Definition (European Call Option)

A European Call Option with expiration $T$ and strike price $K$ is a derivative with payoff $$O_{T}=(S_{t}-K)^{+}$$

Definition (European put option)

A European Put Option with expiration $T$ and strike price $K$ is a derivative with payoff $$O_{T}=(K-S_{t})^{+}$$

Definition (Asian call option)

A Asian Call Option with expiration $T$ and strike price $K$ is a derivative with payoff $$O_{T}=\left( \int\limits_{[0,T]} S_{t} \, dt -K \right)^{+}$$