Why Black-Scholes Implies a Flat Implied Volatility Surface
Summary
The document explains why a single Black-Scholes model with constant volatility implies the same implied volatility across strikes and maturities. The model specifies one stochastic process for the underlying asset, with a common diffusion coefficient. Using different volatility values for separate options would amount to using different underlying models rather than applying one Black-Scholes model consistently.
Implied volatility is the constant volatility that makes a Black-Scholes price match an observed option price. If all market prices were generated using the same Black-Scholes volatility, inversion would recover that value for each option, producing a flat surface. The discussion connects this result to the model’s lognormal terminal-price distribution. It clarifies the model’s implication under its assumptions; it does not claim that real market prices must produce a flat smile or surface.
Key ideas
- Black-Scholes specifies one underlying price process with a constant diffusion coefficient.
- A common model volatility produces the same implied volatility across options with different strikes and maturities.
- Assigning each option its own volatility is inconsistent with valuing all options from one shared underlying model.
- The flat implied volatility result follows when prices are generated by the same Black-Scholes model used to invert them.
- Market implied volatilities need not be flat when observed prices depart from the model’s assumptions.
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Full text
# "Black-Scholes model implies flat implied volatility plots"?
# "Black-Scholes model implies flat implied volatility plots"?
I am a beginner in Finance and I get confused by the statement "Black-Scholes model implies flat implied volatility plots"
Here is one form of this statement: (Dan Stefanica, 150 most frequently asked questions on Quant Interviews, 3.3.12)
> "On the same asset, prices for options with multiple strikes and maturities are quoted and implied volatilities can be computed for each of these options. If the price of the asset had a lognormal distribution - as assumed in the Black-Scholes model - then the resulting plots of implied volatility vs Strike would be flat"
But surely the volatility is assumed constant in the derivation of BS in the first place? It is an assumption, not a consequence of "the price of the asset having a lognormal distribution" ?
Here is another example of my confusion , this time from this paper http://www.columbia.edu%2F~mh2078%2FBlackScholesCtsTime.pdf
As i understand it, in the Black-Scholes model, we fix K,T, we assume $\exists \sigma=\sigma(K,T)$ for those K,T and we come up with the BS formula above
There is nothing in the BS derivation that assumes $\exists \sigma, \forall K,T ....$ ?
## Answer by LocalVolatility (score 5, accepted)
https://quant.stackexchange.com/a/35893
Regarding your second question: Remember that Black/Scholes start by postulating a stochastic model for the dynamics of the underlying asset - a geometric Brownian motion with a constant diffusion coefficient $\sigma$. This asset price process should be the same no matter what option you want to value based on it. Saying that you allow for different values for $\sigma$ for different strikes or maturities essentially means that you use a different model for each vanilla option, which is inconsistent and for sure not what they initially had in mind.
Regarding your first question: The implied volatility $\hat{\sigma}(T, K)$ is the value for the constant diffusion coefficient that results in a Black/Scholes price equal to some observed market price. So obviously, if market prices for all $(T, K)$ were computed using a Black/Scholes model with a common volatility $\sigma^*$, then you would recover this same value as the implied volatility $\hat{\sigma}(T, K) = \sigma^*$ and thus a flat volatility surface. Similarly, this fixed $\sigma^*$ results in log-normally distributed prices. So if implied densities for the terminal spot price of a given maturity were log-normally distributed, then the corresponding implied volatility smile would be flat.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.