Al\`os type decomposition formula for Barndorff-Nielsen and Shephard model.

The objective is to provide an Alos type decomposition formula and an approximate option pricing formula for the Barndorff-Nielsen and Shephard model: an Ornstein-Uhlenbeck type stochastic volatility model driven by a subordinator without drift. Alos (2012) introduced a decomposition expression of the call option prices for the Heston model by using Ito's formula. In this paper, we extend it to the Barndorff-Nielsen and Shephard model. As far as we know, this is the first result on the Alos type decomposition formula for models with infinite active jumps. Moreover, investigating the rate of convergence as the time to maturity tends to 0 for each term in the obtained decomposition formula, we shall present an approximate option pricing formula, and implement numerical experiments, which show that our approximation formula is effective for in-the-money options.