Asymptotic Static Hedge via Symmetrization

TL;DR

This paper extends static hedging techniques for timing risk in diffusion models by employing symmetrization and parametrix methods to construct a universally applicable iterated static hedge, reducing hedging error over time.

Contribution

It introduces a symmetrization-based approach to construct iterated static hedges applicable to any uniformly elliptic diffusion, addressing previous mathematical challenges.

Findings

Successfully constructs a static hedge applicable to all uniformly elliptic diffusions.

Proves the convergence of hedging error to zero through iterative static hedging.

Provides detailed parametrix analysis for the fundamental solution of related PDEs.

Abstract

This paper is a continuation of Akahori-Barsotti-Imamura (2017) and where the authors i) showed that a payment at a random time, which we call timing risk, is decomposed into an integral of static positions of knock-in type barrier options, ii) proposed an iteration of static hedge of a timing risk by regarding the hedging error by a static hedge strategy of Bowie-Carr type with respect to a barrier option as a timing risk, and iii) showed that the error converges to zero by infinitely many times of iteration under a condition on the integrability of a relevant function. Even though many diffusion models including generic 1-dimensional ones satisfy the required condition, a construction of the iterated static hedge that is applicable to any uniformly elliptic diffusions is postponed to the present paper because of its mathematical difficulty. We solve the problem in this paper by relying on the symmetrization, a technique first introduced in Imamura-Ishigaki-Okumura (2014) and generalized in Akahori-Imamura (2014), and also work on parametrix, a classical technique from perturbation theory to construct a fundamental solution of a partial differential equation. Due to a lack of continuity in the diffusion coefficient, however, a careful study of the integrability of the relevant functions is required. The long lines of proof itself could be a contribution to the parametrix analysis.