Payoff Structure and Replication of a Discretely Observed Barrier Option
Summary
The document describes a one-year option observed monthly with a fixed upper barrier. If the asset reaches or exceeds the barrier on an observation date before expiry, the contract pays the positive difference between the asset price and strike at that date and ends. If no trigger occurs, the expiry payoff is the asset price minus strike, independent of the barrier. With strike set at the initial asset price, this leaves the holder exposed to the underlying’s full downside over the term while allowing early redemption after a barrier trigger.
The author proposes dynamic replication using a short position in a one-year put and successive one-month calls, adding calls conditionally after prior periods pass without a knock-out. This is presented as an argument rather than a demonstrated pricing derivation: the document gives no valuation model, market inputs, numerical evidence, or proof that the hedge reproduces all paths and cash flows. Its description is therefore useful for framing the payoff, but the replication claim needs careful verification, particularly around trigger timing and settlement.
Key ideas
- The contract is observed at discrete dates, and crossing the barrier triggers an early payoff based on the asset price and strike.
- If the option remains untriggered until expiry, its payoff is the asset price minus strike regardless of the barrier.
- Setting the strike equal to the initial price leaves the holder exposed to the underlying’s downside until expiry.
- The proposed hedge combines a long sequence of conditional short-dated calls with a short long-dated put, but the document does not establish equivalence.
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# Pricing an exotic with barrier at discrete times
# Pricing an exotic with barrier at discrete times
How would you price the following option on underlying $S$ without dividends?
Time to maturity of option $\tau = 12$ months
Option has a strike $K > 0$ and constant barrier $B > 0$.
$t_0$ is the current point in time, while ${t_1, t_2,...t_{12}}$ are observation (potential trigger) dates and $t_{12}$ is the expiration date.
When the underlying is above the barrier at any of the observation dates, the option is called (similar to up and in call) and the difference between underlying at the observation date $S_{t_i}$ and strike $K$ is payed out.
Assume further for our example:
$ 0 < S_{t_0} < B $
$ K = S_{t_0}$ (ATM)
At each observation date, i.e. ${t_1, t_2,...t_{11}}$, excluding the expiration date, the payoff looks as follows:
$$ \begin{align*} if \quad S_{t_i} \geq B: \quad payoff = max(S_{t_i} - K, 0)\\ elif \quad S_{t_i} < B: \qquad \qquad \qquad payoff = 0 \end{align*}$$
The first case in our setting is per definition positive and implies that option is triggered / redeemed
At maturity / expiration if option hasn't been triggered on one of the observation dates, the payoff is independent of the barrier:
$$ \quad payoff = S_{t_i} - K $$
i.e. you get back the payoff of holding the stock over that the whole maturity when $K = S_{t_0}$
In other words, over the whole maturity the option bears the entire downside potential, but each month the option can be triggered if the stock price level is above the barrier $B$ and is then also redeemed at that observation date $t_i$. At maturity you essentially get back the payoff of holding purely the stock if we assume that $ K = S_{t_0}$.
I´d argue that the profile can be replicated dynamically by shorting a 12 month put and being long 1 month calls at the beginning of each month conditional on being knocked out in the prior period.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.