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Topic Author
Posts: 7
Joined: September 5th, 2010, 10:59 am

### Skew and Kurtosis in Vanilla Option

Given I am using a closed form formula (e.g. B.S.) for a vanilla call option; using implied vol. Where do the skew and kurtosis take parts of the pricing?Apologize if I am not asking it correctly; what I don't understand is that where do we use skew / kurtosis to price a vanilla option......do i use them to get the implied vol from the implied vol surface? Sorry, I did try to search thru topics in this forum, but i'm a bit confused where to start.....

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

### Skew and Kurtosis in Vanilla Option

a BSM model will not allow you to enter skew and kurtosis as separate inputs. The BSM model is a single factor model - that factor being the implied vol. The market sets the prices of options - this includes the markets current best attempt at esimating the prices of skew and kurt.
knowledge comes, wisdom lingers

frolloos
Posts: 1621
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Skew and Kurtosis in Vanilla Option

however take a look at Corrado & Su paper for modification of BS to take into account skew and kurtosis using Gram Charlier expansion.

Topic Author
Posts: 7
Joined: September 5th, 2010, 10:59 am

### Skew and Kurtosis in Vanilla Option

Thanks both.By the way, my understanding is that skew and kurtosis explain the vol-smile / smirk. Is there any paper out there which prove this relationship?

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

### Skew and Kurtosis in Vanilla Option

I dont think anyone can prove such a thing
knowledge comes, wisdom lingers

Alan
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Location: California
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### Skew and Kurtosis in Vanilla Option

QuoteOriginally posted by: jimslimshadyThanks both.By the way, my understanding is that skew and kurtosis explain the vol-smile / smirk. Is there any paper out there which prove this relationship?Well, the Breeden-Litzenberger paper/formula is a proof that if the smile is flat, then the pdf of log S(T) is normal, and thus haszero skewness and excess kurtosis. But skewness & kurtosis are simply statistical measures, so perhaps 'explain' is too strong -- maybe "are associated with" would be better.
Last edited by Alan on December 21st, 2010, 11:00 pm, edited 1 time in total.

Fermion
Posts: 4486
Joined: November 14th, 2002, 8:50 pm

### Skew and Kurtosis in Vanilla Option

QuoteOriginally posted by: jimslimshadyGiven I am using a closed form formula (e.g. B.S.) for a vanilla call option; using implied vol. Where do the skew and kurtosis take parts of the pricing?Apologize if I am not asking it correctly; what I don't understand is that where do we use skew / kurtosis to price a vanilla option......do i use them to get the implied vol from the implied vol surface? Sorry, I did try to search thru topics in this forum, but i'm a bit confused where to start.....First you need to specify what quantity you are referring to that has skew or kurtosis. In finance the most common application of these terms is to the density of returns. However, the term "skew" is also used to describe an asymmetry in implied volatilities. There is no specific connection between these two uses of the term skew, except in the very general sense that any geometric distribution will potentially explain both. (Implied volatility skew tends to disappear when it is plotted against log moneyness.) Kurtosis in returns and the implied volatility smile are also related, but the relationship is less well understood because its exact nature depends on the distribution of returns.A normal distribution is the limit of a pure binomial distribution (pure meaning constant volatility or branching ratios) and has kurtosis = 3. Any binomial model that has node volatilities that increase with distance from the expectation path (i.e. a smiling instantaneous or local volatility) will give "excess" kurtosis (i.e. > 3) in the limit and a smiling instantaneous volatility will typically generate a smiling implied volatility whenever implied volatility can be approximated by some sort of average over paths that end up near the strike.Explicitly modeling skew and kurtosis is a rather artificial way to try to go about valuation. It is easier (and equivalent) to simply model either the SDE or the transition density. However, if you insist on using a skew/kurtosis model, you can use it to generate the transition density explicitly in terms of moments and then do your pricing from that. (There is a paper by Rubinstein which you can find on the internet which essentially does just that.)
Last edited by Fermion on December 21st, 2010, 11:00 pm, edited 1 time in total.

prodiptag
Posts: 124
Joined: September 12th, 2008, 4:41 pm

### Skew and Kurtosis in Vanilla Option

QuoteOriginally posted by: AlanQuoteOriginally posted by: jimslimshadyThanks both.By the way, my understanding is that skew and kurtosis explain the vol-smile / smirk. Is there any paper out there which prove this relationship?Well, the Breeden-Litzenberger paper/formula is a proof that if the smile is flat, then the pdf of log S(T) is normal, and thus haszero skewness and excess kurtosis. But skewness & kurtosis are simply statistical measures, so perhaps 'explain' is too strong -- maybe "are associated with" would be better.Breeden and Litzenberger were followed by Backus, Foresi and Wu, their paper (Accounting for Biases in Black-Scholes, 1997) gives an approximation of implied volatility smile in terms of risk nuetral vol, skewness and kurtosis. Also from the market quotes for collars, strangles around atmfs, it is possible to extract some approximate values for the skewness and kurtosis of the risk neutral distribution. P. Carr had a paper on this for the FX world ("The Information Content of Straddles, Risk Reversals and Butterfly Spreads , 2005). But I think modelling the other way (i.e. assume skewness and kurtosis and then determine your smile) might not make much sense, primarily becuase most of the market players are naturally much more comfortable with smile and not with the underlying RND and their statistical measures, without getting in to the theoritical aspects of such modelling approach

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