A Pricing and Hedging Framework for Infinite-Dimensional Financial Markets

Abstract

In this paper, we provide a comprehensive study of how the classical Black--Scholes-type market framework behaves in an infinite-dimensional setting. The asset prices are modeled by stochastic evolution equations in a separable Hilbert space. The evolution of the risk-free asset and the corresponding discounting operators are governed by a strongly continuous group of bounded linear operators. Accordingly, we consider contingent claims with Hilbert-space-valued payoffs, whose values are discounted through the corresponding operators. Within this framework, we investigate the pricing and hedging of contingent claims, deriving the corresponding Hilbert-space-valued Black--Scholes equations and establishing exact replication results for both strong and mild solutions. We also study the special case where the infinitesimal generator is given by $A=rI$, allowing for real-valued contingent claims. We further investigate no-arbitrage and market completeness, introducing a Hilbert lattice structure to formulate arbitrage and characterizing attainability through operator equations associated with martingale representations. We also establish asymptotic completeness through finite-dimensional noise truncations. We introduce geometric Brownian motions in Hilbert spaces and derive sufficient conditions for the existence and uniqueness of an equivalent risk-neutral measure, together with explicit pricing formulas for some tractable examples. These results extend key aspects of the classical Black--Scholes framework to markets with infinite-dimensional asset-price dynamics. As a concrete application, the model can be interpreted as a segmented market consisting of a sequence of one-dimensional geometric Brownian motions.

arXiv comment38 pages