The Put-Call Parity is an important fundamental relationship between the price of the underlying assets, and a (European) put and call of the same strike and time to expiry.
is the price of the Call,
is the price of the Put,
is the current spot price,
is the strike of the option.
With this relationship, it implies several important characteristics of the call and put options. For example, differentiating the equation with respect to the stock price, we obtain that
This equation can be derived by considering a portfolio which consists of buying a call and selling a put on the same strike. This portfolio is worth . On expiration, this portfolio is going to be worth , where is the future price. If this discount this back to current drives, we obtain . Hence, if these two values are different, we could trade the corresponding options and stock, and wait it out to expiry. At which point, there is potentially strike risk, if the stock trades close to the strike price.
Before Put-Call parity was well understood, some option traders specialized in just trading call options only, or just trading put options only, there were a lot of (almost) risk-free arbitrage trades that could have been done. For example, the 19th century financier used it to create synthetic loans which had higher interest rates than the usury laws of the time would have normally allowed. Nowadays, high frequency traders would have arbitraged away such discrepancy in option prices, ensuring that you receive a fair price.
For American options, due to the potential for Early Exercise, we do not have an equation. Instead, we have the inequality
This arises from constructing a similar portfolio up to the point where the option that you are short gets exercised.