Lesson 19 of 20 · 8 min
Where Black and Scholes is known to be wrong
The model's assumptions are all false in specific, well-documented ways, and the market prices around them.
The model is not approximately true with some noise around it. Its assumptions are individually and identifiably false, and the interesting part is that they are false in known directions, which is why the market can correct for them.
The assumptions, and what actually happens:
- Constant volatility. Volatility clusters — calm periods and turbulent periods come in runs — and it jumps on news. This is why an implied volatility surface exists instead of a number.
- Continuous prices. Prices gap, especially overnight and on news. A hedge that must be adjusted continuously cannot be adjusted through a gap at all.
- Normally distributed returns. Real return distributions have fatter tails: extreme moves happen far more often than the bell curve permits. Track 4's lessons 3 and 4 quantify this.
- Frictionless trading. Spreads, commissions and market impact are real, and lesson 11 of Track 2 showed they can exceed twenty percent of a low-priced contract's value.
- A single known risk-free rate for borrowing and lending. Real borrowing costs more than lending earns, and the gap widens under stress.
- No dividends, in the original form. Handled by extensions, but it is an addition rather than something the base model captures.
The normality assumption is the one worth quantifying, because the failure is not marginal. At 25 percent annual volatility, a daily move is about 1.6 percent. The market's decline on 19 October 1987 was roughly 20 percent in a day, which is about 14 standard deviations. Under a normal distribution the probability of a 14-sigma move is on the order of one in 10 to the 45th power — a number with no physical meaning. It happened anyway, and it was not the only such event.
None of this makes the model useless. It remains the common language for quoting and comparing options, and the smile is precisely how the market patches the tail failure. But it means a model output is a translation of a market price, not an independent verdict on it.
Key terms
- Volatility clustering
- The empirical pattern that turbulent days follow turbulent days. It contradicts the assumption of constant volatility.
- Jump risk
- The possibility of a discontinuous price move. It breaks the continuous-hedging argument the model is built on.
- Model risk
- The risk of loss arising from a model being wrong rather than from markets moving. Distinct from market risk and rarely quantified.
Worked example
How impossible the tails are, under the assumption
25% annual volatility implies a daily standard deviation of 1.6% (25% ÷ √252).
- A 3% daily decline
- about 1.9 sigma
- A 7% daily decline
- about 4.6 sigma
- A 20% daily decline
- about 14 sigma
- Times a decline of that scale has occurred in US equities
- At least once
- Conclusion about the model
- The tail assumption fails by dozens of orders of magnitude
Expected a few times a year.
Normal model: roughly once in 500 years.
Normal model: probability near 1 in 10⁴⁵.
19 October 1987.
The model does not slightly underestimate extreme events. It assigns them probabilities that are effectively zero, and they occur anyway.
Check yourself
Which Black-Scholes assumption does the volatility smile exist to work around, and what does that tell you about using the model?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.