Is the moneyness of a barrier option based on the strike value or the barrier when mapping to a volatility surface?

Question

Say you have a down and in put barrier option with a strike of 100 and barrier at 60. If the stock price sits at 90, which value would you use to determine the moneyness? Is the option in or out of the money in a technical sense? The practical consideration in what I'm asking is how you would map a barrier option to a volatility surface. Additionally, does it matter if the option is path dependent or not, i.e. if the barrier is considered only at maturity or if it only needs to have crossed the barrier at some point during the time the option is alive?

In the case of a down and in put that isn't path dependent, you should be able to replicate it using a number of binary put-option and a normal put option both with strike at the barrier level. So in that sense it would make sense that the barrier is what should be considered when talking about moneyness, but I'm not sure if there's more to it than this or other contradictory cases.

Answer 1

If your barrier is american and your market has any sort of volatility skew then trying to map some sort of moneyness measure to the vol surface will almost certainly fail. That is due to the fact that barriers are sensitive to forward skew, and you need a model to capture that as a continuum of vanilla option prices sadly tells you nothing about the forward skew. As a start I would suggest looking at a local volatility model and pricing the barrier that way.

If you barrier is european on the other hand than what you have in effect a portfolio consisting of a digital option and a put struck at the barrier. This implies that the moneyness you want to look for is determined by the barrier, not the strike - which determines the nominal of the digital