Pls explain and discuss these limitations, and explain which models can I use to overcome these limitations. Alternatively, provide examples of how to modify the original Black Scholes to overcome these limitations.
Actually, handling dividends is fairly easy: http://en.wikipedia.org/wiki/Black-scholes#cite_note-div_yield-3
David mentions this above but "Stock price follows a Weiner [sic] process" is worth a little more discussion. Recently, USDJPY fell 300 pips in just a few minutes. If you accept that USDJPY follows a Wiener process, the odds of this happening even once in a million years are astronomical. USDJPY has done something equally unlikely earlier (250 pips in a few minutes if I remember correctly).
The problem: once something falls "a lot" quickly, it's likely to fall even further. In other words, a loss of 300 pips is 5 minutes is more likely than a loss of 75 pips in 5 minutes.
The solution is to use "fat-tailed" distributions:
but, of course, you then have to decide which fat-tailed distribution to use.
I'm not sure the volatility smile disproves lognormal distribution. My theory on the volatility smile: Why does implied volatility show an inverse relation with strike price when examining option chains?
Technical assumptions are below. I think in practice the most vexing assumptions are (i) Brownian motion assumption that has returns as normal and therefore future price as lognormal (the existence of volatility smiles refutes lognormal prices) and (ii) constant volatility assumption (also empirically refuted). Original BSM is Euro only non-dividend, but many assumptions can be overcome with extensions: American-style, dividends, changing volatility.
Assumptions used to derive BSM differential equation (source: John Hull):
Stock price follows a Weiner process (itself a particular Markov stochastic process) with a constant volatility
Short selling is allowed
No transaction costs and no taxes; securities are perfectly divisible
Dividends are not paid
There are no (risk-less) arbitrage opportunities
Security trading is continuous
The risk-free rate of interest is constant and the same for all maturities
One big limitation is that the BSM doesn't work on long term option pricing, see my blog below: