Why Implied Volatility Smiles Form Around At-the-Money Options
Summary
The document explains why implied volatility often rises away from the at-the-money strike, drawing on three complementary interpretations. One links smile curvature to higher moments of the return distribution: convexity can reflect positive excess kurtosis, while concavity can reflect negative excess kurtosis. A jump example illustrates that smile shapes depend on the modeled distribution and need not always resemble the familiar smile.
Another explanation focuses on volatility changing with market levels. A sharp decline may coincide with higher volatility, making out-of-the-money puts more expensive in Black–Scholes terms and producing a skew. In FX, demand for protection against large moves can also raise the implied volatility of options on the wings. These are qualitative explanations, not a universal rule: the document notes there is no one-to-one mapping between smile shape and distribution moments, and the examples do not establish a single cause across markets.
Key ideas
- Implied volatility smile curvature can reflect higher moments of the underlying return distribution.
- A jump process can produce a concave shape, so smiles are not always convex.
- The leverage effect can raise implied volatility for downside strikes when declines coincide with higher volatility.
- Demand for protection against large FX moves can increase wing premiums.
- Smile shape does not uniquely identify the return distribution or its cause.
Tags
Full text
# At-the-money and volatility smile
# At-the-money and volatility smile
In a volatility smile; why is the ATM point usually or ideally at the bottom? In other words, why is the "smile" smile shaped as opposed to another shape?
## Answer by LocalVolatility (score 4)
https://quant.stackexchange.com/a/30142
The shape of the implied volatility smile is linked to the higher moments of the underlying return distribution though there is no one-to-one relationship. A convex (concave) smile usually indicates a distribution with positive (negative) excess kurtosis.
Here is an example for a concave implied vol. smile. It is often observed in markets where a single large jump is anticipated. The smile in my example was generated from a constant coefficient geometric Brownian motion model with $\sigma = 20\%$, $S_0 = 100.0$, $r = 0\%$ and $T = 1 \text{ month}$. In addition there is a single jump in $T^{\text{J}} = 0.5 \text{ month}$ with a jump size of $\pm 10\%$ and equal probability of the jump being up or down.
## Answer by nbbo2 (score 2)
https://quant.stackexchange.com/a/30119
The issue is that volatility is not constant and is not independent of the market level.
If the market remains near current values until maturity it will almost by definition affect near-the money options the most. Such options can therefore be correctly priced by assuming (well-known BS assumption) that volatility is constant and remains at currently observed level.
Conversely for puts struck well below current market level to come into play, it must be that the market plunges substantially between now and maturity. Experience shows that such a sharp drop will be accompanied by a rise in volatility (so called leverage effect discovered by F. Black in late 1970's). The Black Scholes formula is strictly speaking not applicable in this case, but we can get the right answer by plugging into it an assumed volatility considerably higher than the current, to reflect the expected increase in actual volatility along such a path.
When these B.S. volatilities are plotted against strike we see the familiar smile curve.
## Answer by Randor (score 0)
https://quant.stackexchange.com/a/30116
in fx you have such smiles. the premium on the wings is due to aversion of traders to large moves in the underlying. traders believe the probability of large moves is higher than what is implied through the atm volatility.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.