Graduate Seminar Financial and Actuarial Mathematics LMU and TUM

We apply computational techniques of convex stochastic optimization to optimal operation and valuation of electricity storages in the face of uncertain electricity prices. Our approach is based on quadrature approximations of Markov processes and on the Stochastic Dual Dynamic Programming (SDDP) algorithm which is widely applied across the energy industry. The approach is applicable to various specifications of storages, and it allows for e.g. hard constraints on storage capacity and charging speed. Our valuations are based on the indifference pricing principle, which builds on optimal trading strategies and calibrates to the user's initial position, market views and risk preferences. We illustrate the effects of storage capacity and charging speed by numerically computing the valuations using stochastic dual dynamic programming. If time permits, we provide theoretical justification of the employed computational techniques.

We extend the popular Gibson and Schwartz (1990) and Schwartz and Smith (2000) two-factor models for the spot price of a commodity to include stochastic volatility and correlation. This generalization is based on the Wishart variance-covariance matrix process. For both of the extended models we present the joint characteristic functions of the two state variables. The original models are known to fit the term-structure of implied volatility in futures and options markets very well. However, the extended models are also able to match volatility smiles observed in these markets. Regarding the analysis of financial time series, the assumption of a constant correlation between the state variables is known to be too restrictive. Introducing time-varying correlation via the Wishart process allows us to study its empirical behaviour in commodity markets through the use of filtering techniques.

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.