A Model-Based Intuition for Implied Volatility Smiles
Summary
The document asks whether strike-dependent implied volatility can be explained without relying only on trader preferences. It sketches a model-based intuition using Black–Scholes sensitivities. In that model, options away from at-the-money can have positive curvature of vega with respect to volatility, while the at-the-money option’s corresponding sensitivity is described as zero. A position long a wing option and short a suitable amount of at-the-money options can therefore retain exposure to changes in volatility while neutralizing first-order vega.
If volatility itself is uncertain, demand for that curvature exposure could raise prices of both high- and low-strike options, producing a smile in implied volatility. The answer presents this as an explanatory argument rather than a calibrated pricing model or empirical test, and it acknowledges that volatility risk premia and market behavior remain part of the story. It briefly mentions an analogous sensitivity-based route to explaining skew, but leaves that analysis undeveloped.
Key ideas
- Implied volatility can vary by strike when options have different exposures to changes in volatility.
- In the Black–Scholes explanation, wing options have positive vega curvature while at-the-money options are used to offset vega.
- A vega-neutral position can retain exposure to volatility curvature.
- Demand for that exposure under uncertain volatility may lift implied volatilities at both wings and create a smile.
- The explanation is an intuition and does not provide a calibrated model or empirical evidence.
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Full text
# Question on implied vol (surface) and strikes # Question on implied vol (surface) and strikes there have been loads of papers on skews ATM / OTM, volatility premium and such. Lots of explanations for why iv is different on same stock with different strikes focused on preference of informed traders. But is there a model-based, non-behavioral explanation for the difference of implied vol for different strikes? Could it be approached through state price distribution? Thanks and happy holidays ## Answer by Will Gu (score 1) https://quant.stackexchange.com/a/31631 This is an interesting question. I guess the fundamental belief is that any model serves the purpose of capturing what happens as aggregated behaviors in the market. Believe it or not, model also influences the market, especially when most of the transactions are done by people who utilize models (banks, funds, etc). The basic idea of vol smile is the uncertainty of (realized) vol, and thus the demand for additional risk premia. This is kinda non-behavioral, but not model-based. It's an assumption but it checks out with the empirical observation. There's definitely some evolvement in people's views on the market. For a while in history, the vol was actually flat. If you believe this is true, then the vol smile is an natural derivation from the addition of vol of vol. But I'll try to explain from the model-perspective. Let's start from Black-Scholes model (with some behavioral explanation). By the way, This is model-based, but somewhat behavioral: If you plot the vega-gamma (second derivative of option price w.r.t vol) against the (log) strike from the Black-Scholes model, you would see a two-hump-shape curve. Now if you long an ITM/OTM option, you are effectively long both vega and vega-gamma (since it's positive). And then you can short appropriate amount of ATM option to make vega-neutral (since ATM option has zero vega-gamma in Black-Scholes). Now you are long pure vega gamma, and whenever vol moves, you make money (if you believe that vol does move). In this case, we'd be happy to buy high/low strike options if vol is flat, effectively ATM/OTM option prices are bid up, therefore the implied vol. This works for both high/low strikes, leading to a smile in vol. Similar approach can be used to explain the vol skew (risk-reversal) by looking at vega-dspot, but I'll leave it out for now since it's not exactly what OP was asking about.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.