You are analyzing stock XYZ$.$ The current price of the stock is $$110.$ A one$-$year European call option on XYZ with a strike price of $$110$ trades for $$15,$ while a one$-$year European put option on XYZ with a strike price of $$110$ trades for $$5.$

You are now tasked to price a binary put option. A binary put option with strike price $$120$ pays exactly $$1$ if the stock price is less than or equal to $$120$ and pays nothing if the stock price is above $$120.$ A binary call option with strike price $$120$ pays exactly $$1$ if the stock price is above $$120$ and pays nothing if the stock price is less than or equal to $$120.$

You see that a binary call option on XYZ with strike price $$120$ is trading for $$0\mathrm{.}51.$ Use a no arbitrage relation to find the price of a binary put option on XYZ with strike price $$120.$

Hint: these are not the standard options discussed in class, so the standard put$-$call parity formula will not hold here. However, see if you can form a no$-$arbitrage relation that links the price of a binary call, binary put, and one or more other assets.

$[$Please enter your answer in dollars, without the $ sign, with at least three digits of precision. For example, enter $$1$ as $1\mathrm{.}000.]$