Dutch Math Finance afternoons
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/2202.06666Gergely 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.