Black-Scholes Assumptions And Failure Modes

Every model is built on assumptions, and Black-Scholes states its five plainly: the stock moves continuously with a constant volatility, trading is free and instant, borrowing and lending happen at one known rate, and returns are log-normal. If all five were true, one number — $\sigma$ — would price every option on that stock, at every strike, for every expiry. The fact that real option markets need a different $\sigma$ for almost every strike is not a rounding error. It's the market telling you, loudly, which assumption is false.

A forecast that assumes it never rains sideways

A weather model that assumes rain only ever falls straight down will get most ordinary days roughly right, and be badly wrong exactly on the days that matter — a sideways storm, a freak gust. Black-Scholes is built the same way: it assumes the stock only ever "rains" in small, continuous, evenly-sized moves. Real markets have their version of a sideways storm — an earnings gap, a flash crash, a central bank surprise — and the model has no room in its assumptions to describe one.

The tell: implied volatility isn't flat

$$\sigma_{\text{implied}}(K) = \text{the } \sigma \text{ that makes Black-Scholes match the market price at strike } K$$

In plain English: instead of using one $\sigma$ to price options, traders run Black-Scholes backwards — they plug in the market's actual price and solve for whichever $\sigma$ would have produced it. If the model's assumptions held, that number would come out identical at every strike. It doesn't.

Worked example 1: the cost of ignoring skew. Suppose a 3-month at-the-money option on a $100 stock trades at an implied volatility of 20%, while the 90-strike put — protection against a crash — trades at 25%, and the 110-strike call trades at 18%. This is a textbook equity skew: fear of a drop is priced richer than hope of a rally. A trader who mistakenly uses the flat 20% ATM volatility to price the 90 put, using $r=0$, $T=0.25$, gets $d_1 = (\ln(100/90) + 0.5\times0.04\times0.25)/(0.2\times0.5) = (0.1054+0.005)/0.1=1.104$, $d_2=1.004$, giving a put price around $1.30. Repricing with the correct 25% volatility instead raises $\sigma\sqrt T$ to 0.125, shifts $d_1$ to $0.887$, $d_2$ to $0.762$, and lifts the put price to roughly $2.05 — a 58% underpricing from using the wrong number, on a strike that exists precisely because the model's flat-volatility assumption is false there.

Tilt the surface and watch the skew tilt with it — a flat sheet is what Black-Scholes' single-$\sigma$ assumption predicts; the tilt you can actually create is the market's live rebuttal of it.

Worked example 2: the tell from history, not just skew. Under Black-Scholes' log-normal assumption, a one-day move of 20% is a roughly 20-standard-deviation event at typical equity volatility — a probability so small it would essentially never occur in the lifetime of the universe. The S&P 500 fell 20.5% in a single session on October 19, 1987. That is not a model running slightly hot; it's a model whose "impossible" event happened in real life, because real returns have fatter tails than the log-normal curve allows.

Push the standard deviation slider and look at how little probability mass sits past 4 or 5 standard deviations — real markets put far more weight out there than a normal curve ever will.

What this means in practice

Each broken assumption has a named fix that's its own concept: constant volatility fails, so desks use Local Volatility and Dupire's Formula or stochastic-volatility models that let $\sigma$ move with the strike and with time; continuous trading fails, so hedging is discrete and costs money (see Delta Hedging Frequency And Costs); no-jumps fails, so Merton's Jump-Diffusion Model adds sudden moves on top of the diffusion. None of these replace Black-Scholes outright — they patch one assumption at a time while keeping the same no-arbitrage logic that makes the original model useful in the first place.