How can I calculate the payout of an Asian option? I would like to know how to solve the mean over the entire lifetime of the option. Please show me how to calculate the following formula with the sample parameters which may be replaced if needed for easier calculation. I just want to understand concepts of the option and the payout formula.
$$E\left(\frac1T \left(\int_0^T S_u du-K\right)_+\right)$$
In general this is not possible to do analytically. To simplify this and reduce it to a much neater form, you can use change of numeraire. The calculations for this very type of option are given here: https://courses.maths.ox.ac.uk/node/view_material/48385
Once you obtain the form: \begin{equation} u(x,t) = e^{-r(T-t)} \mathbb{E}_{t}^{\mathbb{Q}} \left[ \left(1-\frac{X_t}{t}\right)^+\right], \end{equation} for the SDE: \begin{equation} dx_t = (1+(y-r)x_t)dt + \sigma x_t dW_t, \end{equation} you may simulate this using Monte Carlo methods to obtain an approximate price.
If you're looking for a similar type of option where you can obtain a closed-form formula, look at geometrically averaged Asian options. Deriving their price is described here: https://en.wikipedia.org/wiki/Asian_option#European_Asian_call_and_put_options_with_geometric_averaging