs) represent a flexible class of Bayesian nonparametric models suitable for dealing with possibly right-censored and left-truncated observations. A reliable estimation of their hyper-parameters is however missing in the literature. We therefore propose 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 is developed: the Expectation-Reinforcement algorithm. Numerical results showing the performance of both algorithms are presented for several analytical examples as well as for a large data set of Canadian annuities.

Joint and Survivor Annuity Valuation with a Bivariate Reinforced Urn Process, April 2021

We introduce a novel way of modeling the dependence of coupled lifetimes, for the pricing of joint and survivor annuities. Using a well-known Canadian data set, our results are analyzed and compared with the existing literature, mainly relying on copulas. Based on urn processes and a one-factor construction, the proposed model is able to improve its performances over time, in line with the machine learning paradigm, and it also allows for the use of experts’ judgements, to complement the empirical data.

About the Estimation of Reinforced Urn Processes under Left-Truncation and Right-Censoring, November 2021

Reinforced Urn Processes (RUPs) represent a flexible class of Bayesian nonparametric models suitable for dealing with possibly right-censored and left-truncated observations. A reliable estimation of their hyper-parameters is however missing in the literature. We therefore propose 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 is developed: the Expectation-Reinforcement algorithm. Numerical results showing the performance of both algorithms are presented using artificial and actual data.

ESR2 Kristoffer Andersson

Deep learning for CVA computations of large portfolios of financial derivatives, 28 October 2020

In this paper, we propose a neural network-based method for CVA computations of a portfolio of derivatives. In particular, we focus on portfolios consisting of a combination of derivatives, withand without true optionality, e.g.,a portfolio of a mix of European- and Bermudan-type derivatives.CVA is computed, with and without netting, for different levels of WWR and for different levels of credit quality of the counterparty. We show that the CVA is overestimated with up to 25%by using the standard procedure of not adjusting the exercise strategy for the default-risk of the counterparty. For the Expected Shortfall of the CVA dynamics, the overestimation was found to be more than 100% in some non-extreme cases.

A Deep Learning Approach for Computations of Exposure Profiles for High-Dimensional Bermudan Options, 2021

In this paper, we propose a neural network-based method for approximating expected ex- posures and potential future exposures of Bermudan options. In a first phase, the method relies on the Deep Optimal Stopping algorithm (DOS) proposed by Becker, Cheridito, and Jentzen (2019), which learns the optimal stopping rule from Monte-Carlo samples of the underlying risk factors. Cashflow paths are then created by applying the learned stopping strategy on a new set of realizations of the risk factors. Furthermore, in a second phase the cashflow paths are projected onto the risk factors to obtain approximations of path- wise option values. The regression step is carried out by ordinary least squares as well as neural networks, and it is shown that the latter results in more accurate approximations. The expected exposure is formulated, both in terms of the cashflow paths and in terms of the pathwise option values and it is shown that a simple Monte-Carlo average yields accurate approximations in both cases. The potential future exposure is estimated by the empirical α-percentile. Finally, it is shown that the expected exposures, as well as the potential future exposures can be computed under either, the risk neutral measure, or the real world measure, with- out having to re-train the neural networks.

Deep Learning for CVA Computations of Large Portfolios of Financial Derivatives, 2021

In this paper, we propose a neural network-based method for CVA computations of a port- folio of derivatives. In particular, we focus on portfolios consisting of a combination of derivatives, with and without true optionality, e.g., a portfolio of a mix of European- and Bermudan-type derivatives. CVA is computed, with and without netting, for different levels of WWR and for different levels of credit quality of the counterparty. We show that the CVA is overestimated with up to 25% by using the standard procedure of not adjusting the exercise strategy for the default-risk of the counterparty. For the Expected Shortfall of the CVA dynamics, the overestimation was found to be more than 100% in some non-extreme cases.

ESR3 Graziana Colonna

GPU Acceleration of the Seven-League Scheme for Large Time Step Simulations of Stochastic Differential Equations

Monte Carlo simulation is widely used to numerically solve stochastic differential equations. Although the method is flexible and easy to implement, it may be slow to converge. Moreover, an inaccurate solution will result when using large time steps. The Seven League scheme [1], a deep learning-based numerical method, has been proposed to address these issues. This paper generalizes the scheme regarding parallel computing, particularly on Graphics Processing Units (GPUs), improving the computational speed.

ESR4 Roberta Simonella

Total Value Adjustment for European Options in a Multi-Currency Setting

In this article we mainly extend to a multi-currency setting some previous works in the literature concerning total value adjustments in a single currency framework. The motiva- tion comes from the fact that financial institutions operate in global markets, so that the financial derivatives in their portfolios involve different currencies. More precisely, in this multi-currency setting we pose the PDE formulations for pricing the total adjustment and the financial derivative with counterparty risk. Moreover, we also formulate the problem in terms of expectations, which allows the use of suitable Monte Carlo techniques that over- come the curse of dimensionality associated to the numerical solution of PDE formulation, when a high number of stochastic factors are involved. Finally, we present some examples to illustrate the performance of the formulations and the proposed numerical techniques.

ESR5 Felix Wolf

Cheapest-to-Deliver Collateral: A Common Factor Approach

The collateral choice option gives the collateral posting party the opportunity to switch between different collateral currencies which is well-known to impact the asset price. Quantifcation of the option's value is of practical importance but remains challenging under the assumption of stochastic rates, as it is determined by an intractable distribution which requires involved approximations. Indeed, many practitioners still rely on deterministic spreads between the rates for valuation. We develop a scalable and stable stochastic model of the collateral spreads under the assumption of conditional independence. This allows for a common factor approximation which admits analytical results from which further estimators are obtained. We show that in modelling the spreads between collateral rates, a second order model yields accurate results for the value of the collateral choice option. The model remains precise for a wide range of model parameters and is numerically effcient even for a large number of collateral currencies.

ESR6 Kevin Kamm

On the Stochastic Magnus Expansion and its Application to SPDEs, 25 February 2020

We derive a stochastic version of the Magnus expansion for the solution of linear systems of Ito stochastic differential equations (SDEs). The goal of this paper is twofold. First, we prove existence and a representation formula for the logarithm associated to the solution of the matrix-valued SDEs. Second,we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature oft he method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

On the Stochastic Magnus Expansion and Its Application to SPDEs

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time τ and provide a novel asymptotic estimate of the cumulative distribution function of τ . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable.We also present numerical tests in order to asses the accuracy of the numerical schemes.

How to handle Negative Interest Rates in a CIR Framework

In this paper, we propose a new model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model without introducing a shift to the market interest rates, because it is defined as the difference of two independent CIR processes. The strength of our model lies within the fact that it is very simple and can be calibrated to the market zero yield curve using an analytical formula.We run several numerical experiments at two different dates, once with a partially sub-zero interest rate and once with a fully negative interest rate. In both cases, we obtain good results in the sense that the model reproduces the market term structures very well. We then simulate the model using the Euler–Maruyama scheme and examine the mean, variance and distribution of the model. The latter agrees with the skewness and fat tail seen in the original CIR model. In addition, we compare the model’s zero coupon prices with market prices at different future points in time. Finally, we test the market consistency of the model by evaluating swaptions with different tenors and maturities.