Another typical pay-off is $$X^{\pm}(S;{K_0},T)=X_0 {\mathbf 1}_{S \gtreqless  {K_0}} {\mathbf 1}_{t \leq  T}.
\label{payb}$$ We can derive this pay-off from (\[Kb\]) and (\[payv\]): it corresponds to binary or digital options. For the sake of simplicity we will set $X_0=1$ hereafter. Vanilla and binary options will be the only instances we will study in practice, although some other contracts may fit our requirements as well. Note that this is not the case of any exotic derivative whose pay-off depends on the past path of the stock, like in Asian options, Lookback options or [*knock-out*]{} options.

When the option can be exercised at the end of the contract lifetime $T$ only, the exercise time is deterministic, and the option is said to be European. If the option can be exercised at any time before expiration it is called American, and $t^*$ becomes an stochastic magnitude as well. Note that the contract is always worthless after maturity: the option buyer must decide under which conditions the option can be optimally exercised before this deadline. The decision will finally depend on the present value of stock price $S(t)$ and the time to expiration, $T-t$. The problem is thus in essence a typical problem of 