Local risk-minimization for Barndorff-Nielsen and Shephard models

TL;DR

This paper derives explicit locally risk-minimizing strategies for options within Barndorff-Nielsen and Shephard models, leveraging Malliavin calculus for Levy processes, and provides numerical illustrations of these strategies.

Contribution

It extends Malliavin calculus-based formulas to Barndorff-Nielsen and Shephard models, verifying conditions and providing explicit strategies and numerical experiments.

Findings

Explicit risk-minimizing strategies derived for the models

Verification of model conditions for Malliavin calculus application

Numerical experiments demonstrating strategy implementation

Abstract

We obtain explicit representations of locally risk-minimizing strategies of call and put options for the Barndorff-Nielsen and Shephard models, which are Ornstein--Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for Levy processes, Arai and Suzuki (2015) obtained a formula for locally risk-minimizing strategies for Levy markets under many additional conditions. Supposing mild conditions, we make sure that the Barndorff-Nielsen and Shephard models satisfy all the conditions imposed in Arai and Suzuki (2015). Among others, we investigate the Malliavin differentiability of the density of the minimal martingale measure. Moreover, some numerical experiments for locally risk-minimizing strategies are introduced.