Upcoming talks

01.07.2024 14:15 Michael Kupper (University of Konstanz): Discrete approximation of risk-based pricing under volatility uncertainty

We discuss the limit of risk-based prices of European contingent claims in discrete-time financial markets under volatility uncertainty when the number of intermediate trading periods goes to infinity. The limiting dynamics are obtained using recently developed results for the construction of strongly continuous convex monotone semigroups. We connect the resulting dynamics to the semigroups associated to G-Brownian motion, showing in particular that the worst-case bounds always give rise to a larger bid-ask spread than the risk-based bounds. Moreover, the worst-case bounds are achieved as limit of the risk-based bounds as the agent’s risk aversion tends to infinity. The talk is based on joint work with Jonas Blessing and Alessandro Sgarabottolo.

Previous talks

within the last year

03.06.2024 14:15 Lorenzo Schönleber (Collegio Carlo Alberto in Turin): Implied Impermanent Loss: A Cross-Sectional Analysis of Decentralized Liquidity Pools

We propose a continuous-time stochastic model to analyze the dynamics of impermanent loss in liquidity pools in decentralized finance (DeFi) protocols. We replicate the impermanent loss using option portfolios for the individual tokens. We estimate the risk-neutral joint distribution of the tokens by minimizing the Hansen–Jagannathan bound, which we then use for the valuation of options on relative prices and for the calculation of implied correlations. In our analyses, we investigate implied volatilities and implied correlations as possible drivers of the impermanent loss and show that they explain the cross-sectional returns of liquidity pools. We test our hypothesis on options data from a major centralized derivative exchange.

03.06.2024 15:00 Maximilian Würschmidt (Universität Trier): A Probabilistic Approach to Shape Derivatives

In this talk, we introduce a novel mesh-free and direct method for computing the shape derivative in PDE-constrained shape optimization problems. Our approach is based on a probabilistic representation of the shape derivative and is applicable for second-order semilinear elliptic PDEs with Dirichlet boundary conditions and a general class of target functions. The probabilistic representation derives from a boundary sensitivity result for diffusion processes due to Costantini, Gobet and El Karoui. Via so-called Taylor tests we verify the numerical accuracy of our methodology.

03.06.2024 16:15 Marco Fritelli, Universität Mailand: Collective Arbitrage, Super-replication and Risk Measures

The theory we present aims at expanding the classical Arbitrage Pricing Theory to a setting where N agents invest in stochastic security markets while also engaging in zero-sum risk exchange mechanisms. We introduce in this setting the notions of Collective Arbitrage and of Collective Super-replication and accordingly establish versions of the fundamental theorem of asset pricing and of the pricing-hedging duality. When computing the Collective Super-replication price for a given vector of contingent claims, one for each agent in the system, allowing additional exchanges among the agents reduces the overall cost compared to classical individual super-replication. The positive difference between the aggregation (sum) of individual superhedging prices and the Collective Super-replication price represents the value of cooperation. Finally, we explain how these collective features can be associated with a broader class of risk measurement or cost assessment procedures beyond the superhedging framework. This leads to the notion of Collective Risk Measures, which generalize the idea of risk sharing and inf-convolution of risk measures.

06.05.2024 14:15 Alexander Merkel TU Berlin: LQG Control with Costly Information Acquisition

Abstract We consider the fundamental problem of Linear Quadratic Gaussian Control on an infinite horizon with costly information acquisition. Specifically, we consider a two-dimensional coupled system, where one of the two states is observable, and the other is not. Additionally, to inference from the observable state, costly information is available via an additional, controlled observation process. Mathematically, the Kalman-Bucy filter is used to Markovianize the problem. Using an ansatz, the problem is then reduced to one of the control-dependent, conditional variance for which we show regularity of the value function. Using this regularity for the reduced problem together with the ansatz to solve the problem by dynamic programming and verification and construct the unique optimal control. We analyze the optimal control, the optimally controlled state and the value function and compare various properties to the literature of problems with costly information acquisition. Further, we show existence and uniqueness of an equilibrium for the controlled, conditional variance, and study sensitivity of the control problem at the equilibrium. At last, we compare the problem to the case of no costly information acquisition and fully observable states. Joint work with Christoph Knochenhauer and Yufei Zhang (Imperial College London).

15.01.2024 14:15 Alessandro Gnoatto, University of Verona: Cross-Currency Heath-Jarrow-Morton Framework in the Multiple Curve Setting

The aim of the present talk is to discuss HJM cross currency models that can serve as the basis for the simulation of exposure profiles in the xVA context. Such models need to take into account the asymmetries that arise in the different currency denominations in view of the benchmark reform: for example, while in the EUR area Euribor is still the dominant interest rate benchmark, the situation in the US is much more complex due to the introduction of SOFR and alternative forward looking unsecured rates such as the Bloomberg BSBY or the Ameribor 90T. The impact of the Libor transition on the structure of cross currency swap is also an aspect we would like to address. In summary we would like to: - Provide, in a HJM setting, a unified treatment of forward looking and backward looking rates with and without a credit/liquidity component, i.e. consider a HJM setting for a general underlying index in each currency area. - Properly link such general single currency HJM models by means of cross currency processes that capture the cross currency basis. - Analyze cross currency swaps with arbitrary combinations of interest rate indexes and collateral rates in the different currency areas i.e. with and without Libor discontinuation. This is a joint work with Silvia Lavagnini (BI Oslo)

15.01.2024 15:00 Mathieu Rosenbaum, Ecole Polytechnique : The two square root laws of market impact and the role of sophisticated market participants

The goal of this work is to disentangle the roles of volume and participation rate in the price response of the market to a sequence of orders. To do so, we use an approach where price dynamics are derived from the order flow via no arbitrage constraints. We also introduce in the model sophisticated market participants having superior abilities to analyse market dynamics. Our results lead to two square root laws of market impact, with respect to executed volume and with respect to participation rate. This is joint work with Bruno Durin and Grégoire Szymanski.

15.01.2024 16:15 Christian Bender: Entropy-Regularized Mean-Variance Portfolio Optimization with Jumps

Motivated by the tradeoff between exploitation and exploration in reinforcement learning, we study a continuous-time entropy-regularized mean variance portfolio selection problem in the presence of jumps. A first key step is to derive a suitable formulation of the continuous-time problem. In the existing literature for the diffusion case (e.g., Wang, Zariphopoulou and Zhou, Mach. Learn. Res. 2020), the conditional mean and the conditional covariance of the controlled dynamics are heuristically derived by a law of large numbers argument. In order to capture the influence of jumps, we first explicitly model distributional controls on discrete-time partitions and identify a family of discrete-time integrators which incorporate the additional exploration noise. Refining the time grid, we prove convergence in distribution of the discrete-time integrators to a multi-dimensional Levy process. This limit theorem gives rise to a natural continuous-time formulation of the exploratory control problem with entropy regularization. We solve this problem by adapting the classical Hamilton-Jacobi-Bellman approach. It turns out that the optimal feedback control distribution is Gaussian and that the optimal portfolio wealth process follows a linear stochastic differential equation, whose coefficients can be explicitly expressed in terms of the solution of a nonlinear partial integro-differential equation. We also provide a detailed comparison to the results derived by Wang and Zhou (Math. Finance, 2020) for the exploratory portfolio selection problem in the Black-Scholes model. The talk is based on joint work with Thuan Nguyen (Saarbrücken).

24.11.2023 10:00 Dr. Ari-Pekka Perkkiö, Niklas Walter, Niklas Weber: Christmas Workshop 2023 in Stochastics and Finance


03.07.2023 14:00 William Lim: Optimal Investment under Terminal Wealth Constraints

We study two aspects of making optimal investment decisions for pension investors in the savings phase. First, we explore the impact of an investor’s perception towards inflation risk on their investment strategy. We find that mis-specifying inflation risk reduces the expected utility of the risk averse investors, and more risk averse investors face larger reductions. For investors who adopt terminal wealth constraints (e.g. minimum guarantee), ignoring inflation results in real wealth not adhering to the real constraints. The conclusion is that investors ignore inflation at their peril. Secondly, we compare the retirement outcomes derived from the risk averse and loss averse utility functions. We use a numerical dynamic programming approach and a model that includes ongoing pension contributions to savings, prohibits short-selling and borrowings, and, when applicable, includes wealth constraints. We find that the loss averse utility function, without wealth constraints, naturally results in a more favourable retirement income distribution that peaks at the investor's chosen income goal with some level of robustness. We conclude that the investor can benefit from adopting a loss aversion-derived optimal investment strategy to target a sufficient level of income at retirement.

03.07.2023 14:45 Gaurav Khemka: A Simple Lifecycle Strategy that is Near-Optimal and Requires No Rebalancing

We propose a simple lifecycle strategy entailing contributions made during accumulation being invested entirely into a risky portfolio until pre-specified ‘switch age’ and then entirely into a risk-free portfolio after the switch age, followed by withdrawing during decumulation from both portfolios based on annuitization rates that vary with age according to remaining life expectancy. First, we show analytically that the strategy is optimal for range of investors with HARA risk preferences, and derive the dynamics of the investment strategy. Second, we show numerically that the proposed strategy delivers limited loss of utility versus an optimal solution for investors with CRRA preferences and low risk aversion, while significantly outperforming strategies commonly used in practice. The proposed strategy offers an attractive alternative for use in practical settings as it is simple to follow and removes the need for portfolio rebalancing.

03.07.2023 15:45 Anna Battauz: On the valuation of executive stock options with vesting periods and liquidation penalties

We develop a simple and flexible technique to price executive stock options (ESOs) with vesting periods and liquidation penalties. The vesting period implies that the ESO is activated when a designed performance measure triggers a prespecified barrier. The performance measure is usually an accounting figure, such as the ROE or the EBITDA, normally correlated with the stock price. Once the option is activated, the holder has the right to buy the stock whenever she wants during the residual life of the option. The bivariate strutucture of the ESO, whose payoff depends jointly on the performance measure and the stock, makes usual lattice techniques difficult to apply. We first reduce the ESO to a compound forward-starting American call option on the stock. We then show how to evaluate the ESO option by means of an intuitive hybrid method that uses simulation to determine the bivariate distribution of the foward-starting date of the option and the corresponding price of the stock, and lattice techniques to retrieve the initial value of the activated call option. Liquidation penalties are common in ESOs, aiming at lowering the chances of selling the ESOs and the underlying company shares. We show that the presence of even mild liquidation penalties triggers the existence of optimal exercise opportunities for the ESOs that are absent when the option can be fully liquidated. Joint with M. De Donno and Alessandro Sbuelz

For talks more than one year ago please have a look at the Munich Mathematical Calendar (filter: "Oberseminar Finanz- und Versicherungsmathematik").