Understanding Volatility Smile and Skew in Financial Markets

In this blog post, we will delve into the concepts of volatility smile and volatility skew, which are crucial for candidates preparing for the FRM Part 2 exam. These concepts help in understanding implied volatility and its implications in the options market.

Recap of Implied Volatility

Implied volatility is a key input in the Black-Scholes model, which is widely used for valuing European call options. The model requires six inputs, five of which can be directly observed or reliably forecasted. However, the implied volatility input is unique because it is not directly observable. Instead, it is adjusted to ensure that the model price matches the market price of the option.

Definition of Implied Volatility

Implied volatility is defined as the volatility input that equates the model price of an option to its market price. Here are four important aspects to remember about implied volatility:

1. Future Uncertainty: Unlike historical volatility, which measures past price movements, implied volatility reflects future uncertainty regarding the underlying asset.
2. Model Dependency: The value of implied volatility can vary depending on the model used. Different models may yield different implied volatility values for the same option.
3. Option Dependency: Implied volatility varies based on the specific option from which it is derived. Different options with varying strikes and expiration dates will have different implied volatilities.
4. Market Behavior: In an ideal market without transaction costs, the implied volatility from call options should equal that from put options, given they have the same strike and expiration, due to put-call parity.

What is Volatility Smile?

Volatility smile refers to the pattern of implied volatility as it relates to the strike price of options for a given expiration date. It illustrates how implied volatility changes with different strike prices. To visualize this, we can plot implied volatility against strike prices:

• At-the-Money (ATM): The point where the strike price equals the spot price.
• Out-of-the-Money (OTM) Puts/In-the-Money (ITM) Calls: Strikes less than the spot price.
• Out-of-the-Money (OTM) Calls/In-the-Money (ITM) Puts: Strikes greater than the spot price.

Patterns of Volatility Smile

In the market, two primary patterns are observed:

1. Volatility Smile: Implied volatilities for OTM puts and OTM calls are higher than for ATM options. As one moves away from the ATM point, implied volatility increases.
2. Downward Sloping Skew: Implied volatilities for OTM puts are higher than ATM options, while OTM calls have lower implied volatilities. This indicates a decrease in implied volatility as one moves from OTM puts to OTM calls.

Understanding the Downward Sloping Skew

The downward sloping skew is particularly relevant in equity options markets. Institutional investors often hedge against downside risks by purchasing OTM put options. This heightened demand for OTM puts drives up their prices, leading to increased implied volatilities. Conversely, OTM calls may see lower demand, resulting in decreased implied volatilities.

Market Sentiment and Implied Volatility

If market sentiment is bullish, demand for OTM calls may rise, leading to increased implied volatilities for those options, resulting in a smile pattern. Conversely, if the market sentiment is bearish, the downward sloping skew will prevail.

Reassessing Probabilities of Extreme Moves

Understanding implied volatility is essential for assessing the probabilities of extreme price movements in the underlying asset. The Black-Scholes model assumes a log-normal distribution for the asset price at a future time. However, market-derived implied volatility can suggest different probability distributions.

Comparing Probability Distributions

1. Downward Sloping Skew: This pattern indicates a probability distribution with a fat left tail and a slim right tail, suggesting a higher probability of significant downward moves compared to the log-normal distribution.
2. Volatility Smile: This pattern indicates a probability distribution with both fat left and right tails, suggesting a more balanced view of extreme price movements in either direction.

Implications for Risk Assessment

When comparing the two distributions:

• The probability of a significant downward move is greater according to the market's implied distribution than the Black-Scholes model.
• Conversely, the probability of a significant upward move is lower in the market's implied distribution compared to the Black-Scholes model.

Conclusion

In summary, understanding volatility smile and skew is crucial for analyzing market behavior and assessing risk in options trading. These concepts provide insights into market sentiment and the probabilities of extreme price movements, which are essential for effective risk management in financial markets.