Barrier Call Payoffs and Path-Dependent Knockout Conditions

Summary

The document raises a notation question about up-and-out and down-and-out call payoffs. It defines the terminal call payoff using the strike and multiplies it by an indicator based on whether the price path remains on the permitted side of a barrier. The path’s infimum and supremum summarize its lowest and highest values over the option’s life.

For an up-and-out call, survival requires the path’s maximum to stay below the upper barrier; for a down-and-out call, survival requires the path’s minimum to stay above the lower barrier. The question’s quoted down-and-out condition instead uses the maximum, which would not express that survival event. The excerpt contains the question but no answer, so it does not discuss barrier-touching conventions, monitoring frequency, rebates, or other contract details that can affect exact payoff definitions.

Key ideas

• A barrier option’s payoff depends on both the terminal underlying price and the path followed before expiry.
• An up-and-out option survives only if the path’s maximum remains below its upper barrier.
• A down-and-out option survives only if the path’s minimum remains above its lower barrier.
• Exact payoff indicators depend on contract conventions such as barrier monitoring and treatment of a touch.

Tags

# Payoff of barrier options


# Payoff of barrier options












I was reading a research paper recently and the author defined payoffs of Up-and-Out and Down-and-Out barrier call options as max[0, ST - K]I(m < H) and max[0, ST - K]I(M > H) respectively.

K is the strike price, H is the barrier, spot price process is defined as S = {St, 0 < t < T}, m is defined as inf{St, 0 < t < T} and M is defined as sup{St, 0 < t < T}. What I do not understand is the conditions of the indicators. For UO shouldn't it be I(M < H) because if the supremum is below the barrier then the price on the whole interval is below the barrier and the Up-and-Out option is active? And for DO it seems logical to me to use I(m > H) instead of I(M > H).

Can anyone please clarify it?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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