Dutch Math Finance afternoons

The next meeting will take place in Amsterdam on Friday 7 June 2024.

Location: Science Park 904 G3.02, University of Amsterdam

Registration: here

Schedule: 

• 14:00 - 14:30 : Zhipeng Huang

• 14:30 - 15:00 : Jasper Rou

• 15:00 - 15:30 : Josha Dekker

• 15:30-16:00: Coffee Break

• 16:00 - 16:30 : Malvina Bozhidarova

• 16:30 - 17:00 : Leonardo Perotti

• 17:00 - 17:30 : Jia He

• 17:30-20:00: Closing and dinner

Speakets, Titles and Abstracts:

• Malvina Bozhidarova (UN / TUD): Describing financial crisis propagation through epidemic modelling on multiplex networks
  
  Abstract:  We introduce an innovative approach for modelling the spread of financial crises in complex networks, combining financial data, Extreme Value Theory and an epidemiological transmission model. We address two critical aspects of contagion: one driven by direct financial connections (fundamentals-based contagion) and another triggered by broader global impacts (pure contagion). Using data from 398 companies' stock prices, geography, and economic sectors, we create a multiplex network model on which a Susceptible- Infected-Recovered transmission model is defined. By studying stock prices during the 2008 and 2020 crises, we test our model's ability to predict crisis spread. The results show its effectiveness in forecasting crisis propagation, emphasizing the importance of each layer in the network for predicting specific impacts during crises.

• Josha Dekker (UvA): Optimal stopping with randomly arriving opportunities to stop
  
  Abstract: We develop methods to solve general optimal stopping problems with opportunities to stop that arrive randomly. Such problems occur naturally in applications with market frictions. Pivotal to our approach is that our methods operate on random rather than deterministic time scales. This enables us to convert the original problem into an equivalent discrete-time optimal stopping problem with natural number-valued stopping times and a possibly infinite horizon of which we establish the theoretical properties. To numerically solve this problem, we design a random times least squares Monte Carlo method. We also analyze an iterative policy improvement procedure in this setting. We illustrate the efficiency of our methods and the relevance of randomly arriving opportunities in a few examples.

• Jian He (UvA): The calibration of the credit migration models
  
  Abstract: In this paper we develop Maximum likelihood (ML) based algorithms to calibrate the model parameters in credit rating transition models. Since the credit rating transition models are not Gaussian linear models, the celebrated Kalman filter is not suitable to compute the likelihood of observed migrations. Therefore, we develop a Laplace approximation of the likelihood function and as a result the Kalman filter can be used in the end to compute the likelihood function. This approach is applied to so-called high-default portfolios, in which the number of migrations (defaults) is large enough to obtain high accuracy of the Laplace approximation. By contrast,  low-default portfolios have a limited number of observed migrations (defaults). Therefore, in order to calibrate  low-default portfolios, we develop a ML algorithm using a particle filter (PF) and Gaussian process regression. Experiments show that both algorithms are efficient and produce accurate approximations of the likelihood function and the ML estimates of the model parameters.

• Zhipeng Huang (UU): Generalized convergence for the Deep BSDE method
  
  Abstract: We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long  (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to fully-coupled drift coefficients, and give sufficient conditions for convergence under standard assumptions. The resulting conditions are directly verifiable for any equation. Consequently, our convergence analysis enables the treatment of FBSDEs stemming from stochastic optimal control problems. In particular, we provide a theoretical justification for the non-convergence of the deep BSDE method observed in recent literature, and present direct guidelines for when convergence can be guaranteed in practice. Our theoretical findings are supported by several numerical experiments in high-dimensional settings. This is a joint work with Balint Negyesi and Cornelis W. Oosterlee.

• Leonardo Perotti (UU): Modeling and Replication of the Prepayment Option of Mortgages including Behavioral Uncertainty
  
  Abstract: Prepayment risk embedded in fixed-rate mortgages is a significant fraction of a financial institution's exposure, and it receives particular attention because of the magnitude of the underlying market. The embedded prepayment option (EPO) bears the same interest rate risk of an exotic interest rate swap (IRS) with a suitable stochastic notional. We investigate the effect of relaxing the assumption of a deterministic relationship between the market interest rate incentive and the prepayment rate. A non-hedgeable risk factor is used to capture the uncertainty in mortgage owners' behavior, leading to an incomplete market. We show that including behavioral uncertainty reduces the exposure's value. We observe that, due to the incomplete economy, the true price of prepayment is not known. However, bounds are computed based on the risk appetite/aversion of the financial institution. The exposure resulting from the embedded prepayment option is statically replicated with IRSs and swaptions. We observe that swaps only cannot control the right tail of the exposure distribution, while including swaptions allows that. The replication framework is flexible and focusing on different features of interest in the exposure distribution (such as, e.g, its tail behavior) is achieved using additional terms in the objective of the optimization.

• Jasper Rou (TUD): Convergence of Deep Gradient Flow Methods for Option Pricing
  
  Abstract: In this research, we consider the convergence of neural network algorithms for option pricing partial differential equations (PDE). More specifically, we consider a Time-stepping Deep Gradient Flow method, where the PDE is solved by discretizing it in time and writing it as the solution of minimizing a variational problem. A neural network approximation is then trained to solve this minimization using stochastic gradient descent. This method reduces the training time compared to, for instance, the Deep Galerkin Method. We prove two things. First, as the number of nodes of the network goes to infinity that there exists a neural network converging to the solution of the PDE. This proof consists of three parts: 1) convergence of the time stepping; 2) equivalence of the discretized PDE and the minimization of the variational formulation and 3) convergence of the neural network approximation to the solution of the minimization problem by using a version of the universal approximation theorem. Second, as the training time goes to infinity that stochastic gradient descent will converge to the neural network that solves the PDE.