I am doing some research involving black-scholes model and got stuck with dividend-paying stocks when evaluating options. What is the real-world approach on handling the situations when an underlying pays dividend? Thank you
The simplest common approach is to assume a continuous dividend yield. This is treated mathematically in the same way as foreign interest rates on FX options, and the necessary changes to the formula can be found with a quick web search. If the options are European exercise then you might be perfectly happy with this.
A more robust treatment of dividends involves using dynamic programming (such as trinomial trees) to price the option, where dividends are treated as partly proportional to stock price, and partly fixed according to taste. In this case, the dividend is treated as a boundary condition on your solution grid. It's quite a bit more difficult to get that right. You can see how it works in Hull's book.
Assuming that by "real-world", you mean "with dividends", you can find extensions of the Black-Scholes models which include dividends on this wikipedia page.
As @TheBridge mentioned in his answer, there are several assumptions that are made within the BS framework, so your model can become more complicated depending on the assumptions you make.
Well the real issue (IMO) is not dividend but rather estimating the forward price of the underlying at exercise date (in the BS setting for vanillas which I think is your setting).
So my answer, with which you will not be really satisfied I think, is that any model (continuous, discrete, even stochastic dividend) will do the trick for pricing purposes as long as you get the correct forward of the underlying stock of your option (for hedging this is a different issue).