Why Implied Volatility Varies Across Option Strikes
Summary
The document explains why Black–Scholes implied volatility can vary with an option’s strike, forming a smile or skew. It begins from the model’s assumptions of constant volatility and continuous price paths, then frames implied volatility as the Black–Scholes volatility that makes the model’s option value match a value based on the underlying asset’s actual probability distribution. If that distribution is not lognormal, the fitted volatility need not be the same across strikes.
The answers offer possible reasons for the shape: option prices may reflect the market’s assessment of extreme or more frequent jumps, and demand may differ between at-the-money and out-of-the-money contracts. These are presented as explanations rather than settled conclusions. The document gives no empirical analysis or specific calibration method, and its central point is that implied volatility is a model-dependent quoting measure that reveals departures from Black–Scholes assumptions, not a direct observation of constant realized volatility.
Key ideas
- Black–Scholes assumes constant volatility and smooth price paths, conditions that market prices may not satisfy.
- Implied volatility is the volatility input that makes the model price match the observed option price.
- A non-lognormal underlying distribution can produce different implied volatilities across strikes.
- Jump risk and strike-specific option demand are proposed as contributors to the smile, but are not established as definitive causes.
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# Why implicit volatility has the shape of a "smile"?
# Why implicit volatility has the shape of a "smile"?
Two of the conditions for an asset price to have a lognormal distribution are:
- The volatility of the asset is constant.
- The price of the asset changes smoothly with no jumps.
In practice, neither of these conditions is satisfied for an exchange rate, but I don't understand why these reason determine the form of a "smile" for implied volatility?
## Answer by lostAstronaut (score 2, accepted)
https://quant.stackexchange.com/a/34729
There are plenty of theories but no one is 100% certain.
Theory 1: Options which have strike prices increasingly far away from the spot price are including extreme movements in the market such as black swan events into the price. When an event like this happens there is a sharp change in price of the option and extreme change in volatility.
Theory 2: Investors may have the same demand in terms of expiration date but not strike price. Investors tend to prefer in-the-money and out-of-the-money options over ATM. Since demand pushes up the prices of options, implied volatility also becomes higher.
Theory 3: Similar to 1, implied volatility can include jump predictions, but this theory includes all types of jumps, small and medium ones as well as large ones. These jumps are accompanied by large kurtosis as stated in the link @LocalVolatility commented.
## Answer by will (score 4)
https://quant.stackexchange.com/a/34730
The smile is there exactly because the model is wrong.
The reason it's used though (despite being wrong) is that it provides a convenient space to look at the underlying - the vol*
The (undiscounted) value of an option is given by:
$$ \int_0^\infty \mathrm{PDF}(s) (s-K)^+ \mathrm{d}s $$
where $\mathrm{PDF}(x)$ is the real probability distribution of the underlying. This is model independent.
Under BS, the value is the following:
$$ \int_0^\infty \mathrm{PDF_{LN}}(F,\sigma)(s) (s-K)^+ \mathrm{d}s $$
where $\mathrm{PDF_{LN}}(F,\sigma)$ is the pdf of a lognormal variable with an expected value of $F$ and vol of $\sigma$.
So now, we just need to make them match - and we have one parameter to solve for - $\sigma$. This gives us the BS vol. If the distribution truly were lognormal, you'd obtain the same vol everywhere. Since it's not, you get a changing vol.
*BS vol.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.