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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.

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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:

http://en.wikipedia.org/wiki/Fat_tail#Applications_in_economics

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?

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  • @"I'm not sure the volatility smile disproves lognormal distribution."

    It does, by definition.

     – quant_dev Mar 27 '11 at 02:48
  • The volatility smile just tells us what investors think. You'd have to look at the actual values of the underlying to see if it does or doesn't follow a lognormal distribution. –  Apr 06 '11 at 15:47
  • If the stock prices followed the lognormal process, there would be no consistent smile for over 20 years. – quant_dev Apr 06 '11 at 16:39
  • There could be. There's no reason to believe option traders can predict the future better than anyone else. –  Apr 07 '11 at 01:33
  • If the stock prices consistently followed a different process than the option traders believed, for 20 years, than the first option trader who'd bother to look at historical data and adjust her expectations would reap enormous profits.

    So: yes, it might happen, but it's like saying that a pink unicorn might happen (in the sense that you can't make a formal proof that pink unicorns do not exist).

     – quant_dev Apr 07 '11 at 05:57
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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

David Harper
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  • When there were temporary restrictions on short selling in European markets after 2008, did it observably change the stock price dynamics? – quant_dev Mar 26 '11 at 09:30
  • I always wondered why don't they price it with $e^{-\int_0^t r(s)ds}$ instead of $e^{-rt}$ as the discount factor to account for your last point? – Jase Nov 14 '12 at 15:10
  • I don't think stock prices is a Wiener process, otherwise they might become negative. Instead, the relative change of stock prices is a Wiener process. That is $dS_t/S_t \sim \cal{N}(r , dt, \sigma^2 , dt)$, where $S_t$ is the price of a stock at time $t$. – wsw Dec 01 '12 at 22:23
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One big limitation is that the BSM doesn't work on long term option pricing, see my blog below:

http://value2get.blogspot.com/2011/03/why-doesnt-black-scholes-model-work-in.html