Vol. Skew (Smirk) - Question

[emilioalzamora1]

Hi All,

perhaps quite a simple question. I just read through the following CFA article

http://www.cfapubs.org/doi/full/10.2469/dig.v41.n1.2

and now I am wondering whether Vol. Skew and Vol. Smirk can be used 1:1 interchangeably? Is there any difference between these two terms?

Vol Skew = Vol Smirk?

E. Derman in his book 'Volatility Smile' does not even mention the vol smirk.

I have also consulted the following book by L. Shover 'Trading Options in turbulent markets' (by the way, it's a good background reading for those interested), but the definition (skew = smirk) is not 100% clear.

http://www.wiley.com/WileyCDA/WileyTitle/productCd-1118343549.html

Thank you!

 

[emilioalzamora1]

I got an answer to this myself in Jorion's handbook where he says:

For equity index options, the effect is more asymmetrical, with very high ISDs for low strike prices. Because of the negative slope, this is called a volatility skew. A skewed smile is sometimes called a smirk. In other words, out-of-the-money (OTM) put options are priced with a higher ISD than at-the-money (ATM) or even in-the-money (ITM) put options. Before the stock market crash of October 1987, this effect was minor. Since then, it has become more pronounced.

What remains is the question: do both equity options AND equity index options (e.g. S&P) show a volatility skew? Or is just equity options which have a skew pattern?

 

[emilioalzamora1]

I would like to quote J. James book 'FX Option Performance' here - she mentions the following:

'We have already suggested that very sharp downward moves are more likely than very sharp upward moves in the index. The shape of the volatility skew for equity index options simply confirms that option markets are priced to reflect this; you must pay more for downside protection than for upside participation.'

According to J. James we would argue that both equity index options and plain equity options would have a volatility skew pattern.

What's surprising and in contrast (or in more detail to J. Hull's explanation) is the fact that FX options can have a skew as well

'For a currency pair that includes a ‘hard’ currency such as the US dollar and a ‘soft’ currency (e.g. an emerging market currency) safe-haven flows are most likely to mean the selling of the latter and increased demand for the former. Thus, in cases where a currency pair contains a clear safe-haven currency we may expect the shape of the volatility smile to be skewed so that it resembles an equity index implied volatility ‘smirk’, with volatility higher for strike prices that involve a strong depreciation of the more risky currency. Depending on how we quote the exchange rate, the smirk may slope in the opposite direction; we should expect the exchange rate to rise in times of crisis if the US dollar is specified as the base currency'

David, can we please have your opinion on this?

 

[David Harper CFA FRM]

Hi @emilioalzamora1 Very interesting! In terms of semantics, I can just tell you being anchored in John Hull, I tend to connote implied volatility skew as downward sloping, however in his glossary he defines "Volatility Skew: A term used to describe the volatility smile when it is nonsymmetrical." So, for example, when it comes to a practice question or any rigorous statement, I think "skew" or "smirk" simply implies (at a minimum) non-symmetrical, and in practice requires further clarification. (a "frown" would be upside down, on the other hand). So, personally, I don't assume skew/smirk means upward- or downward ....

However, clearly we expect the skew to be downward sloping in the case of equity options because we presume out of the money (OTM) puts, which have lower strike prices, tend to have higher implied volatilities than in-the-money puts. A downward sloping equity smile (aka, skew) is consistent--is implied by--a theory of demand for insurance instruments. I hope that's helpful, thank you for sharing some key links!

 

[sebseb5557]

Hi David,

Refering to your study note problem 20.3 : What volatility smile is likely to be caused by jumps in the underlying asset price?
You answers : "Jumps tend to make both tails of the stock price distribution heavier than those of the lognormal distribution."
However in the above part " Explain the impact of asset price jumps on volatility smiles" it is explained that anticipated large jumps generate volatility frown which is the opposite of volatility smile.
I'm a bit confused. Is this due to the difference between "anticipated" and "resulting from" jumps ?

Many thanks

Seb

 

[emilioalzamora1]

I am quoting @David Harper CFA FRM here (post back in 2012) even if I would like to know more about this myself:

Hi sassing, good point. The study note should be revised because it is trying to capture (summarize) two separate points by Hull (it is unclear which the AIMs means to refer to) :

• Section 19.8 (When a Single Large Jump is Anticipated): the single large jump is illustrated with "true distribution is bimodal." Based on a binomial pricing model, Hull then shows that "a single large jump [i.e., $50 either up to 58 or down to 42]" produces a bimodal distribution and the prices imply an implied volatility frown or "inverted smile" (lower implied vol left/right, higher implied vol in the middle)
• The this question, from Hull, appears to make a different point about multiple "jumps" versus a single binomial, where the jumps create fat tails. However, I don't see it explicitly sourced in the reading. (except for the reference that jumps imply a mixture-of-lognormals, which itself would imply heavy tails, as mixtures tend to imply heavy tails). But it's a little unclear how the question above is explicitly sourced. I hope that helps,

Tagged for:

1. Submit to GARP for discrepancy; i.e., do jump(s) imply smile or frown? if both, insufficient differentiation
2. Subsequent revision to note

Updated by Nicole to add link where quoted text was retrieved: https://forum.bionicturtle.com/threads/hull-19-03.6391/#post-26868

 

[emilioalzamora1]

E. Derman has written a book called 'Volatility Smile' last year. I will check and see whether there is something valuable in there with regard to your question.

 

[emilioalzamora1]

Even if this is not the final solution to the question, but Derman writes:

'Why are we interested in jump models? Because we observe jumps in reality. Most security prices don’t just diffuse smoothly as time passes; their movements are punctuated by jumps. Stocks and indexes definitely jump. Currencies sometimes jump. Commodity prices jump, too. What separates a jump from normal diffusion? There is no precise, universally accepted definition of a jump, but it usually comes down to magnitude, duration, and frequency. A jump is a large return that happens over a very short time period. By “a very short time period”we almost always
mean intraday, and by “large” we mean a move that is large compared to sigma*Sqrt(t), the expected standard deviation over that time period. Really large jumps happen rarely in equity index markets (the frequency is usually of the order of one jump per several years), but when they do happen they have important economic, financial, and especially psychological effects. In equity
markets, indexes mostly suffer negative jumps, while individual stocks tend to undergo both positive and negative jumps. As an explanation of the volatility smile, jumps are attractive because they provide an easy way to produce the persistently steep short-term negative skew that we observe in equity index markets. In fact, this persistent skew first appeared soon after the jump/crash of 1987.
Unfortunately (from a theoretical point of view), jumps are inconsistent with arbitrage-free risk-neutral pricing, the bedrock of all the modeling we’ve done up to this point. The inconsistency stems from our inability to instantaneously hedge an option whose underlier can undergo many different jumps of different sizes. The alternative to risk-neutral pricing—economic models that depend on an individual’s subjective risk tolerance— are unattractive in that they demand detailed behavioral modeling. To avoid this, most jump-diffusion models simply assume risk-neutral pricing without convincing justification. Though they may be difficult to model, there have been and will be jumps in asset prices. Even if we can’t fully hedge them, we still need to understand how jumps impact option prices and the volatility smile.'