Oberseminar Finanz- und Versicherungsmathematik

Venue

Technical University of Munich (TUM) (Room 2.02.01, Parking 11, Garching)

Timetable

Titles and Abstracts

Lane Hughston - 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).