$(20$ points$)$ This is an open ended question and you need to do some research to get the answers:

Is arbitrage good or bad? Justify.

What are some problems with the Black$-$Scholes$-$Merton model?

Suggest some improvements of the Black$-$Scholes$-$Merton model.

In the class, we have derived the Black$-$Scholes$-$Merton formula for European options. Is is possible to derive something similar for American options?