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Barrier Option Moneyness and Volatility Surface Mapping

Article Quant Q&A · Author: Oscar
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Summary

The document asks how to choose moneyness when mapping a barrier option to a volatility surface. Its example is a down-and-in put with a strike above its barrier and a spot price between them. It also asks whether the answer changes when the barrier is checked only at maturity or can be crossed during the option’s life.

The response distinguishes a European barrier from an American-style, path-dependent one. In the European case described, the payoff can be represented using a digital option and a put struck at the barrier, so the barrier determines the relevant moneyness while the original strike sets the digital’s nominal. For American barriers in markets with volatility skew, a single surface moneyness measure may fail because barrier value depends on forward skew. The response suggests local volatility as a starting model; it gives no numerical comparison or complete calibration procedure.

Key ideas

  • For the described European down-and-in put, the barrier level determines relevant moneyness for surface mapping.
  • The original strike determines the nominal amount associated with the digital component in the stated replication.
  • American-style barriers can be sensitive to forward skew, which a simple moneyness mapping may not capture.
  • A local volatility model is suggested as a starting point for pricing path-dependent barriers.
  • The discussion does not provide a complete calibration method or numerical test.

Tags

OptionsVolatilityDerivatives pricingMarket microstructure
Full text
# Is the moneyness of a barrier option based on the strike value or the barrier when mapping to a volatility surface?


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












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 by river_rat (score 3, accepted)

https://quant.stackexchange.com/a/58940

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.