How much Black-Scholes survives broken assumptions
Question
How robust is the Black-Scholes model when its assumptions are violated?
Short answer
Surprisingly robust. Its assumptions (constant volatility, continuous hedging, no costs, lognormal returns) all fail, but the hedging error from using the wrong volatility grows smoothly with the gap between realised and implied, discrete hedging error averages out, and modest costs can be absorbed by adjusting volatility. It fails badly with large jumps, very illiquid markets or wildly wrong volatility, which is why traders use it as a quoting language plus a smile.
Adapted from Wilmott, Frequently Asked Questions in Quantitative Finance, FAQ 45, p.206; question and answer rewritten by SelfMadeQuant.com.
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Open the worked solutionLessons behind this question
- The Black-Scholes PDE, Formula and Implied Volatility
- Derivation — From Binomial Trees to Black-Scholes in the Continuous Limit
- The Black-Scholes Formula Derived via Change of Measure
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