The fourth meeting took place in Delft on Friday 13 December 2024.
Location: Lecture Hall F, Building 36, TU Delft
Registration: here
Schedule:
14:00 - 14:30 : D. Brus
14:30 - 15:00 : K. Chatziandreou
15:00 - 15:30 : B. Négyesi
15:30-16:00: Coffee Break
16:00 - 16:30 : K. Gubbels
16:30 - 17:00 : T. Maessen
17:00 - 17:30 : F. Mies
17:30-20:00: Closing and dinner
Speakets, Titles and Abstracts:
Daniël Brus (UU): Environmental Value Adjustments for asset pricing
Abstract: Climate change is a global externality from an economical perspective, despite the widely recognized urgency of its mitigation. To internalize climate change in the economy, financial incentives to reduce greengouse gas emissions, the primary driver of climate change, should be created. In this thesis, we present three environmental value adjustment (EVA) models that integrate environmental effects in asset valuation: We present a model to capture long-term climate risk in credit value adjustment (CVA) calculations, a model to price the enabled emissions of assets, and a valuation model specifically for voluntary carbon credits based on regular CVA frameworks.
Konstantinos Chatziandreou (UvA): Optimal Execution strategies in short-term energy markets under (Marked) Hawkes processes
Abstract: This thesis develops theoretical tools for the risk management and optimization of intermittent renewable energy in short-term electricity markets. The first part introduces a stochastic model based on a mutually exciting marked Hawkes process to capture key empirical characteristics of Germany’s intraday energy market prices. The model effectively reflects the increasing market activity, volatility patterns, and the Samuelson effect observed in the realized mid-price process as time to delivery approaches. By fitting the empirical signature plot of the mid-price process, the model provides a robust calibration method through a closed-form solution, using least squares to match the empirical data. Building on this foundation, the second part of the thesis explores optimal execution strategies for energy companies managing large trading volumes, whether from outages, renewable generation, or trading decisions. The study employs a linear transient price impact model combined with a bivariate Hawkes process, which models the flow of market orders, to solve a meta-order execution problem. The model determines an optimal trading trajectory, allowing traders to react to the actions of other market participants. The thesis concludes with a back-testing transaction cost analysis, comparing the proposed optimal strategy against benchmark execution strategies like Time Weighted Average Price (TWAP) and Volume Weighted Average Price (VWAP). The results confirmed that the optimal strategy is cost-efficient, significantly reducing transaction costs compared to traditional methods like TWAP and VWAP. Further analysis of individual hourly products revealed that cost reductions were particularly substantial for early trading hours (1h-4h) and stabilized thereafter, with a slight decrease in the mid-day products (12h-16h). This indicates that cost savings are negatively correlated with the average traded volume for each product, suggesting that less volatile, more liquid products might benefit less from the optimal strategy compared to less liquid ones, where improvements are more pronounced.
Koos Gubbels (U Tilbung & Achmea): Principal Component Copulas for Capital Modeling and Systemic Risk
Abstract: We introduce a class of copulas that we call Principal Component Copulas (PCCs). This class combines the strong points of copula-based techniques with principal component-based models, which results in flexibility when modelling tail dependence along the most important directions in high-dimensional data. We obtain theoretical results for PCCs that are important for practical applications. In particular, we derive tractable expressions for the high-dimensional copula density, which can be represented in terms of characteristic functions. We also develop algorithms to perform Maximum Likelihood and Generalized Method of Moment estimation in high-dimensions and show very good performance in simulation experiments. Finally, we apply the copula to the international stock market in order to study systemic risk. We find that PCCs lead to excellent performance on measures of systemic risk due to their ability to distinguish between parallel market movements, which increase systemic risk, and orthogonal movements, which reduce systemic risk. As a result, we consider the PCC promising for internal capital models, which financial institutions use to protect themselves against systemic risk. This is joint work with Jelmer Ypma (Achmea) and Kees Oosterlee (UU).
Thijs Maessen (UvA): A universal approximation theorem for signatures of Banach space valued rough paths
Abstract: Solutions of differential equations along rough paths be expressed as functions on these paths. In particular, when the rough paths come from Brownian motion, such a function would help us understand the solution an SDE. Although it is often possible to express solutions as a continuous function on rough paths, it is not always clear what this function looks like, how it behaves and how to calculate this function. For this reason, we are interested in approximating the function. This we do with a so-called Universal Approximation Theorem (UAT). A UAT, is theorem that states a class of functions can be uniformly approximated by a simpler class of functions. It has been proven that we can approximate continuous functions on a compact domain of finite dimensional rough paths with linear functions on the signature of the paths: the signature of a rough path is a series of iterated integrals along this rough path. In the talk, I will elaborate on these signatures, and this UAT. I will also talk about my attempts to generalize this UAT to the case where the rough paths can take values in Banach spaces. Possible applications of the UAT include pricing options on forwards in energy markets, as these are typically modelled on infinite dimensional spaces. This project is a joint work with Sonja Cox and Asma Khedher.
Fabian Mies (TUD): Estimation of mixed semimartingales
Abstract: The mixed fractional Brownian motion is a stochastic process given as the sum of a Brownian motion and a fractional Brownian motion with Hurst index H>3/4. It was introduced by Cherιdito (2001) as a continous-time model for financial asset prices, which does not allow for arbitrage, but exhibits long memory. However, the model can not be estimated consistently from data as it can not be statistically distinguished from a pure Brownian motion. We show that, surprisingly, the statistical estimation is recovered by introducing cross-correlation between the two process components. Motivated by this qualitative observation, we present an asymptotically normal estimator based on high-frequency asset returns and construct confidence intervals. A simulation study illustrates the potential and pitfalls of the model from an econometric perspective. This talk is based on joint work with Carsten Chong (HKUST) and Thomas Delerue (Helmholtz Munich).
Bálint Négyesi (TUD): A Deep BSDE approach for the simultaneous pricing and delta- gamma hedging of large portfolios consisting of high-dimensional multi- asset Bermudan options
Abstract: Direct financial applications of the methods introduced in Negyesi et al. (2024), Negyesi (2024) are presented. Therein, a new discretization of (discretely reflected) Markovian backward stochastic differential equations is given which involves a Gamma process, corresponding to second-order sensitivities of the associated option’s price. The main contributions of this work is to apply these techniques in the context of portfolio risk management. Large portfolios of a mixture of European and Bermudan derivatives are cast into the framework of discretely reflected BSDEs. The resulting system is solved by a neural network regression Monte Carlo method proposed in the aforementioned papers. Numerical experiments are presented on high-dimensional portfolios, consisting of several European, Bermudan and American options, which demonstrate the robustness and accuracy of the method. The corresponding Profit-and-Loss distributions indicate an order of magnitude gain in the number of rebalancing dates needed in order to achieve a certain level of risk tolerance, when the second-order conditions are also satisfied. The hedging strategies significantly outperform benchmark methods both in the case of standard delta- and delta-gamma hedging.
The third meeting will took 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.
The second meeting took place in Utrecht on Friday 24 November 2023.
Location: Room 2.02, Minnaert Building, Utrecht University
Schedule:
14:00 - 14:30 : Nestor Parolya (TUD): "Two is better than one: Regularized shrinkage of large minimum variance portfolios"
14:30 - 15:00 : Gergerly Bodo (UvA): "Stochastic integration with respect to cylindrical Lévy processes in Hilbert space"
15:00 - 15:30 : Felix Wolf (ULB, guest in UU): "Consistent asset modelling with parameter uncertainty across time regimes"
15:30-16:00: Coffee Break
16:00 - 16:30 : Gijs Mast (TUD): "Fast calculation of Potential Future Exposure and XVA sensitivities using Fourier series expansion"
16:30 - 17:00 : Sven Karbach (UvA): "Modelling Renewable Energy Markets using CARMA Processes"
17:00 - 17:30 : Thomas van der Zwaard (UU, and Rabobank): "Incorporating Smile in Valuation Adjustments Through the Randomization of Short-Rate Models"
17:30-20:00: Closing and dinner
Titles and Abstracts:
Nestor Parolya: Two is better than one: Regularized shrinkage of large minimum variance portfolios
Abstract: In this paper we construct a shrinkage estimator of the global minimum variance (GMV) portfolio by a combination of two techniques: Tikhonov regularization and direct shrinkage of portfolio weights. More specifically, we employ a double shrinkage approach, where the covariance matrix and portfolio weights are shrunk simultaneously. The ridge parameter controls the stability of the covariance matrix, while the portfolio shrinkage intensity shrinks the regularized portfolio weights to a predefined target. Both parameters simultaneously minimize with probability one the out-of-sample variance as the number of assets p and the sample size n tend to infinity, while their ratio p/n tends to a constant c>0. This method can also be seen as the optimal combination of the well-established linear shrinkage approach of Ledoit and Wolf (2004, JMVA) and the shrinkage of the portfolio weights by Bodnar, Parolya and Schmid (2018, EJOR). No specific distribution is assumed for the asset returns except of the assumption of finite 4+epsilon moments for any small epsilon>0. The performance of the double shrinkage estimator is investigated via extensive simulation and empirical studies. The suggested method significantly outperforms its predecessor (without regularization) and the nonlinear shrinkage approach by Ledoit and Wolf (2020, Annals of Stats) in terms of the out-of-sample variance, Sharpe ratio and other empirical measures in the majority of scenarios. Moreover, it obeys the most stable portfolio weights with uniformly smallest turnover. A very first draft can be found here: https://arxiv.org/abs/2202.06666
Gergely Bodó: Stochastic integration with respect to cylindrical Lévy processes in Hilbert space
Abstract: In this talk, we introduce a theory of stochastic integration with respect to arbitrary cylindrical Lévy processes in Hilbert space. One possible motivation for introducing such an integration theory comes form the mathematical modelling of commodity markets, where forward price dynamics can be defined as mild solutions of certain SPDEs driven by a cylindrical Lévy process in Hilbert space. Since cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach is based on a decoupling inequality for the tangent sequence of the Radonified increments. This enables us to characterize the largest space of predictable Hilbert-Schmidt operator-valued processes which are integrable with respect to a cylindrical Lévy process. As a by-product of our construction, we obtain an explicit analytic condition for the integrability of a predictable process, which is expressible in terms of the cylindrical characteristics of the integrator. Due to their importance in financial modelling, as an application, we consider separately the special case of standard symmetric α-stable cylindrical Lévy processes. Here, our theory simplifies significantly and it is possible to identify the largest space of predictable Hilbert-Schmidt operator-valued integrands with the collection of all predictable processes that have paths in the Bochner space L^α.
Felix Wolf: Consistent asset modelling with parameter uncertainty across time regimes
Abstract: By replacing deterministic parameters with random variables in stochastic processes, we gain greater modelling flexibility, allowing us to account for parameter uncertainty. However, this "randomization" approach initially results in challenging mixture models. To address this, we employ the quadrature integration method to convert these mixture models into local volatility models. These local volatility models are well-defined within the familiar setting of stochastic differential equations and benefit from crucial properties of the original processes, including analytical solutions. In this presentation, we extend the approach by introducing time-dependent parameter uncertainty, represented through a switching model that transitions between different randomized models. We first consider deterministic switching rules, for which we derive a local volatility model. Subsequently, we shift to stochastic switching rules, intriguingly analyzed using the same quadrature methods, again capturing randomness (in this case, the random variables governing the switches) in a local volatility model. Finally, we derive a classical regime-switching model, where a Markov chain controls the transitions between randomized processes, and compare it to the quadrature randomization results.
Gijs Mast: Fast calculation of Potential Future Exposure and XVA sensitivities using Fourier series expansion
Abstract: A new approach to calculate risk metrics of counterparty credit risk (CCR) such as Potential Future Exposure (PFE) and Expected Exposure (EE) is developed based on the Fourier cosine (COS) method. The approach can accommodate a large class of models where the joint distribution of risk factors is analytically or semi-analytically tractable, thus including almost all the CCR models used in practice. A clear advantage of the COS method over the Monte Carlo simulation method is that COS remains particularly efficient when the portfolio includes a large number of trades. Numerical tests are conducted using portfolios consisting of linear interest rate and foreign exchange products involving three risk factors, with varying portfolio sizes from 100 to 10,000 trades. Test results demonstrate that the COS method achieves a much higher level of accuracy than the Monte Carlo method, while the computation time is about 20 to 100 times shorter. The observed error convergence rates are also consistent with the theoretical predictions. Thus, it seems promising that the COS method can serve as a significantly more efficient alternative to the Monte Carlo method for calculating PFEs and EEs (and thus XVAs) on portfolio level, at least for liquid linear portfolios involving relatively few risk factors.
Sven Karbach: Modelling Renewable Energy Markets using CARMA Processes
Abstract: In this presentation, we explore the potential of Continuous-Time Autoregressive Moving Average (CARMA) processes for modeling various dynamic aspects of renewable energy markets. We demonstrate the versatility of the CARMA class by reviewing CARMA-based models of key market variables, such as electricity prices, wind power generation levels, temperature, and volatility. Furthermore, we highlight the tractability and interpretability of these CARMA-based models, underscoring their potency as robust modelling tool for pricing and hedging derivatives in renewable energy markets.
Thomas van der Zwaard: Incorporating Smile in Valuation Adjustments Through the Randomization of Short-Rate Models
Abstract: Short-rate models, falling in the category of Affine Diffusion (AD) processes, are frequently used for the computation of Valuation Adjustments (xVA). However, these models cannot be calibrated to the entire market volatility surface. We use the class of Randomized AD (RAnD) models to extend standard short-rate models to generate and control skew and smile from the model, to match the volatility patterns observed in financial markets. RAnD allows for exogenous stochasticity to be defined while remaining within the AD class for individual realizations of stochastic parameters, and therefore preserving analytic tractability. Prices under RAnD are recovered as a weighted sum of prices at quadrature points, where for each of the underlying prices the fast analytic pricer can be used. This allows us to calibrate the randomized short-rate model to the market efficiently. Furthermore, we can now assess the effect of market smile on xVA in an efficient way, in a practically relevant setting, with portfolios containing derivatives in multiple currencies for multiple underlyings.
The first meeting took place at TU Delft on Friday 31 March 2023.
Location: EEMCS-Lecture Hall Pi, Mekelweg 4, 2628 Delft (map)
Schedule:
13:30-13:45: Welcome
13:45-14:15: Jori Hoencamp
14:15-14:45: Chenguang Liu
14:45-15:15: Kristoffer Andersson
15:15-15:45: Coffee Break
15:45-16:15: Luis Souto Arias
16:15-16:45: Fenghui Yu
16:45-17:15: Matteo Michielon
17:15-20:00: Closing and dinner
Titles and Abstracts:
Kristoffer Andersson (Utrecht): "A robust deep FBSDE method for stochastic control"
Luis Souto Arias (Utrecht): "Option pricing with self-exciting jump processes"
Jori Hoencamp (Amsterdam): "The impact of stochastic volatility on initial margin and MVA in a post LIBOR world"
Chenguang Liu (Delft): "Convergence rates for BSDEs driven by Lévy processses"
Matteo Michielon (Amsterdam): "Implied risk-neutral default probabilities via conic finance"
Fenghui Yu (Delft): "Dynamic Optimal Statistical Arbitrage Strategies"
Abstract: Statistical arbitrage is a class of market-neutral investment strategies that exploits temporal price differences between similar assets by employing mean reversion models, and has been widely used by traders and hedge funds. Pairs trading is one of the most popular example of statistical arbitrage strategies. In this talk, I will first discuss the dynamic optimal pairs trading strategies which involve holding a long position and a short position simultaneously to take advantage of inefficient pricing in correlated securities. Explicit analytical trading strategies are obtained under a dynamic mean-variance framework. More general cases of statistical arbitrage will also be presented in the end.