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.