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.

• Similar question has already been asked. See quant.stackexchange.com/questions/17083/…. The price is $X^2/H$ under the Black-Scholes' setting. Mar 18 '16 at 16:10
• Thank you for the comment, I have seen the answer you gave on the link. However, there is a point I don't quite understand in your explanation: given the density $f(t)$, how do you show that the integral of $f(t)$, i.e., $\mathbb{P}(T<\infty)$ is equal to $e^{2\nu y}$? Do you know a short evaluation of this integral? Thank you for your comment. PS So do you suggest the book of Jeanblanc and Yor? Mar 18 '16 at 18:36
• Yes. It was based on the book of Jeanblanc and Yor. For the expectation, you integrate with respect to $t$ from 0 to $\infty$. Mar 18 '16 at 18:40
• That's precisely what I cannot figure out about that integral. Do you know if the integral is calculated explicitly in that book to give $X^2/H$? Mar 18 '16 at 18:48
• Check Formulas 3.3.3 and 3.3.4 in that book. Mar 18 '16 at 19:57