Dutch pension funds are under the supervision by De Nederlandsche Bank (DNB) and they must adhere to the Financial Assessment Framework (FTK), which outlines the methods for calculating liabilities, buffer reserves, and risk factors. As part of the FTK, a feasibility test must be performed, which is a scenario-based analysis of pension funds' investment strategies and their pension policies based on economic scenario sets projected 100 years into future. Every quarter, DNB publishes these economic scenarios (P-scenarios) and equivalent risk-neutral scenarios (Q-scenarios) are used to calculate the effects of the pension fund reform in the Netherlands. These scenarios are generated by a stochastic model, which is revised every 5 years by a commission appointed by the Dutch government, called the Commission Parameters. The model currently in use is based on Commission Parameters 2022 and is referred to as CP2022.

The CP2022 model is an affine and arbitrage-free model framework for correlated interest rate, inflation rate, stock index and consumer price index, each being driven by Heston's stochastic volatility process. To generate realistic economic scenarios, the model is calibrated quarterly using bond and option market data. Currently, as part of the calibration process, option prices are computed under the dynamics stated by CP2022 using a Monte Carlo-based approach. As an alternative, we resort to a Fourier method called the COS method, which relies on the availability of the characteristic function of the state factors. We propose an efficient option valuation scheme for pricing European call and put options, and cliquet options under CP2022. Compared to the Monte Carlo method, our scheme demonstrates superior accuracy and computational efficiency.

In an attempt to find alternatives for solving partial differential equations (PDEs)
with traditional numerical methods, a new field has emerged which incorporates
the residual of a PDE into the loss function of an Artificial Neural Network. This
method is called Physics-Informed Neural Network (PINN). In this thesis, we study dense neural networks (DNNs), including codes developed in the context of this bachelor project. We derive the backpropagation equations necessary for training and use different configurations in a DNN to test its interpolating accuracy. We distinguish between a-PINNs which use automatic differentiation to evaluate a PDE, and n-PINNs which approximate differential operators in a PDE with numerical differentiation. We compare both PINNs on the harmonic oscillator, the 1D heat equation and the 1-soliton and 2-soliton solutions of the Korteweg-De Vries (KdV) equation. Both PINNs could accurately converge to the solution, except to the 2-soliton solution, where the a-PINN outperformed the n-PINN. Furthermore, we tested a highly nonlinear problem of the KdV equation, which can be described by a train of solitons. We observed that PINNs are inaccurate if insufficient training samples are used for training. Adding training samples on the interior from a numerical solution leads to a good qualitative agreement, though more effort is required to find a better network configuration to obtain more accurate predictions.
Additionally, PINNs were used for inverse problems to derive an unknown coefficient in a PDE and proved to be highly accurate for noiseless data. When we
generated training samples with 10% noise from a uniform distribution, the PINN
results’ relative error stayed within a margin of under 2%. However, inverse PINNs are much more inefficient compared to nonlinear least squares methods like the Levenberg–Marquardt algorithm.
As of now, PINNs are still very early in development and stand no match against
traditional numerical methods to a known PDE. They may, however, provide a
useful alternative in the future as they are constantly being improved.