In the presence of an implied volatility smile, it is not clear which volatility to use when pricing barrier options in the Black-Scholes framework. A simple, although theoretically not justified, tweak is an approach related to outside barrier options. For these contracts, the payoff is determined by one asset while the barrier trigger is linked to another.
In the two-volatility model, these two assets are perfectly correlated. They have the same initial price and drift parameters but different diffusion coefficients. The payoff is typically determined by a geometric Brownian motion with a volatility equal to the Black-Scholes implied volatility at the strike price. The barrier trigger is determined by the second asset whose diffusion term is equal to the implied volatility at the barrier. Note, that this model is internally inconsistent for pricing barrier options and has to be regarded as a pure tweak to incorporate the volatility smile in some ad-hoc way. If the volatility smile is negatively sloped, as it is typical for equity index options, then the volatility for a downside barrier is higher than the strike volatility. The perfect correlation implies that the barrier asset is strictly below the strike asset when the barrier is being triggered. The model thus assigns zero probability to high payoffs close to the maximum possible intrinsic value, while these events have non-zero probability in reality. Similarly, the model assigns a positive probability to some up-and-out call payoffs that are greater than the difference between barrier and strike.
The general outside barrier pricing formula can e.g. be found in Haug’s book “The complete Guide to Option Pricing Formulas” (McGraw-Hill, 2006). The pricing formula for the down-and-out put that we obtain is the limiting case when the correlation goes to one.