20.01.2025 14:15 Lane Hughston (Goldsmiths University of London): Valuation of a financial claim contingent on the outcome of a quantum measurement
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.
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20.01.2025 15:00 Muneya Matsui (Nanzan University): Regular variation of multivariate GARCH models and tail dependence measure, extremogram and cross-extremogram
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.
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20.01.2025 16:00 Alessandro Sgarabattolo : LIMITS OF MULTI-STAGE RISK MEASURES BASED ON OPTIMAL TRANSPORT
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.
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