Pricing Claims That Pay at a Random Barrier Hitting Time

Summary

The document formulates a contingent claim on an underlying price process that starts between two barriers. The claim pays a specified function of the underlying value at the first time the process exits the interval. This differs from the barrier options the questioner has encountered, which may activate after a barrier event but are not framed as paying precisely at that event.

The proposed valuation is the risk-neutral expected payoff at the exit time, conditional on the initial price. The document asks whether such claims are traded, but supplies no answer, pricing derivation, market convention, or evidence of availability. It is therefore most useful as a concise statement of a first-exit payoff structure and valuation question; practical analysis would need to specify the process dynamics, payoff, and assumptions about whether and when the barriers are reached.

Key ideas

• The claim pays a function of the underlying value at the first exit from a bounded price interval.
• The payoff time is the barrier hitting time itself, rather than a later activation period.
• The proposed value is a risk-neutral conditional expectation of the payoff at exit.
• The document does not establish whether this structure is traded or provide a pricing method.

Tags

# Looking for Options Which Pay Exactly When A Random Barrier is Reached


# Looking for Options Which Pay Exactly When A Random Barrier is Reached












Supper that I fix two barriers $a<b$ and I consider a price process $X_.$ starting in the interval $(a,b)$. Let V be a payoff function and let $\tau:= \inf \{t>0: X_t\not\in (a,b)\}$.

Are there options which pay $V(X_{\tau})$? That is they pay some function V of the underlying but only at the moment when the price reaches either the lower or upper barriers?

All the barrier options Ive seen only start to pay when a barrier is reached, but they do not activate at that exact time.

Specificically, I want to compute the following (where $\mathbb{Q}$ is an ELMM for the discounted price process $X_{\cdot}$) $$ v(x):=\mathbb{E}_{\mathbb{Q}}\big[V(X_{\tau})|X_0=x\big] $$ but I'm wondering if such options are really traded?

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