The famous Black-Scholes (BS) model used in the option pricing theory
contains two parameters - a volatility and an interest rate. Both
parameters should be determined before the price evaluation procedure
starts. Usually one use the historical data to guess the value of these
parameters. For short lifetime options the interest rate can be estimated
in proper way, but the volatility estimation is, as well in this case,
more demanding. It turns out that the volatility should be considered
as a function of the asset prices and time to make the valuation self
consistent. One of the approaches to this problem is the method of
uncertain volatility and the static hedging. In this case the envelopes
for the maximal and minimal estimated option price will be introduced.
The envelopes will be described by the Black - Scholes - Barenblatt
(BSB) equations. The existence of the upper and lower bounds for the
option price makes it possible to develop the worse and the best cases
scenario for the given portfolio. These estimations will be financially
relevant if the upper and lower envelopes lie relatively narrow to each
other. One of the ideas to converge envelopes to an unknown solution
is the possibility to introduce an optimal static hedged portfolio.
Högskolan i Halmstad/Sektionen för Informationsvetenskap, Data- och Elektroteknik (IDE) , 2007.