In R40-P2-T5 Hull - Problem 20.3 on volatility smile & Jumps in asset price I'm confused since when I read the part about Jumps in the asset price, it made me think of the learning objective below and a volatility frown.

However the answer to the problem is a normal (happy looking?) volatility smile.

Are they referring to 2 different kinds of jumps?

- Vol frown = A single expected up or down jump but don't know which way it will go so at the money options are most volatile
- Vol smile = more random, larger & frequent jumps which make the distribution tails heavier.

Problem 20.3

Problem 20.3

What volatility smile is likely to be caused by jumps in the underlying asset price? Is the pattern

likely to be more pronounced for a two-year option than for a three-month option?

Answer:

Jumps tend to make both tails of the stock price distribution heavier than those of the lognormal

distribution. This creates a volatility smile similar to that in Figure 20.1. The volatility smile is

likely to be more pronounced for the three-month option.

Figure 20.1

**Learning objective I was thinking of:**

__Explain the impact of asset price jumps on volatility smiles.__

This concerns a situation where a large jump—either up or down—is anticipated. In such a case, the actual distribution is not lognormal but rather bimodal; i.e., it has two camel-like humps. Under such a distribution, an at-the-money option has a higher volatility than both an out-of-the-money option and an in-the-money option.

The implication of this inverted situation is an upside-down volatility smile that peaks in the middle, where the strike price equals the stock price; this is also known as a “volatility frown.”

Thanks

Karim