What's the difference between binomial trees pricing and black sholes model pricing?The reason we do monte carlo or whatever experience is to get the underlying stock price path?

I think both Monte-Carlo and binomial tree are numerical techniques while Black Schole formulacan give you a closed form solution. But for some security there is no closed form solution. Inthis case you have to use numerical technique and in many cases you will use Monte-carlo insteadof binomial tree because the tree will be very complicated.Correct me if I am wrong, I am a beginner.

Thank you bquant, it makes sense. I also think that it's the difference between discrete and continuous, european option and american and exotic options. About monte carlor I know, such as box-mullar method is to convert the unifrom distribution to normal distribution and generate random number. Is that the only reason using monte carlo? And how does the monte carlo random number ralated to stock price path?

As I understand, what you mentioned (i.e.,random number generator) is just the fundamental partof the Monte carlo simulation. That is not the reason of monte carlo. To see why use monte carlo,we can imagine what other alternatives we have got, say finite method for PDE and tree. Asfor finite method , you cannot do it when the problem is in high dimension. Tree isalso impractical for complicated security. But monte carlo is powerful, since according tothe Feynman-Kac formula, the price of a security is an expectation of something. Computingthe expectation is exactly the work Monte carlo can do. Monte carlo can take you a lot of time,but it can give you a good accuracy. As for the relationship to path of stock price. If your security is path-dependent, you will need tosimulate the path of stock price, otherwise you donot need to simulate the path.

That's interesting. Thank you bquant.