Mean-variance hedging based on an incomplete market with external risk factors of non-Gaussian OU processes

TL;DR

This paper develops a mean-variance hedging strategy for incomplete markets with external non-Gaussian Ornstein-Uhlenbeck risk factors, proving its global risk optimality through analytical and numerical validation.

Contribution

It introduces a semi-explicit hedging strategy in incomplete markets with non-Gaussian OU processes and establishes its global risk optimality with rigorous proofs and examples.

Findings

Proves no-arbitrage condition for the market

Constructs the variance-optimal martingale measure density

Demonstrates the optimality of the hedging strategy

Abstract

In this paper, we prove the global risk optimality of the hedging strategy of contingent claim, which is explicitly (or called semi-explicitly) constructed for an incomplete financial market with external risk factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) processes. Analytical and numerical examples are both presented to illustrate the effectiveness of our optimal strategy. Our study establishes the connection between our financial system and existing general semimartingale based discussions by justifying required conditions. More precisely, there are three steps involved. First, we firmly prove the no-arbitrage condition to be true for our financial market, which is used as an assumption in existing discussions. In doing so, we explicitly construct the square-integrable density process of the variance-optimal martingale measure (VOMM). Second, we derive a backward stochastic differential equation (BSDE) with jumps for the mean-value process of a given contingent claim. The unique existence of adapted strong solution to the BSDE is proved under suitable terminal conditions including both European call and put options as special cases. Third, by combining the solution of the BSDE and the VOMM, we reach the justification of the global risk optimality for our hedging strategy.