Classes of elementary function solutions to the CEV model I

Abstract

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 β, the elasticity of volatility. The CEV model subsumes some of the previous option pricing models. For β = 0, β = −1/2, and β = −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. The pricing formulas for these financial instruments give concrete information about the pricing of options only after the employment of some intermediate approximation scheme. 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. Furthermore we do not allow ourselves to be drawn into any rash generalizations or inferences from the vanilla European case by prematurely focusing on those cases and we obtain concrete information for the pricing of options without needing to introduce any intermediate approximation schemes. 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 β, all solutions "in quadratures" of the CEV ordinary differential equation. These solutions give rise, by separation of variables, 1 emelas@econ.uoa.gr to simple solutions to the CEV partial differential equation. In particular, when β = ..., − 5 2 , −2, − 3 2 , −1, 1, 3 2 , 2, 5 2 , ..., we obtain four classes of of denumerably infinite elementary function solutions, when β = − 1 2 and β = 1 2 we obtain two classes of of denumerably infinite elementary function solutions, whereas, when β = 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.

Elementary functions solutions to the Bachelier model generated by Lie point symmetries

“…In this paper we find the Lie point symmetries of the Bachelier PDE and we use these symmetries in order to generate new classes of solutions from the solutions we derived in [15]. We anticipate that both the solutions we found in [15] and the solutions we find in this paper describe various types of new interesting financial instruments. This paper is organised as follows: In Section 2 we give the bare essentials of the Bachelier model and we derive the Bachelier PDE.…”

“…As pointed out in [15], since the Bachelier PDE is linear an immediate cosequence of Theorem 1 is the following…”

“…Our new classes of elementary function solutions to the Bachelier model have been derived with a two−stage Research Program: First, in [15], we obtained Liouvillian solutions to the Bachelier model by studying its differential Galois Group. Then, in this paper, we use the Lie point symmetries of the Bachelier model in order to generate new solutions from the solutions which were derived at the first stage of the Program.…”

“…In [15] by studying the differential Galois group of Bachelier ODE we obtained four classes of denumerably infinite elementary function solutions to the Bachelier model which are expressed via products of polynomials and exponential functions. In this paper we find the Lie point symmetries of the Bachelier PDE and we use these symmetries in order to generate new classes of solutions from the solutions we derived in [15]. We anticipate that both the solutions we found in [15] and the solutions we find in this paper describe various types of new interesting financial instruments.…”

“…In Section 3 we find the Lie point symmetries of the Bachelier PDE. In Section 4 we revise the elementary functions solutions to the Bachelier model we found in [15]. In Section 5 we generate new classes of elementary function solutions to the Bachelier model.…”