什么是波动率微笑？

What is volatility smile?

The volatility smile is a common graph that is obtained by plotting the strike price versus implied volatility for a set of options on the same underlying asset and expiration date. The volatility smile is so named because it is shaped like a smiling face. When the underlying asset of an option is more out-of-the-money (OTM) or in-the-money (ITM), the implied volatility increases and is lower relative to at-the-money (ATM) options. Not all options have a volatility smile.

Key Points

• When options have the same expiration date and the underlying asset at different strike prices, the graph of implied volatility tends to take on a smile shape.
• The smile curve shows that options that are in the money (ITM) or out of the money (OTM) have the highest implied volatility.
• Options with the lowest implied volatility usually have strike prices close to or around the money (ATM).
• Not all options will exhibit an implied volatility smile. Short-dated options and currency-related options are more likely to see a volatility smile.
• The implied volatility of an individual option may also exhibit a volatility smile as it moves toward more in-the-money or out-of-the-money options.
• While implied volatility is a factor in option pricing, it is not the only factor. Traders also need to pay attention to other factors that affect option prices and volatility.

What does the volatility smile convey?

The volatility smile is caused by changes in implied volatility as the underlying asset moves in or out of value. The more in-the-money or out-of-the-money an option is, the higher its implied volatility is usually, while at-the-money options tend to have the lowest implied volatility.

The volatility smile does not fit the Black-Scholes model, one of the main formulas used to price options and other derivatives. The Black-Scholes model predicts that when the implied volatility curve is plotted for different strike prices, the curve will be flat. According to this model, the implied volatility of all options expiring on the same day and with the same underlying asset should be consistent, regardless of the strike price. However, in reality this is not the case.

Since the 1987 stock market crash, the volatility smile has begun to appear in options pricing. This phenomenon has not previously occurred in the U.S. market, suggesting that the market structure is more consistent with the predictions of the Black-Scholes model. After 1987, traders realized that extreme events could occur and the market experienced significant skew. So, in reality, implied volatility increases or decreases as options move toward more in-the-money or out-of-the-money options.

Furthermore, the existence of the volatility smile shows that demand for in-the-money and out-of-the-money options generally exceeds that of at-the-money options. Demand drives price, which affects implied volatility. This can partly be attributed to the reasons mentioned above. Extreme events can cause significant moves in option prices, and the possibility of large moves is factored into the calculation of implied volatility.

Example of using Volatility Smile

A volatility smile can be observed when comparing various types of options with the same underlying asset and expiration date but different exercise prices. If implied volatility is plotted for different strike prices, a U-shape may appear. The U shape is not always as perfect as the above figure.

To get a rough estimate of whether an option is U-shaped, look at an option chain that lists the implied volatility at each strike price. If the option is U-shaped, the ITM and OTM options should have roughly the same implied volatility. The further the strike price is away from at-the-money, the higher its implied volatility, while options near at-the-money have the lowest implied volatility. If this is not the case, then the option does not qualify as a volatility smile.

The implied volatility of an individual option can also be plotted as a time series plot as the price of the underlying asset moves. Implied volatility can take on some kind of U-shape as prices move inward or outward.

This is useful when looking for options with low implied volatility. In this case, options that are close to the money can be chosen. If you are looking for higher implied volatility, you should choose options that are more in the money or out of the money. However, keep in mind that as the underlying asset price moves closer or further away from the strike price, this will affect the implied volatility. Therefore, maintaining a portfolio of options at a specific implied volatility will require constant adjustment.

Not all options have a volatility smile. Before using Volatility Smiles to help with trading decisions, be sure to check whether the option's implied volatility actually fits this model.

The difference between volatility smile and volatility skew/smile

While short-term options and FX options tend to be more consistent with the volatility smile, index options and longer-term equity options tend to be more consistent with the volatility skew. The skew/smile indicates that in-the-money or out-of-the-money options may have higher implied volatility.

Limitations of using volatility smiles

First, it is important to determine whether the options you are trading are truly consistent with the volatility smile. The volatility smile is a model that some options may fit, but implied volatility may also fit more closely with an inverse or positive skew/smile.

Additionally, the volatility smile (if applicable) may not be a clean U-shape (or smile) due to other market factors (such as supply and demand). It may have a basic U-shape, but it may be quite uneven, and some options may have higher or lower implied volatility than the model expects.

The Volatility Smile highlights which direction traders should look when looking for higher or lower implied volatility, but there are many other factors to consider when making options trading decisions.

References

[1] Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein. “Explaining Asset Pricing Puzzles Associated with the 1987 Market Crash,” Page 1. Journal of Financial Economics, September 2011.

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