Pricing an American Cash-or-Nothing Barrier Call

Summary

The document poses a pricing problem for an American-style cash-or-nothing call: the holder receives a fixed cash amount if the stock first reaches a specified level above its current price. It asks whether the option’s early-exercise feature makes it more valuable than a European binary call, and notes that the interest rate is zero.

No pricing method or answer is provided. The problem statement highlights why the usual equivalence between American and European vanilla calls cannot simply be assumed for a path-dependent digital payoff. A unique price cannot be inferred from the stated information: the maturity is unspecified, and the document supplies no model for the stock’s price dynamics or other valuation assumptions. It is therefore useful as a problem framing, but not as a worked pricing example or evidence for a particular valuation.

Key ideas

• The payoff is a fixed cash amount triggered when the stock first reaches a barrier above its current price.
• The question concerns early exercise and whether American and European binary calls have equal value.
• The document does not provide a solution or a stock-price model.
• Without a maturity and valuation assumptions, the stated information is insufficient to determine a unique price.

Tags

# Binary American Call Option (Cash or Nothing)


# Binary American Call Option (Cash or Nothing)












Suppose we have a stock with current price $S(0)=X$ and the interest rate is zero. When the stock reaches level $\$ H$ for the first time ($H>X$), the option can be exercised and its payoff is $\$ X$. What is the current price of such option?

I have realized that this is an example of an American binary call option, of the type "cash or nothing". Furthermore, the interest rate is zero, which should simplify things. However, it seems clear to me that for such American binary option, the rule that European call is worth as American call, valid for vanilla options, is not valid anymore: this American binary option should definitely carry more rights than its European counterpart. Does anybody know how to price such an option? Thanks. PS in the problem it is not specified the time to maturity.

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