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.

• This is not the definition of the binary option. You are critically missing some maturity $T$ which limits the life of this option. If I understand well, you're trying to value the option using a replicating portfolio is that right? – SRKX Feb 11 '15 at 8:11
• Yes that is correct. Assume there is no expiration date $T$. – FinForFun Feb 11 '15 at 22:28
• But if there is no $T$ your option doesn't really make sense. In what context are you seeing that? If you assume the stock follows a GBM and $\sigma > 0$, then your stock will indeed for sure hit $x$ at some point in the future. If there is no discount, then you know that this option is worth 1 today indeed. – SRKX Feb 12 '15 at 1:29

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.