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.