Black–Scholes Assumptions, Empirical Limits, and Possible Extensions
Summary
The document outlines assumptions behind the original Black–Scholes–Merton option model and highlights practical problems with them. The baseline setup assumes continuous trading, no transaction costs or taxes, frictionless short selling, no dividends, a constant risk-free rate, and an underlying price following a continuous process with constant volatility. In practice, volatility is not constant, and observed returns can have more extreme moves than a normal-return model predicts. The discussion cites sharp currency moves as an illustration of tail risk and notes volatility smiles as evidence challenging the simplest lognormal price assumptions.
Possible adjustments include accounting for dividends, using models with changing volatility, and choosing fat-tailed return distributions, though the appropriate distribution must be decided. Extensions can also address American-style options. The document does not specify particular alternative models or compare their performance, and it cautions that the smile alone may not settle the distribution question. Its claims about long-term option pricing are raised without supporting analysis in the included answer.
Key ideas
- The original model assumes constant volatility and a continuous price process with lognormal future prices.
- Real markets can show large moves that are poorly represented by a normal-return assumption.
- Volatility smiles and changing volatility motivate extensions to the baseline model.
- Dividends and American exercise can be handled through model extensions.
- Fat-tailed distributions are one possible response to extreme returns, but selecting one requires judgment.
Tags
Full text
# What are the main limitations of Black Scholes? # What are the main limitations of Black Scholes? 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. ## Answer by user59 (score 4, accepted) https://quant.stackexchange.com/a/825 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? ## Answer by David Harper (score 3) https://quant.stackexchange.com/a/822 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 ## Answer by user673 (score 0) https://quant.stackexchange.com/a/847 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
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