Projecting Forward Volatility Surfaces

[David Brown-27]

Greetings All-

 

   Given a portfolio of american options on one stock, let's suppose that we wanted to figure out what this portfolio was worth in an arbitrary number of days, suppose 30.  One could take the stock, let's suppose that we could project the pdf of the price forward 30 days perfectly with no loss of fat tails.  (This is another problem in itself.)  No we have this accurate pdf, let's say composed of 50000 MCMC simulations.  Every one of these values could be plugged into the options we hold and now and semi-accurate values could be obtained using whatever we priced the option with in the first place, baroni-adesi-whaley / whatever.  The only matter now is that we don't have the slightest idea of what the volatility would be in this arbitrary number of days.

   How do we begin to solve this problem?  My approach given my own thoughts and a number of papers, including Peter Carr and Jim Gatheral, is to find the market implied volatility surface at time zero and figure out using historical data what the surface should approximately look like in the future.  This is way easier said than done.  Dupire has a slide-show that discusses the different types of local volatility and stochastic volatility surfaces, but his book on volatity only has a short chapter, number 8, called Dynamics of the Skew / Smile or something to that effect.  He doesn't really seem to answer the question as to what must be done to solve this problem accurately.

   So I've raised this question here, because I think this is the only place where there is a possibility of this problem being solved given the expertise of the parties involved in similar projects.  Peter Carr addressed this a little bit with his FMLS model which uses stable distributions to fit the market smile, but the model has yet to be built for stochastic volatility.  I contacted Johnathon Nolan on the matter and he told me a good source to try would be Hugh McCullough's papers.  I tried his papers and it appears that he hasn't fully solved the matter either.  Peter Carr himself said that some type of PDE could be used to solve for the speed of the volatility surface, but from what I gather, this was his form of conjecturing.  Perhaps not?

   Any advice on how to proceed with this problem?  I am highly interested in programming this into Quantlib.

 

 

Best Wishes,

David Brown

 

 

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[Joseph Wang-2]

One way of doing this is the Carr-Mandan technique.  If you have a set of
liquid vanilla european option prices at a particular time, you can use a
damped Fast Fourier Transform to get the volatility surface for a particular
time and a given set of parameter values.  You then minimize least squares
over the parameter values.

The problem is that this only works if you have a set of liquid European
option.  The major piece of infrastructure that isn't in QuantLib is the
ability to do fast fourier transforms plus maybe some C++ data structures to
handle the data.

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[Luigi Ballabio]

Hi David,
        I'm afraid I can't help with the problem. However...

On Wed, 2007-08-29 at 22:03 -0400, David Brown wrote:
>    So I've raised this question here, because I think this is the only
> place where there is a possibility of this problem being solved given
> the expertise of the parties involved in similar projects.

...you might try the Wilmott forums as well.

Later,
        Luigi


--

Green's Law of Debate:
Anything is possible if you don't know what you're talking about.



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