Dear all,
Thank you for the interesting comments by Klaus Pinn, Jessica James and
Giulia Iori.
Here are some answers. A more detailed discussion can be found in
our book `Theory of Financial Risk', which has just been completed and
will be available free from www.science-finance.fr the 25th of May,
1999.
1) About hedging (K. Pinn's comment). The risk minimisation procedure
of Bouchaud-Sornette leads to the optimal strategy in terms of the
variance. The general result is that the optimal hedge is NOT the
Black-Scholes Delta hedge. BUT, since one is at a quadratic minimum,
this means that a small error in the hedge (say delta phi), will lead
to a SECOND ORDER increase in the risk, in delta phi^2.
Numerically, delta phi is indeed found to be rather small, leading in
many cases to an insignificant increase of the residual risk, as
found by K. Pinn. However:
a) This depends on the parameters, and for certain highly kurtic
markets,
or for short maturities, the difference can be appreciable.
b) The dependence of the optimal strategy phi^* with the price of the
underlying is systematically smaller for our optimal strategy than for
the Delta hedge. This means that rehedging is less frequent or involves
less assets, and this leads to a non negligible decrease of the
transaction
costs.
2) About J. James comment. We strongly disagree with the common view
that:
`` the historical distribution is irrelevant when it comes to option
pricing. Only the implied distribution (that which can be derived from
option prices in the market) is relevant ''
which is the dogma of modern finance. This is simply not true. The
historical
distribution is relevant, and actually in many cases very close to the
implied
distribution, provided one has a good model for it. The Black Scholes
like
models are so lousy that indeed they look very different from the
implieds.
Of course, on some special occasions, the implieds have an extra piece
of information not contained in the historical. But from our experience
as a company doing applied research on smile generators, we feel that
these
are exceptional situations, and that the historical distribution is the
correct
benchmark.
The view that the historical distribution is irrelevant is an
unfortunate
consequence of the Black-Scholes and the CRR models, which are very
peculiar. A fuller discussion of this subtle but important point can be
looked at in our book.
In summary: even if the hedging performances of our optimal strategy are
not significantly better than the one used by the market, we feel that
our framework provides a much more pedagogical and transparent starting
point for explaining options and their associated risks, and also for
devising better option pricing models using historical data in an
adequate
way.
Also our formalisation allows one to compute the residual risk
associated with option writing, a quantity that does not even exist in
the Black and Scholes world.
Best to all,
Jean-Philippe Bouchaud and Marc Potters
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