  The Black-Scholes model anticipates rather well the observed prices for
options in the case of a strike price that is not too far from the current
price of the underlying asset. Some useful extensions can be obtained by an
adequate modification of the coefficients in the Black-Scholes equation. We
investigate from a mathematical point of view an extension directly related to
the quantum harmonic oscillator. In the considered case, the solution is the
sum of a series involving the Hermite-Gauss functions. A finite-dimensional
version is obtained by using a finite oscillator and the Harper functions. This
simplified model keeps the essential characteristics of the continuous one and
uses finite sums instead of series and integrals.
