What are the Black-Scholes model assumptions, and which ones are most violated in real markets?

The Black-Scholes-Merton (BSM) model is elegant but relies on assumptions that real markets regularly violate. Understanding these gaps is essential for CFA Level II.

The Six Key Assumptions

1. Stock price follows geometric Brownian motion with constant volatility — In reality, volatility clusters and changes over time (GARCH effects). This is arguably the most violated assumption.

1. No dividends during the option's life — Easily corrected by using the dividend-adjusted BSM formula (reduce S₀ by PV of dividends).

1. Continuous trading is possible — Markets close overnight and on weekends. Large gap moves occur at the open.

1. No transaction costs or taxes — Trading options involves bid-ask spreads, commissions, and margin costs. Delta hedging in practice is expensive.

1. Risk-free rate is constant and known — Rates change, and the appropriate rate isn't always clear (repo rate? T-bill rate?).

1. No arbitrage opportunities — Generally reasonable for liquid markets but can break down in crises.

The Volatility Assumption: Where BSM Fails Most

If BSM were perfectly correct, implied volatility would be the same for all strikes and expirations on the same underlying. But in practice, we observe:

• Volatility smile/skew: Out-of-the-money puts have higher implied volatility than at-the-money options (post-1987 crash phenomenon)
• Volatility term structure: Near-term options often have different implied vol than long-dated options

Example:

Crestfield Options Desk prices SPX options and finds:

If BSM assumptions held perfectly, all implied vols would be 17.2%. The skew exists because market participants price in the possibility of large downside moves — something the log-normal distribution of BSM underestimates.

Real-World Impact:

• Fat tails: BSM underestimates the probability of extreme moves. A 5-sigma event under BSM should occur once every 14,000 years — in reality, markets experience such moves every few years.
• Volatility clustering: Periods of high vol tend to follow periods of high vol, violating the constant-volatility assumption.

CFA Exam Focus: Expect questions asking you to identify which assumption is violated given a specific scenario (e.g., 'the stock pays a large special dividend' → dividend assumption, 'implied volatility varies by strike' → constant volatility assumption).

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