Hull and White and Al\`os type formulas for barrier options in stochastic volatility models with nonzero correlation

Two novel closed-form formulas for the price of barrier options in stochastic volatility models with zero interest rate and dividend yield but nonzero correlation between the asset and its instantaneous volatility are derived. The first is a Hull and White type formula, and the second is a decomposition formula similar in form to the Al\`os decomposition for vanilla options. A model-free approximation is also given.