Generalized integrands and bond portfolios: Pitfalls and counter examples

TL;DR

This paper reveals critical flaws in the concept of generalized portfolios in zero-coupon bond markets driven by cylindrical Brownian motion, showing that prices can be infinite or arbitrarily convergent, highlighting potential pitfalls in such models.

Contribution

It demonstrates fundamental issues with generalized portfolios in bond markets, including unbounded prices and convergence anomalies, which challenge existing financial modeling assumptions.

Findings

Existence of bounded smooth variables with infinite risky part prices

Sequences of approximating portfolios can converge to any extended real number

Generalized portfolios may have significant modeling pitfalls

Abstract

We construct Zero-Coupon Bond markets driven by a cylindrical Brownian motion in which the notion of generalized portfolio has important flaws: There exist bounded smooth random variables with generalized hedging portfolios for which the price of their risky part is $+\infty$ at each time. For these generalized portfolios, sequences of the prices of the risky part of approximating portfolios can be made to converges to any given extended real number in $[-\infty,\infty].$