Title: Classes of elementary function solutions to the CEV model. I,
Abstract: The CEV model subsumes some of the previous option pricing models. An
important parameter in the model is the parameter b, the elasticity of
volatility. For b=0, b=-1/2, and b=-1 the CEV model reduces respectively to the
BSM model, the square-root model of Cox and Ross, and the Bachelier model. Both
in the case of the BSM model and in the case of the CEV model it has become
traditional to begin a discussion of option pricing by starting with the
vanilla European calls and puts. In the case of BSM model simpler solutions are
the log and power solutions. These contracts, despite the simplicity of their
mathematical description, are attracting increasing attention as a trading
instrument. Similar simple solutions have not been studied so far in a
systematic fashion for the CEV model. We use Kovacic's algorithm to derive, for
all half-integer values of b, all solutions "in quadratures" of the CEV
ordinary differential equation. These solutions give rise, by separation of
variables, to simple solutions to the CEV partial differential equation. In
particular, when b=...,-5/2,-2,-3/2,-1, 1, 3/2, 2, 5/2,..., we obtain four
classes of denumerably infinite elementary function solutions, when b=-1/2 and
b=1/2 we obtain two classes of denumerably infinite elementary function
solutions, whereas, when b=0 we find two elementary function solutions. In the
derived solutions we have also dispensed with the unnecessary assumption made
in the the BSM model asserting that the underlying asset pays no dividends
during the life of the option.