Classes of elementary function solutions to the CEV model I

摘要

In the equity markets the stock price volatility increases as the stock price declines. The classical Black-Scholes-Merton (BSM) option pricing model does not reconcile with this association. Cox introduced the constant elasticity of variance (CEV) model in 1975, in order to capture this inverse relationship between the stock price and its volatility. An important parameter in the model is the parameter 0, the elasticity of volatility. The CEV model subsumes some of the previous option pricing models. For beta = 0, beta = -1/2, and beta = -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. However, there are simpler solutions to both models than those pertaining to the standard calls and puts. Mathematically, it makes Sense to investigate the simpler cases first. In the case of BSM model simpler solutions are the log and power solutions. Similar simple solutions have not been studied so far for the CEV model. We use a group-theoretic method, Kovacic's algorithm, which has not been used before to problems of Finance or Economics and obtain new classes of elementary function solutions to the CEV model for all half-integer values of beta. In particular, when beta = ... , -5/2, -2, -3/2, -1, 1, 3/2, 2, 5/2, ... , we obtain four new classes of denumerably infinite elementary function solutions, when beta = -1/2 and beta = 1/2 we obtain two new classes of denumerably infinite elementary function solutions, whereas, when beta = 0 we find two elementary function solutions. (C) 2019 Elsevier B.V. All rights reserved.