[Quantlib-dev] Projecting Forward Volatility Surfaces

[David B. <doc...@gm...>]

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