Local risk-minimization for Barndorff-Nielsen and Shephard models with volatility risk premium

TL;DR

This paper develops a method to compute local risk-minimization for options in Barndorff-Nielsen and Shephard models, extending previous work by relaxing the volatility risk premium constraint using Malliavin calculus.

Contribution

It introduces a new approach to local risk-minimization in BNS models by removing the previous restriction on the volatility risk premium parameter.

Findings

Derived explicit representations for local risk-minimization

Extended previous models by relaxing the volatility risk premium constraint

Applied Malliavin calculus under the minimal martingale measure

Abstract

We derive representations of local risk-minimization of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian rnstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two parameters: volatility risk premium $\beta$ and leverage effect $\rho$. Arai and Suzuki (2015, arxiv:1503.08589) dealt with the same problem under constraint $\beta=-\frac{1}{2}$. In this paper, we relax the restriction on $\beta$; and restrict $\rho$ to $0$ instead. We introduce a Malliavin calculus under the minimal martingale measure to solve the problem.