Option smile, also known as volatility smile, is a curve describing the relationship between implied volatility of options and exercise price.

The Black-Scholes option pricing model assumes that the stock price volatility remains unchanged, but the implied volatility curve of the option price in the actual market presents an upward half-moon shape with a low middle and high sides, that is, the volatility of the out-of-price option (imaginary option) and the in-price option (real option) is higher than that of the in-price option, like a smiling mouth.

If you draw a picture, the horizontal axis is options with different strike prices and the vertical axis is the volatility calculated according to the market price of options, then it looks like a smile.

Second, the emergence of the smile option.

The calculation method of implied volatility is to substitute the option price in the market into BS formula. When calculating, it is assumed that all the assumptions of BS are true, and then the volatility is regarded as an unknown number, and the others are known numbers. Therefore, the market price of each option can be calculated as volatility.

The fluctuation of option price in the market will lead to the change of implied volatility. The higher the option market price, the higher the implied volatility.

The volatility of out-of-price (imaginary) options and in-price (real) options is higher than that of in-price options (that is, option smile), indicating that out-of-price and in-price options in the market are higher when they are upside down.

The out-of-consideration and in-consideration options in the market give higher pricing, which is higher than that given by BS model, resulting in higher implied volatility, and then a volatility smile chart is produced.

If the pricing of various options given in the market is not too high and just equal to the pricing of BS model, then the calculated implied volatility is a horizontal line. Because the BS model often uses constant volatility when pricing options.

Third, to further understand the reasons for choosing a smile

When the actual market situation deviates from the theoretical model (BS formula), we have two thinking directions: one is that the market is wrong and the market is invalid; One is that the model is wrong, and some assumptions of the model are wrong.

Comrades who think that the market is wrong explain that market participants overestimate the probability of large fluctuations in stock prices, so they are more willing to buy the results of large fluctuations in stock prices when buying options, that is, to buy options (deep/imaginary options) with a far cry from the current actual stock prices and wait for winning. This behavior will cause the option price at both ends to be extra raised, thus forming a "high on both sides and low in the middle"

This is somewhat similar to the psychology of people who buy lottery tickets. Although they know that winning the lottery is a small probability event, they actually overestimate the winning probability because they can't really perceive how small it is.

Comrades who think the model is wrong explain that the assumption that the stock price walks randomly and the yield obeys normal distribution in BS model is wrong. The probability of small probability events (both ends) in the model is much larger in practice, and black swans are not so rare.

They think that the real rate of return may obey the thick-tailed distribution. Therefore, when the normal distribution model is used to measure the world with thick tail distribution, both ends are underestimated, which makes the theoretical price (deep real value and imaginary value) of options calculated by both ends lower than the actual price in the real world.