In response to some of the preceding messages regarding performance tests
for option pricing models, I would like to make some comments about
- how option pricing models are used by the market
- wby the magnitude of the "pricing error".
However, this procedure ignores how option pricing models such as the
Black-Scholes models, are used by market practitioners. Indeed, the
"Black-Scholes
option pricing formula" is a misnomer: the Black-scholes
formula for the price of a European call is never used as above to
calculate the price
of a call option from historical volatilities! It is rather used the other
way round,
to obtain Black-Scholes implied volatilities from market prices of call/
put
options which are fixed, asable for evaluating a derivatives position
but it is not used directly by the market for pricing options.
The emphasis on using implied volatilities obtained from liquid options as
in the case of the Black-scholes model
the delta hedging strategy reduces the random payoff of a call to a
(non-random)
payoff of zero : F becomes achastics, 1999) or in general to minimize some
other function U
("expected utility" or objective function) of the final payoff. In fact
the preceding cases are particular examples of utility functions:
U(x)=x^2 for the
minimal variance hedging and U(x)= step function for quantile hedging.
So, in order to compare two option pricing/hedging models, one can:
- calibrate the model parameters to empirical options prices
- apply the hedging strategy proposed by the models
- examine the distribution of the payoff of the hedged positions
- use a criterion to compare the two distributions in terms of their risk:
which
one has higher variance? which one has higher quantiles? which one has
heavier tails? etc.
In particular, two distributions F1 and F2 are not comparable in a
non-ambiguous way so
different criteria of comparison will give different "performance
assessments" for option pricing models.
Cordially,
Rama Cont
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Rama CONT
Centre de Mathematiques Appliquees - CNRS UMR 7641
Ecole Polytechnique
F-91128 Palaiseau, France.
Fax: 00 33 1 69 33 30 11.
E-mail: [log in to unmask]
WWW: http://www.cmap.polytechnique.fr/~rama/
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