the obligation, to buy (call) or sell (put) shares of the underlying stock at some prearranged price, the [*strike*]{} price $K$, within a certain period or on a specific date, the maturity or expiration time $T$. Sometimes $K$ is a parameter but in general, depending on the contract specifications, it will be a function involving some other constants: $$K^{\pm}=S(t)\mp X_0 {\mathbf 1}_{S(t) \gtreqless {K_0}}.
\label{Kb}$$ We will use the generic sign ${\mathbb{K}}$ as a shorthand for all the contract parameters. As a consequence of their privileged position, option holders will only exercise their rights if they obtain a net benefit. In other words, we can see options as contingent claims with their present value determined by the discounted value of the expected profit under our risk-neutral measure: $$P(t_0,S(t_0);{\mathbb{K}})={\mathbb{E}}[X(S(t^*);{\mathbb{K}}) e^{-r(t^*-t_0)}|{\mathcal{F}\/}(t_0)],$$ where $X(S(t);{\mathbb{K}})$ is the pay-off function and $t^*$ is the actual exercise time.

The notation we use in the definition of the pay-off function is not incidental. We will concentrate our attention on those contracts for which the pay-off is a function of the current value of the asset, like in the case of vanilla calls (+) and puts (-) where $$X^{\pm}(S;K,T)=\max(\pm(S-K),0) {\mathbf 1}_{t \leq  T},
\label{payv}$$ and $K$ is constant. 