I am trying to understand the no arbitrage argument for to determine the price of an option that pays $1$ when the stock hits $\$H$ for the first time. The current price of the stock is $\$1$. The argument goes as follows. I can buy $1/H$ of the stock now, which will give me $\$1$ when the stock becomes $\$H$. Thus the option can not be more than $1/H$. On the other hand, if the option price is $C$ less than $\$1/H$ then I can buy one option by borrowing $C$ shares of the stock. Once it hits $\$H$, exercise the option to make profit $1 - CH > 0$. So the price of the option can not be less than $1/H$.
What I don't understand about this argument is that how come one doesn't take into account the possibility that the stock never gets to $H$. In both cases, the argument assumes that the stock gets hits $H$ and deduces the conclusion. Any clarification would be appreciated. Thank you
For simplicity let's change the face value of the option payout to be $H$ when the stock hits the upper level $H\,.$ The price for this option today is clearly the stock price today. Let's call this $S_0\,.$
Proof (I guess that's what you had in mind). If the option price were less than $S_0$ you could buy the option for less than $S_0$ and borrow a stock, sell it at at $S_0$ and pocket a profit. When the stock hits $H$ you get $H$ from the option which you use to buy back the stock and return it to the lender (zero sum game at the end). Profit was pocketed at the beginning. $\quad\quad\quad\quad\quad\Box$
Clearly, when the face value is one the option price will be $S_0/H\,.$
The reason that this works without volatility is because we silently assumed that the option has a perpetual maturity. In reality such options expire after a while which changes the picture completely:
Using the well-known Reiner & Rubinstein (1991b) formula for this option with maturity $T$ which we can find in [1] we get the following price-maturity relationship that depends on volatility $\sigma$ but -as expected- approaches the stock price of $S_0=100$ for large $T$:
The Reiner & Rubinstein (1991b) option pricing formula is $$ H\Big(\frac{H}{S_0}\Big)^{\mu+\lambda}\Phi(-z)+H\Big(\frac{H}{S_0}\Big)^{\mu-\lambda}\Phi(-z+2\lambda\sigma\sqrt{T}) $$ where $\mu=\frac{r-q+\sigma^2/2}{\sigma}$ and $\lambda=\sqrt{\mu^2+\frac{2r}{\sigma^2}}$ and $z=\frac{\log(H/S_0)}{\sigma\sqrt{T}}+\lambda\sigma\sqrt{T}\,.$
Here $q$ is the continuous dividend yield of the stock. In our case it must be zero. If not, the lender will typically demand a fee which we have ignored in the above proof. A non zero dividend yield will change the option price even for very large maturities.
[1] E.G. Haug, The Complete Guide to Option Pricing Formulas.