Making Mean-Variance Hedging Implementable in a Partially Observable Market

TL;DR

This paper develops a practical approach for mean-variance hedging in partially observable markets, providing explicit solutions and simulation schemes that facilitate real-time hedging and analysis.

Contribution

It introduces explicit, implementable solutions for mean-variance hedging under partial observation using BSDEs, ODEs, and asymptotic expansions.

Findings

Semi-closed form solutions via simple ODEs enable quick numerical evaluation.

Hedging problems reduce to European option pricing under a new measure.

Explicit asymptotic expressions allow real-time hedging updates and terminal distribution analysis.

Abstract

The mean-variance hedging (MVH) problem is studied in a partially observable market where the drift processes can only be inferred through the observation of asset or index processes. Although most of the literatures treat the MVH problem by the duality method, here we study a system consisting of three BSDEs derived by Mania and Tevzadze (2003) and Mania et.al.(2008) and try to provide more explicit expressions directly implementable by practitioners. Under the Bayesian and Kalman-Bucy frameworks, we find that a relevant BSDE yields a semi-closed solution via a simple set of ODEs which allow a quick numerical evaluation. This renders remaining problems equivalent to solving European contingent claims under a new forward measure, and it is straightforward to obtain a forward looking non-sequential Monte Carlo simulation scheme. We also give a special example where the hedging position is available in a semi-closed form. For more generic setups, we provide explicit expressions of approximate hedging portfolio by an asymptotic expansion. These analytic expressions not only allow the hedgers to update the hedging positions in real time but also make a direct analysis of the terminal distribution of the hedged portfolio feasible by standard Monte Carlo simulation.