Serving the Quantitative Finance Community

• 1
• 2

gjbiren
Topic Author
Posts: 10
Joined: December 31st, 2010, 6:29 am

### Deriving pdf of underlying from option prices

I am an Econ PhD doing an internship and a project I have been given is to derive the underlying pdf of a futures contract based upon option prices. For the sake of simplicity, we are ignoring time considerations, insomuch as we are only looking at one contract with one expiration, i.e. I do not need to construct an entire volatility surface and then derive the corresponding pdf's for the the different expirations. I have read, and partially understood, Jim Gatheral's book "The Volatility Surface" and have read numerous papers on the topic of volatility and local volatility, so my knowledge is somewhat better than beginner, but I will make no claims of complete comprehension or understanding.Therefore, my question is twofold:1) What would be the easiest way to approximate the pdf of the underlying from option prices (should I use a local volatility model such as Dupire or Derman first then try to calculate the moments of the volatility etc?)2) What is typically the optimal way to perform such a task?My goal is first to achieve any type of results so I would prefer focus on the easiest way first. After having accomplished that, I would like to look at more techniques that would theoretically provide better goodness of fit or just be more robust in general.Any direction or input would be greatly appreciated.Glenn

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

### Deriving pdf of underlying from option prices

hi, i think what you might be looking for is deriving the pdf using Breeden-Litzenberger's formula (so google Breeden & Litzenberger). basically this is deriving the pdf from tight butterfly spreads.

gjbiren
Topic Author
Posts: 10
Joined: December 31st, 2010, 6:29 am

### Deriving pdf of underlying from option prices

Thanks for the response.I actually have an old paper from Risk magazine in 1993 by David Shimko that uses the Breeden and Litzenberger methodology. It uses the second derivative of the option price wrt strike price but that is a rough analog of butterfly pricing. Due to the age of the paper I was wondering if there were any new methodologies that were being used, but if it is the standard, then that is the way I will go.

pimpel
Posts: 363
Joined: May 12th, 2006, 5:26 pm
Location: Warsaw

### Deriving pdf of underlying from option prices

QuoteOriginally posted by: frollooshi, i think what you might be looking for is deriving the pdf using Breeden-Litzenberger's formula (so google Breeden & Litzenberger). basically this is deriving the pdf from tight butterfly spreads.That is the right thing to do, though you need to have a good vol interpolation/extrapolation formula. What is the market you are working on? Try fitting SVI model and from it using BL formula determine numerical derivatives for deriving pdf.

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

### Deriving pdf of underlying from option prices

QuoteOriginally posted by: pimpelQuoteOriginally posted by: frollooshi, i think what you might be looking for is deriving the pdf using Breeden-Litzenberger's formula (so google Breeden & Litzenberger). basically this is deriving the pdf from tight butterfly spreads.That is the right thing to do, though you need to have a good vol interpolation/extrapolation formula. Try fitting SVI model and from it using BL formula determine numerical derivatives for deriving pdf.Absolutely. As Pimpel writes the tricky part is the inter- and extrapolation without introducing arbitrage. SVI is good, although I am not entirely sure SVI is always arbitrage free?

pimpel
Posts: 363
Joined: May 12th, 2006, 5:26 pm
Location: Warsaw

### Deriving pdf of underlying from option prices

QuoteOriginally posted by: frolloosQuoteOriginally posted by: pimpelQuoteOriginally posted by: frollooshi, i think what you might be looking for is deriving the pdf using Breeden-Litzenberger's formula (so google Breeden & Litzenberger). basically this is deriving the pdf from tight butterfly spreads.That is the right thing to do, though you need to have a good vol interpolation/extrapolation formula. Try fitting SVI model and from it using BL formula determine numerical derivatives for deriving pdf.Absolutely. As Pimpel writes the tricky part is the inter- and extrapolation without introducing arbitrage. SVI is good, although I am not entirely sure SVI is always arbitrage free?There is a number of restrictions on parameters available, which should prevent arbitrage. I wouldn't go for any spline algos, as this for sure on illiquid markets will imply bumpy or even negative densities. The other approach would be SABR, but this in some cases can lead to negative densities in tails (at least while using the standard formula from original paper).

MCarreira
Posts: 1724
Joined: July 14th, 2002, 3:00 am

### Deriving pdf of underlying from option prices

Agree with Pimpel and frolloos, the Malz paper is also worth reading.Start with a model you can understand first (but avoiding arbitrage).

gjbiren
Topic Author
Posts: 10
Joined: December 31st, 2010, 6:29 am

### Deriving pdf of underlying from option prices

Gatheral in The Volatility Surface says that he uses splines with SVI. I am assuming that was because he was looking at the SPX which is obviously very liquid. You correctly assumed that my market is not. I am not nearly at that point yet, but will I be able to use SVI without using splines?

Alan
Posts: 10656
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Deriving pdf of underlying from option prices

Yes, you just fit the 5 parameters to a given implied vol. smile. No splines are used/needed. (Gatheral was talking about somethingyou're not doing -- fitting the whole implied vol surface).However, you need to use some judgments in constraining the parameters. In my experience, it took about 15 minutes to code up my first SVI fit, but more than a month of fitting various underlyings to get theconstraints reasonably 'tweaked'. Also, remember that you need *European-style* option prices, so if they aren't, you need to estimate what they would be.
Last edited by Alan on November 7th, 2011, 11:00 pm, edited 1 time in total.

pimpel
Posts: 363
Joined: May 12th, 2006, 5:26 pm
Location: Warsaw

### Deriving pdf of underlying from option prices

QuoteOriginally posted by: AlanIn my experience, it took about 15 minutes to code up my first SVI fit, but more than a month of fitting various underlyings to get theconstraints reasonably 'tweaked'. Also, remember that you need *European-style* option prices, so if they aren't, you need to estimate what they would be.Have you published those anywhere, or is it your proprietary product (the constraints)? I guess you are talking about smiles for commodities, where quoted options are american style? Smile fit very well on those markets to SVI as I remember.

Alan
Posts: 10656
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Deriving pdf of underlying from option prices

Re constraints: will have some discussion in my upcoming (no date yet set) "Option Valuation under Stochastic Volatility II".Re exercise-style: actually, I was thinking about equity options, but the issue is, of course, general.

Gmike2000
Posts: 801
Joined: September 25th, 2003, 9:49 pm

### Deriving pdf of underlying from option prices

why infer the pdf from option prices? what are you going to do with it? by smoothing the implied vol just to get nice 2nd derivatives you lose information.good example of not seeing the forest for the trees, in my view

Gmike2000
Posts: 801
Joined: September 25th, 2003, 9:49 pm

### Deriving pdf of underlying from option prices

..and also, the tails of the distribution, what about them. there are many fancy methods to create this missing information. so you loose information in the middle, by smoothing, and create information in the tails. also, some options are marked at minimum tick prices that overstate implied vol because these things cannot be marked at zero. i think i have posted these comments like 100 times over the past 10 years...this is like groundhog day.

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

### Deriving pdf of underlying from option prices

i think the OP is just trying to learn stuff, not trading yet.but i am curious, what is the way to go in your opinion Gmike?

gjbiren
Topic Author
Posts: 10
Joined: December 31st, 2010, 6:29 am

### Deriving pdf of underlying from option prices

I wanted to clarify that I am actually a PhD student and that this is a project that I have been assigned at an internship. With those caveats in place, I again want to thank all of you for your input and ask for some further elaboration:Regarding splines, esp. @ Alan and pimpel, couldn't I just look at adjusting the tangents of a cubic Hermite spline to preserve monotonicity and therefore reduce if not eliminate arbitrage concerns? I am thinking of strict monotonicity but this is just a intuitive guess rather than one based upon any practical experience.@Gmike2000 I share many of your concerns. Any smoothing or interpolation for missing strikes will be problematic. Especially as I am then going to use a parametric model, e.g. SVI, to determine that implied distribution. It seems like I will be basically smoothing a smoothing, hence my goodness of fit remark. I am looking at in terms of entropy (kind of Shannon Entropy and the entropy rate for a stochastic process. This assumes a stationary process but I am already making huge assumptions so why not one more?) I am unsure of the ultimate use of this research but I also would share your concerns about trading/making markets from it. I would suppose it could be used for marking to market existing positions etc. but I would not want to go much further, especially in light of the relative illiquidity of the underlying.