  There is a well developed framework, the Black-Scholes theory, for the
pricing of contracts based on the future prices of certain assets, called
options. This theory assumes that the probability distribution of the returns
of the underlying asset is a gaussian distribution. However, it is observed in
the market that this hypothesis is flawed, leading to the introduction of a
fudge factor, the so-called volatility smile. Therefore, it would be
interesting to explore extensions of the Black-Scholes theory to non-gaussian
distributions. In this contribution we provide an explicit formula for the
price of an option when the distributions of the returns of the underlying
asset is parametrized by an Edgeworth expansion, which allows for the
introduction of higher independent moments of the probability distribution,
namely skewness and kurtosis. We test our formula with options in the brazilian
and american markets, showing that the volatility smile can be reduced. We also
check whether our approach leads to more efficient hedging strategies of these
instruments.
