According to Black-Scholes model assumptions, all options with different strikes and exercise dates have the same implied volatility, i.e. implied volatility does not depend on either the option's moneyness or the expiry date. In reality, however, the implied volatility of options that are observed on exchanges differs depending on the strike price and expiration date. A graph showing the relationship between the strike price and implied volatility is called the volatility skew. The relationship between the expiry date and implied volatility is a term structure of volatility.

The volatility slope takes into account implied volatility among options with different strikes or different deltas with the same expiry date.

**Fig. 1. Vol skew of S&P 500
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The slope of volatility is very important from a practical point of view, as it indicates the amount of risk in the market or the expected realized volatility at different potential prices of the underlying asset.

For example, if the implied volatility of the lower strikes (both for call and put options) is elevated relative to the volatility of the high strikes, then the market places a high probability that the price of the underlying asset will fall. And falls, as a rule, occur with higher volatility (in the stock market) than growth.

Lower-strike put options are usually traded with higher implied volatility than high-strike put options or ITM options. This happens because market participants are willing to pay relatively higher prices to protect the underlying asset (stock) from a significant fall than to protect it from a similar upward movement.

**Historical evidence **

The market crash in 1987 taught investors that while the distribution of share market returns has a negative asymmetry, the share market returns distribution chart also has thick tails on the left side. Therefore, the probability of significant drops is very high, which is observed in reality. However, in the long run, stock markets tend to grow slowly from year to year. One shouldn't forget about price moves that take place during a fall. Since options provide protection against such surges, investors are willing to pay a higher price for lower strikes. Now options with lower strike prices (i.e. OTM puts) have higher implied volatility than OTM call options.

**Fear**

Falling markets usually generate increased uncertainty, leading to higher volatility in stock prices and further sell-offs - especially with high leverage of investment funds and banking sector. The volatility skew, which is calculated as the difference between the implied volatility of OTM put and OTM call, tends to rise as the price of risky assets (stocks, corporate bonds) falls, and falls as the markets rise.

**Leverage of companies**

A decline in the share price usually indicates a deteriorating financial position, which could potentially lead to bankruptcy. As a result, a relatively low share price implies higher leverage and therefore the value of the company's equity capital becomes more volatile.

Thus, in the stock option market, the implied volatility graph presents a curve with a negative slope, which becomes positive at higher strikes. The implied volatility graph curves upwards at high strike prices, as these call options provide protection (or participation) in a very rapid growth, the probability of which should not be underestimated. And since this growth will be a very volatile event, traders incorporate it into option prices. In the slang of traders, the form of the volatility skew chart is called a “smirk”.

**Fig. 2. Smirk
**

However, there are also other forms of implied volatility curves - the ideal “smile” of volatility, which has the form of a parabola, is characteristic of the currency options market; and the form of “skew” of volatility represents a curve with a negative slope coefficient.

**Fig. 3. Smile**

**Fig. 4. Skew**

Usually among traders, the term “volatility skew” is used to explain that each strike has its own level of implied volatility.