Graduate Seminar Financial and Actuarial Mathematics LMU and TUM

This paper develops a framework for cost-sensitive training of probabilistic machine learning models that predict the direction of aggregate stock returns. We introduce a bi-objective loss function that augments conventional log-loss optimization with an expected error-cost objective, using the option-implied conditional value-at-risk as a forward-looking measure of misclassification cost. We illustrate how cost-sensitive learning produces economically sensible predictions. Market timing strategies constructed using forecasts from elastic-net logistic regression and gradient boosted decision trees trained using our framework improve out-of-sample risk-adjusted returns and substantially reduce downside risk.

We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in extensions of the Black-Merton-Scholes model under progressively enlarged filtrations. It is assumed that the information available from the market is modelled by Brownian filtrations progressively enlarged with the random times at which the underlying process attains its global maximum or minimum, that is, the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval, which are supposed to be progressively observed by the holders of the contracts. We show that the optimal exercise times are the first times at which the asset price process reaches certain lower or upper stochastic boundaries depending on the current values of its running maximum or minimum depending on the occurrence of the random times of the global maximum or minimum of the risky asset price process. The proof is based on the reduction of the original necessarily three-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and either normal-reflection or normal-entrance conditions for the value functions at the optimal exercise boundaries and the edges of the state spaces of the processes, respectively. This is a joint work with Libo Li (Sydney).

The exposure of sovereigns to climate risks is priced and can affect credit ratings and debt servicing costs. I will discuss how integrated assessment models (IAMs) can be linked with stochastic debt sustainability analysis to inform our understanding of climate risks to sovereign debt dynamics and assess the available fiscal space to finance climate policies. The narrative scenario architecture developed within the IPCC and the transition scenarios of NGFS can be adopted to bring structure and transparency to the analysis. The analysis is complicated by deep uncertainty —risks, ambiguity, and mis-specifications— of climate change. Using scenario trees, narrative scenarios, and model ensembles, respectively, we can address these three challenges. I will discuss two applications combining well-known climate models with a high-realism debt sustainability analysis model, and discuss (i) sovereign debt effects of the transition to low-carbon economies on a large panel of countries, and (ii) effects of damages and adaptation. Results show that adaptation pays, but raises the question of who will pay for it.

Real-life financial time series exhibit heavy tails and clusters of extreme values. In this talk we will address models that exhibit these stylized facts. This is the class of regularly varying time series, introduced by Davis and Hsing (1995, AoP) and further developed by Basrak and Segers (2009, SPA). The marginal distribution of a regularly varying time series has tails of power-law type, and the dynamics caused by an extreme event in this time series is described by the spectral tail process. The perhaps best known financial time series models of this kind are Engle’s (1982) ARCH process, Bollerslev’s (1986) GARCH process and Engle’s and Russell’s (1998) Autoregressive Conditional Duration (ACD) model. The length and magnitude of extremal clusters in such a series can be described by an analog of the autocorrelation function for extreme events: the extremogram. The extremal index is another useful tool for describing expected extremal cluster sizes. Both objects can be expressed in terms of the spectral tail process and allow for statistical estimation. The probabilistic and statistical aspects of regularly varying time series are summarized in the recent monograph by Mikosch and Wintenberger (2024) “Extreme Value Theory for Time Series. Models with Power-Law Tails”. The talk is based on joint work with Olivier Wintenberger (Sorbonne).

Let π∈Π(μ,ν) be a coupling between two probability measures μ and ν on a Polish space. In this talk we propose and study a class of nonparametric measures of association between μ and ν, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between ν and the disintegration πx1 of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures μ and ν. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2022]. Our approach applies to probability laws on general Polish spaces.

We study how to construct a stochastic process on a finite interval with given `roughness'. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization, we provide a better (path wise) estimator of Hölder exponent. Furthermore, we study the concept of (generalized) p-th variation of a real-valued continuous function along a sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.

The Wasserstein distance W is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein AW distance extends this theory to laws of discrete time stochastic processes in their natural filtrations, making it particularly well suited for analyzing time-dependent stochastic optimization problems. While the topological differences between AW and W are well understood, their differences as metrics remain largely unexplored beyond the trivial bound W

Stochastic optimal control problems naturally arise in contexts such as optimal investment, optimal consumption, and economic growth. Moreover, many fundamental models in robust finance - such as G-Brownian motion, G-diffusions, or G-semimartingales - can be translated to frameworks of stochastic control. A central aspect of these problems is the connection between value functions and Hamilton-Jacobi-Bellman (HJB) equations. For controlled diffusions with sufficiently regular coefficients, this link is typically established either through the comparison method, relying on a comparison principle for discontinuous viscosity solutions, or via the verification approach, which requires the existence of classical or Sobolev solutions. In this talk, we consider a general class of controlled diffusions for which these traditional methods break down. We present a new approach that connects stochastic control problems and HJB equations by combining probabilistic and analytic techniques. Furthermore, we discuss uniqueness results, leading to stochastic representations of HJB equations in terms of control problems, and provide stability results for associated value functions.

In this talk, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multimarginal couplings, as a generalization of well-known correlation statistics such as the Pearson correlation. The first main result states that even an arbitrarily small positive dependence between losses can result in perfectly correlated tails beyond a certain threshold and seemingly complete independence before this threshold. In a second step, we focus on the aggregation of individual risks with known marginal distributions by means of arbitrary nondecreasing left-continuous aggregation functions. In this context, we show that under an arbitrarily small positive dependence, the tail risk of the aggregate loss might coincide with the one of perfectly correlated losses. A similar result is derived for expectiles under mild conditions. In a last step, we discuss our results in the context of credit risk, analyzing the potential effects on the value at risk for weighted sums of Bernoulli distributed losses.

Martingales associated with path-dependent payoff functions are intrinsically linked to path-dependent PDEs. While this connection is typically established via a functional Itô formula, in this talk we present a semigroup-theoretic framework for the analytic characterization of martingales with path-dependent terminal conditions. Specifically, we show that a measurable adapted process of the form V(t) - ∫_0^t Ψ(s)ds is a martingale if and only if a time-shifted version of V is a mild solution to a final value problem (FVP) involving a path-dependent differential operator. We establish existence and uniqueness of solutions to such FVPs using the concept of evolutionary semigroups on path space. We also discuss the relationship between semigroups on path space, nonlinear expectations and their penalty functions. The talk is based on joint work with David Criens, Robert Denk and Markus Kunze.

We present a multi-agent and mean-field formulation of a game between investors who receive private signals informing their investment decisions and who interact through relative performance concerns. A key tool in our model is a Poisson random measure which drives jumps in both market prices and signal processes and thus captures common and idiosyncratic noise. Upon receiving a jump signal, an investor evaluates not only the signal's implications for stock price movements but also its implications for the signals received by her peers and for their subsequent investment decisions. A crucial aspect of this assessment is the distribution of investor types in the economy. These types determine their risk aversion, performance concerns, and the quality and quantity of their signals. We demonstrate how these factors are reflected in the corresponding HJB equations, characterizing an agent's optimal response to her peers' signal-based strategies. The existence of equilibria in both the multi-agent and mean-field game is established using Schauder's Fixed Point Theorem under suitable conditions on investor characteristics, particularly their signal processes. Finally, we present numerical case studies that illustrate these equilibria from a financial-economic perspective. This allows us to address questions such as how much investors should care about the information known by their peers.

We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine expansion (COS) method. The classical COS method is numerically very efficient in one-dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a cosine series. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.