Minicourses
Christoph Reisinger:
tba
Almut Veraart:
Advances in ambit stochastics and continuous-time network processes: Applications to energy markets and high-frequency financial data
This mini-course presents recent developments in ambit stochastics and continuous-time network stochastic processes, with applications to high-frequency financial data and energy markets. The first part focuses on trawl processes, Lévy semistationary processes, and ambit fields, highlighting their ability to model complex dependence structures and stochastic volatility in continuous time. Particular attention is given to the probabilistic properties of trawl processes and to statistical inference, including pairwise maximum likelihood estimation and nonparametric methods. The course also illustrates how this methodology can be applied to high-frequency limit order book data. The second part of the course is devoted to continuous-time stochastic processes on networks. After introducing graph Ornstein-Uhlenbeck processes, the course considers graph supOU extensions and discusses their use in modelling network-dependent phenomena in energy systems, with a particular application to wind energy capacity factors. Special invited lectures
Eva Lütkebohmert:
Empirical performances of the Bayesian generalized recovery
We propose a novel approach for recovering physical densities from quoted option prices by explicitly accounting for data errors in the implied volatility surface. Building on the generalized recovery theorem of Jensen et al. (2019), we employ Bayesian calibration with sequential Monte Carlo methods to account for model uncertainty. Unlike existing approaches, our framework incorporates the full posterior distribution of model parameters rather than relying on point estimates. We demonstrate that our methodology produces robust estimates of the physical density that can be exploited to construct profitable investment strategies. Overall, our findings indicate that Bayesian recovery delivers more accurate and informative estimates with stronger predictive performance than conventional recovery methods. Joint work with Riccardo Brignone and Sven Knaust.
Marcel Nutz:
Optimal Fees for Liquidity Provision in Automated Market Makers
Passive liquidity providers (LPs) in automated market makers (AMMs) face losses due to adverse selection (LVR), which static trading fees often fail to offset in practice. We study the key determinants of LP profitability in a dynamic reduced-form model where an AMM operates in parallel with a centralized exchange (CEX), traders route their orders optimally to the venue offering the better price, and arbitrageurs exploit price discrepancies. Using large-scale simulations, we analyze how LP profits vary with market conditions such as volatility and trading volume, and characterize the profit-maximizing AMM fee at fixed liquidity. We then endogenize liquidity through competitive LP entry and characterize the fee that maximizes equilibrium total value locked (TVL). We highlight the mechanisms driving these relationships through extensive comparative statics, and confirm the model's relevance through market data calibration. A key trade-off emerges: fees must be low enough to attract volume, yet high enough to earn sufficient revenues and mitigate arbitrage losses. We find that under normal market conditions, the profit-maximizing AMM fee is competitive with the trading cost on the CEX and remarkably stable, whereas in periods of very high volatility, a high fee protects passive LPs from severe losses. Similarly, under competitive entry, the TVL-maximizing fee typically lies slightly below the CEX trading cost, and equilibrium liquidity falls sharply with volatility. Our findings suggest that a threshold-type dynamic fee schedule is robust to market conditions and improves LP outcomes.
Sergio Pulido:
When do Volterra processes have a finite-dimensional Markovian representation?
Finite-dimensional Markovian representations of Volterra processes are important for practical applications. In this talk, we provide a characterization of the kernels that enable such representations, based on geometric arguments in an infinite-dimensional lifting framework. We also highlight connections with finite-dimensional realizations in HJM fixed income models. Joint work with Alexandre Pannier. Short lectures
Konstantinos Chatziandreou:
Semi-Static hedging of volumetric risk in energy markets
Power purchase agreements (PPAs) play a pivotal role in the green energy transition by locking in energy prices for uncertain future production from renewable plants, often over horizons of 10–15 years. The value of a PPA is largely determined by the joint distribution of future renewable generation and forward electricity prices. In this talk, we present quantitative methods for pricing PPAs and hedging their inherent risks. We develop a coupled model for forward electricity prices and renewable power production indices. The use of a Wishart-type stochastic covariance model allows us to capture the complex covariance structure between future production volumes and forward electricity prices. In addition, we deploy a semi-static variance-optimal hedging strategy, where the dynamic component involves trading in liquid electricity forward contracts, while the static buy-and-hold component consists of a basket of contingent claims. Key tools in our approach are quanto indices and weather derivatives, which provide tailored risk management solutions by capturing stochastic price–production covariance. By jointly addressing volume risk and price risk within specific payoff structures, such as swaps, puts, and calls, these products can be used to implement risk management strategies that increase revenue stability. Finally, we explore the effectiveness of this integrated approach in mitigating the volume and price risks intrinsic to PPAs.
Serena Della Corte:
tba
Bud Schiphorst:
Continuous-time embedding algorithms for absorbing Markov chains.
Markov chains are widely used for modelling stochastic state transitions across various disciplines, notably for modelling credit rating migration in finance. A continuous-time Markov chain (CTMC) is especially useful, as it allows for modelling state transitions for any future horizon. However, in practice, only a pre-calibrated discrete-time model may be available, represented by a transition probability matrix. In that case, embedding algorithms can conveniently provide a CTMC that approximates the existing discrete-time model. Existing embedding algorithms typically rely on generic loss functions that prioritize single-step transitions and treat all states as equally important. We argue that this is insufficient for many practical applications, because it ignores the importance of multi-step survival probabilities in absorbing Markov chains. In this work, we propose a novel embedding algorithm that more effectively preserves the discrete-time survival information of the original absorbing Markov chain. Specifically, we introduce a generalized framework for constructing Expectation-Maximization (EM) algorithms that minimize divergence-based embedding loss functions. Using this framework, we develop a specialized EM algorithm designed to maintain the original probability distribution of survival times. Numerical experiments based on empirical credit rating transition matrices demonstrate that our approach strikes a robust balance between preserving survival probabilities and fitting overall transition dynamics across various time horizons. We benchmark our method against earlier EM-based embedding algorithms and standard approaches, including regularization methods and distance minimization.
Nguyen N.T. Truong:
Single- and Multi-Level Fourier-RQMC Methods for Multivariate
Shortfall Risk
Multivariate shortfall risk measures provide a principled framework for quantifying systemic risk and determining capital allocations prior to aggregation in interconnected financial systems. Despite their well-established theoretical properties, the numerical estimation of multivariate shortfall risk and the corresponding optimal allocations remains computationally challenging, as existing Monte Carlo–based approaches can be numerically expensive due to slow convergence. In this work, we develop a new class of single- and multi-level numerical algorithms for estimating multivariate shortfall risk and the associated optimal allocations, based on a combination of Fourier inversion techniques and randomized quasi–Monte Carlo (RQMC) sampling. Rather than operating in physical space, our approach evaluates the relevant expectations appearing in the risk constraint and its optimization in the frequency domain, where the integrands exhibit enhanced smoothness properties that are well suited for RQMC integration. We establish a rigorous mathematical framework for the resulting Fourier-RQMC estimators, including convergence analysis and computational complexity bounds. Beyond the single-level method, we introduce a multilevel RQMC scheme that exploits the geometric convergence of the underlying deterministic optimization algorithm to reduce computational cost while preserving accuracy. Numerical experiments demonstrate that the proposed Fourier–RQMC methods outperform sample average approximation and stochastic optimization benchmarks in terms of accuracy and computational cost across a range of models for the risk factors and loss structures. Consistent with the theoretical analysis, these results demonstrate improved asymptotic convergence and complexity rates relative to the benchmark methods, with additional savings achieved through the proposed multilevel RQMC construction. To the homepage of the Winter School on Mathematical Finance |