Hi
I am a little confused with the recent discussion so please allow me
some thoughts on distinct issues.
First:
> Is it possible to use a risk neutral valuation in
> association
> with a non-Gaussian underlying process ?
There are many-many processes, like general Markov processes, binomial
trees like Cox-Ross-Rubinstein, etc., which are not Gaussian and
certainly there is a martingale measure.
> If the underlying is not Gaussian
> (better to say quasi-Brownian) then basicly the market is not complete
> and the measure is not unique. In this situation....
I am sorry but I do not quite agree with that, for the reasons I stated
previously.
Second point to Marco:
> An implied volatility surface is used to to price exotic options
> consistently with the market expectations represented by a
> set of known european options values.
> Since the implied volatility surface is usually not flat, it is
> used to represent deviation from Gaussian behavior of the underlying,
> while preserving risk-neutral valuation.
My understanding, shared also among various academics, is that the use
of implied volatilities is not a consistent method for pricing options.
Independently of whether the process is Gaussian or not (see comment
above) there is NO REASON why it is risk-neutral. It is in fact easy to
see why. The volatility surface changes with time (you estimate it today
but then tomorrow the market can change it). In other words, by using
implied volatilities one has no idea of the risk exposed, and no idea of
how to construct a replicating portfolio. Previsibility is lost. There
might be a martingale measure, but simply we do not know it.
Please correct me if I am wrong.
I know that many practisioners use implied volatilities, but at the
moment there is no mathematical justification for it and it is a very
difficult theoretical problem. My colleagues here told me that there are
many experts are looking on the issue at the moment.
Best regards
Vassilis
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Vassilis Koulovassilopoulos
I.N.R.I.A. Rocquencourt
Project META-2: MATHEMATICAL FINANCE,
Batiment 12
Domaine de Voluceau-Rocquencourt
B.P. 105
78153 LeChesnay Cedex
FRANCE
http://gogol2.inria.fr/~vk/
Tel: +33 1 39 63 51 01 (work)
+33 6 83 97 04 16 (mobile phone)
Fax: +33 1 39 63 57 86
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