Graduate Seminar Financial and Actuarial Mathematics LMU and TUM

In this interdisciplinary study at the interface of finance theory and quantum theory, we consider a rational agent who at time 0 enters into a financial contract for which the payout is determined by a quantum measurement at some time T > 0. The state of the quantum system is given in the Heisenberg representation by a known density matrix p. How much will the agent be willing to pay at time 0 to enter into such a contract? In the case of a finite dimensional Hilbert space H, each such claim is represented by an observable X where the eigenvalues of X determine the amount paid if the corresponding outcome is obtained in the measurement. We use Gleason's theorem to prove, under reasonable axioms, that there exists a pricing state q which is equivalent to the physical state p such that the pricing function Π takes the linear form Π(X) = P0T tr(qX) for any claim X, where P0T is the one-period discount factor. By ‘equivalent’ we mean that p and q share the same null space: that is, for any |ξ⟩ ∈ H one has p|ξ⟩ = 0 if and only if q|ξ ⟩ = 0. We introduce a class of optimization problems and solve for the optimal contract payout structure for a claim based on a given measurement. Then we consider the implications of the Kochen–Specker theorem in this setting and we look at the problem of forming portfolios of such contracts. This work illustrates how ideas from the theory of finance can be successfully applied in a non-Kolmogorovian setting. Based on work with Leandro Sánchez-Betancourt (Oxford). The paper can be found at J. Phys. A: Math. Theor. 57 (2024) 285302.

We explain the multivariate regular variation and a tail dependence measure "extremogram and cross-extremogram", taking a bivariate GARCH model as an example. We show that the tails of the components of a bivariate GARCH(1,1) process may exhibit power law behavior but, depending on the choice of the parameters, the tail indices of the components may differ. Then, we derive asymptotic theory for the extremogram and cross-extremogram of a bivariate GARCH(1,1) process. Moreover, we discuss what GARCH models can and cannot do in terms of the tail modeling, while comparing stochastic volatility models. We also mention limitations of the current notion of multivariate regular variation and we pose several problems to be solved.

Abstract. In the first part of the talk, we discuss convex risk measures with weak optimal transport penalties. We show that these risk easures allow for an explicit representation via a nonlinear transform of the loss function. We also discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, divergence risk measures, uncertainty on path spaces, moment constraints, and martingale constraints. In the second part, we focus on classical transport penalties and consider a suitable multi-stage iteration of these risk measures. Such multi-stage iteration naturally defines a dynamically consistent risk measure which takes into account uncertainty around the increments of an underlying Lévy process. Using direct arguments, we show that passing to the limit in the number of iterations yields a convex monotone semigroup, and obtain explicit convergence rates for the approximation. Finally, we associate this semigroup with the solution to a drift control problem.

There has been increasing interest in the estimation of risk capital within enterprise risk models, particularly through Monte Carlo methods. A key challenge in this area is accurately characterizing the distribution of a company’s available capital, which depends on the conditional expected value of future cash flows. Two prominent approaches are available: the “regress-now” method, which projects cash flows and regresses their discounted values on basis functions, and the “regress-later” method, which approximates cash flows using realizations of tractable processes and subsequently calculates the conditional expected value in a closed form. This paper demonstrates that the left and right singular functions of the valuation operator serve as robust approximators for both approaches, enhancing their performance. Furthermore, we describe situations where each method outperforms the other, with the regress-later method demonstrating superior performance under certain conditions, while the regress-now method generally exhibiting greater robustness.

Reinforcement learning (RL) is a version of stochastic control in which the system dynamics are unknown (up to the type of dynamics such as Markov chains or diffusion processes). There has been an upsurge of interest in RL for (continuous-time) controlled diffusions in recent years. In this talk I will highlight the latest developments on theory and algorithms arising from this study, including entropy regularized exploratory formulation, policy evaluation, policy gradient, q-learning, and regret analysis. Time permitting, I will also discuss applications to mathematical finance and generative AI.

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.

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.

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.

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.

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).