Replicating a Perpetual Up-and-In Binary with Stock

Summary

The document considers a perpetual binary claim that pays one dollar when a stock first reaches an upper barrier. It challenges the idea that the claim must be worth one dollar merely because a random-walk stock may eventually reach the barrier. With spot at one hundred and a barrier at one hundred ten, the response proposes holding one divided by the barrier price in shares and selling them when the barrier is touched. That position then produces the stated one-dollar payoff and costs about 0.91 dollars initially.

This is a replication argument for a particular payoff under the stated assumptions, including no interest rate and no bankruptcy. The answer gives no treatment of dividends, transaction costs, barrier monitoring, jumps, or whether the stock is guaranteed to hit the level under a more general price process. It therefore illustrates why eventual payout probability alone does not determine present value, without providing a complete pricing framework.

Key ideas

• A perpetual binary that pays one dollar upon first reaching an upper barrier can be replicated with a fixed number of shares in the stated setup.
• The share position is sized by dividing the one-dollar payoff by the barrier price.
• The proposed replication costs less than one dollar when the current stock price is below the barrier.
• An eventual barrier hit with probability one does not by itself imply a present value of one dollar.
• The argument relies on simplifying assumptions and does not address trading frictions or broader price dynamics.

Tags

# How to value a Binary Option using market data?


# How to value a Binary Option using market data?












Is there a way to calculate the price of a binary option (i.e., an option that pays out 1 dollar when the stock price hits $x$ amount) using market call/put option prices, forward prices, etc. for a stock? Assume no interest rate.

Provided that the company never goes bankrupt, shouldn't the value of this option be 1 dollar? If the stock price follows a random walk, it will hit $x$ in finite time with probability 1.

## Answer by mbison (score 1)

https://quant.stackexchange.com/a/19467

Sorry to disagree but if interest rates is 0, the binary is still not worth $1 now.

Suppose spot $S(0) = 100$, assume $x = 110$ and upon touch (whenever it happens as the option has no maturity) you receive one dollar.

Suppose I buy 1 stock. If the barrier hits, i sell the stock and receive 110 USD. What if I buy N stocks at t=0? upon hit of barrier i sell my stock and pocket N*110. Now pick N = $\frac{1}{110}$. When i sell my N stocks upon hit, I will hold $N*110 = 1$ USD.

So this strategy of holding $N = 1/110$ stocks replicates the payout. Furthermore at iniation it costs me $N*S(0) = 100/110 = 0.909090$. Note that this is cheaper than your price of 1 USD.

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.