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Could you please share your opinions about Volatility Smile? What does it tell us when it gets more convex or when its level changes over time or any other change on it. Any paper/work/blog recommendations are also welcome.

Thanks,

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closed as too broad by LocalVolatility, Attack68, Lliane, Helin, JejeBelfort Mar 22 at 2:13

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    $\begingroup$ Although a good question, I'm afraid it's akin to asking what is the meaning of life. Books and countless papers have been written on this topic. You might want to narrow the question down a bit. $\endgroup$ – ilovevolatility Mar 15 at 6:24
  • $\begingroup$ @ilovevolatility just convexity and level change then? $\endgroup$ – TryingtobeQuant Mar 15 at 6:27
  • $\begingroup$ At its most basic a level change could indicate market participants anticipating either heightened / more subdued market moves. Changes in convexity and skew are more complex to interpret and explain. Also given the multiple definitions of convexity and skew. So, when you say convexity what do you mean with it? $\endgroup$ – ilovevolatility Mar 16 at 18:41
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As Nick said, the BS model does not explain the smile. Moreover, there is an interesting historical fact: While the smile was present for forex options quite the whole time, stock option haven‘t showed any volatility smile until the stock market crash 1987. Afterwards, the smile was even present there. So, the bottom line is: it‘s all about market participants.

An option at the money (atm) would have a lower vola than options in the money (itm) or out the money (otm). Why? So atm an investor is quite calm as chances and risks are offset. So bigger market events could lead to gains but also to losses (the premium). However, if you are itm or otm it is a bit different. A massive market movement could lead to huge losses (itm) or to a higher chance to make gains (otm). In fact, investors itm or otm are more sensible re market movements and, therefore, the IMPLIED volatility of these options is higher. By the way, the main reasons why the implied volatility is used as a stress index in financial markets.

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Volatility smile basically derives its meaning from implied volatility for a series of option that have the same expiration dates. When implied volatilities are plotted against strike prices, the resultant curve slopes upwards on either end. it's good to note that the smiles is never expected in the realm of Black-scholes option theory. Given an a particular expiration date, options whose strike price differs substantially from the underlying asset's price command higher prices than what is suggested by black-scholes option pricing model. These options are either described to be deep in the money or out of the money.
Volatility smiles should never occur in the context of BS theory, this is because BS requires a completely flat volatility curve. In reality graphing implied volatility against strike prices for a given expiry yields a skewed lines that slopes upwards, hence the term 'smile'. One of the ways to look at the volatility smiles is that it brings out the deficiencies in the standard models of option pricing which assumes constant volatility and log-normal distributions of underlying asset returns. In reality, empirical asset returns distributions exhibits kurtosis and skewness. Take for example, the implied volatility for upside equity options is typically lower than at the money equity for at-the-money equity options. However the implied volatilities of options on foreign exchange contracts tend to rise in both the downside and upside directions. In equity markets, a small tilted smile is often observed near the money as a kink in the general downward sloping implicit volatility graph. Practitioners in the market generally use the term implied-volatility to indicate the volatility parameter for the at the money option.

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The existence of a volatility smile is basically telling us that the assumptions behind the Black Scholes model do not hold true under all circumstances and underlyings. The volatility smile is the market adjustment to the assumptions made by the model. The main assumptions that the existence of the volatility smile is telling us is not true is that returns are normally distributed and that market participants are risk neutral. In real life the underlyings exhibit fat tails (and skew) and traders are risk averse/seeking. The more convex the smile could be due to either one (or a combination) of these two assumptions changing.

There was an article I read ages ago called “The holes in Back-Scholes” that might be helpful.

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