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.