Infinite-Horizon Barrier Claims and the Limits of Replication

Summary

The document poses a pricing puzzle for a claim that pays one dollar when a stock first reaches a barrier above its current price. Under the stated risk-neutral assumption, the barrier is reached eventually with probability one, so the discounted expected payoff at zero interest is one dollar. A proposed static hedge buys enough shares to be worth one dollar at the barrier, giving a lower cost today and apparently the same payoff at the crossing time.

The two calculations highlight an important issue in pricing claims tied to an unbounded horizon: matching the payoff at the barrier does not by itself establish that the stock position replicates the claim in every relevant state or at the same time. The document provides the setup but no answer resolving the discrepancy. Its assumptions and missing treatment of what happens after the crossing, admissible trading strategies, and pricing conditions limit any firm conclusion.

Key ideas

• The claim pays when the stock first reaches a fixed barrier above its current value.
• The assumed risk-neutral probability of eventual barrier crossing is one.
• The expected-payoff method prices the claim at one dollar under the stated zero-rate assumption.
• A static stock holding has the same value as the payoff at the crossing but costs less initially.
• The document presents the conflict without establishing which pricing argument applies.

Tags

# Infinite Horizon Barrier Option Paradoxe


# Infinite Horizon Barrier Option Paradoxe












I've came across this question which is puzzling me. Imagine that interest rates are zero and that you observe a stock $S_t$ whose value today $S_0$ is equal to 1\$. We consider the derivative that pays 1\$ at the time $\tau$ where the stock crosses a fixed barrier $B > 1\$$.

We will suppose that the risk-neutral probability of the event

$\mathbb{Q} \left(\exists t \geq 0, \quad S_t\geq B\right)=1$

Let us now consider the following pricing approaches :

- Martingale Approach :

Using this approach, we know that the value of the derivative today is the expected (discounted, but rates are 0.0) payoff under the risk-neutral probability. The payoff is exactly equal to the indicator function

$1_{\exists t \geq 0,S_t\geq B}$

So the price, is simply the probability of that event which is 1.

- Replication Approach

Imagine that one wants to replicate the option using the underlying, he can buy $1/B$ of the stock today. At the time where the stock value is equal to $B$, the portfolio is worth $1/B * B = 1$ which is the payoff of the option.

$\implies$ A portfolio of $1/B$ shares today replicates the payoff of the option at each state of the world. By no arbitrage the value of the option should be $1/B$

The two approaches yield different pricing results, yet I cannot find the logical flow behind one method or the other. Can you please help ?

Thanks!

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