What is the value of an option?

Options are financial contracts depending on underlying assets such as stocks. The simplest example is a European Call Option which is of the form:

"The holder of this contract has the right to buy one stock of type A at time T for the price K."

If T is some time into the future and you have an interest in stock A, what should you be willing to pay for such a contract today? We shall study this question in a toy model, but using concepts and techniques that generalize to more sophisticated settings.