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Model-Free Variance and Black–Scholes Implied Volatility Are Different

Article Quant Q&A · Author: user853717
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Summary

The question compares a model-free volatility expression based on option prices across strikes with implied volatility obtained by inverting the Black–Scholes pricing formula. The answer clarifies that the two quantities represent different concepts. The model-free expression corresponds to the implied volatility of a log contract and is associated with the fair strike of a variance swap. Black–Scholes implied volatility is the volatility input that makes the model price of a vanilla option match its observed market price.

The discussion therefore resolves the apparent confusion without deriving an iterative calculation or giving a numerical example. Its main limitation is scope: it identifies the distinction but does not explain implementation details, assumptions behind the model-free formula, or how to estimate either quantity from a particular option surface. The two values should not be treated as interchangeable merely because both are described using implied volatility terminology.

Key ideas

  • The model-free expression describes the implied volatility of a log contract and relates to a variance swap's fair strike.
  • Black–Scholes implied volatility is the model parameter that matches a vanilla option's theoretical price to its market price.
  • The two volatility measures have distinct meanings and should not be conflated.
  • The explanation does not address formula implementation or the assumptions needed for estimation.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Implied volatility model-free


# Implied volatility model-free












I know that $\operatorname{IV-model \space free}=2 \int_{0}^{+\infty}\frac{c_0(T,Ke^{r(T-t)})-c_0(t,Ke^{r(T-t)})}{K^2}\operatorname{d}K$ is calculated using an iterative procedure, i.e. setting a volatility value and comparing theoretical price obtained by software (with that volatility value) and the market price: if theoretical price is less than market price we have to increase the volatility (and vice-versa). This would be the reason:

"You can not invert the BS-equation because implied volatility is repeated both in 1) the PDF of Normal standard like standard deviation of log-returns distribution and 2) the argument of Normal standard in $d_1$ and $d_2$ terms."

Deriving BS-close form I have $$\varphi(S_t)=e^{-r(T-t)}\int_{z_0}^{+\infty}((S_te^{(r-\frac{\sigma^2}{2})(T-t)+\sigma\sqrt{T-t}z})-K)^+\Phi(z)\operatorname{d}z$$ with $z_0=\frac{\operatorname{ln}(\frac{S_t}{K})+(r+\frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$ and $\Phi(z)\doteq \frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}$. So, knowing that $d_1:=-z_0+\sigma\sqrt{T-t}$ and $d_2:=d_1-\sigma\sqrt{T-t}$ (so the point 2) is trivially true), what is the $\sigma=\operatorname{IV}$ to which it refers the point 1)?

Thanks in advance for any help!

## Answer by user34971 (score 2)

https://quant.stackexchange.com/a/60342

The "IV model-free" formula you wrote is the implied volatility of the log contract, which is the fair strike of a variance swap. That is the meaning of your IV model-free.

The IV in the Black-Scholes vanilla options formula is the volatility parameter you need to input into the formula to match the market price of vanilla options.

So these are two different things.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.