On a Generalized Squared Gaussian Diffusion Model for Option Valuation

In financial mathematics, option pricing models are vital tools whose usefulness cannot be overemphasized. Modern approaches and modelling of financial derivatives are therefore required in option pricing and valuation settings. In this paper, we derive via the application of Ito lemma, a pricing model referred to as Generalized Squared Gaussian Diffusion Model (GSGDM) for option pricing and valuation. Same approach can be considered via Stratonovich stochastic dynamics. We also show that the classical Black-Scholes, and the square root constant elasticity of variance models are special cases of the GSGDM. In addition, general solution of the GSGDM is obtained using modified variational iterative method (MVIM).


Introduction
In modern finance, the importance of options in pricing theory cannot be overemphasized as they can be used for control risk and hedging. This therefore requires the attention of financial analysts when dealing with finance, actuarial sciences, and other related areas of applied sciences [1][2][3]. Hence, the involvement of stock options in the study of option pricing and valuation theory [4]. Black and Scholes [5,6]  Black and Scholes based their model on some assumptions such as arbitrage-free opportunities, no allowance of dividend-yield paying stock, lack of transaction costs, constant mean rate and constant volatility parameter among others [7]. To be addressed here among the assumptions is that of constant volatility, that is the stock price standard deviation. The constant nature of the volatility as assumed by Black and Scholes was to make the model a linear type in order to obtain analytical solution easily. Contrary to this, constant volatility appears unrealistic in practical settings. We therefore intend to resort to a more accommodating model driven by constant elasticity of variance (CEV).

Basic Definitions, Theorem(s) and Existing Model
The assumption of constant volatility as in (1.1) has drawn the attention of many researchers; see [8][9][10] and the references therein for more details. Delbean and Shirakawa [11] proved that the CEV model permits arbitrage opportunities when the stock price is based on strict positive conditions. Cox and Ross [12] considered the CEV diffusion process governed by the SDE: whose solution is t S , [ represents an elasticity rate, while , , and t W P V remain as earlier defined.
Beckners [13] considered the CEV and its implications for option pricing on the basis of empirical studies and concluded that the CEV class could be a better descriptor of the actual stock price in terms of behaviour than the traditionally used lognormal model. MacBeth and Merville [14] proposed a three-stage-procedure on how to estimate and In what follows, the Black-Scholes model will be modified using the SDE associated with the CEV model in (2.1) to get the the GSGDM as a proposed model.

Note:
In this work, we take all stochastic processes to be square Gaussian (SG).
Equation (2.5) is called an Ito lemma. [17,18] Considering the general nonlinear PDE of the form:

VIM and the Modified VIM
, , , , , , R is a linear operator whose partial derivatives are w.r.t. x , , Nu x t is a nonlinear term associated to (3.1) and , f x t is a source term (which may be homogeneous or inhomogeneous), thus by the classical VIM, the solution of (3.1) is expressed as: Remark: Using (3.4) for the solution of special kind of nonlinear differential equations involves the calculation of unrequired terms, repeated calculations, and timeconsumption, hence, the need for meaningful modification of the VIM. The modified VIM as proposed by [17] gives the iterative formula as follows:

Derivation of the Generalized Model (GSGDM)
Suppose the stock price, t S at time t , satisfies the SDE in (2.1), with all parameters as earlier defined, and that the value of the contingent claim , therefore by Ito lemma, we have: Thus, for: we therefore write (3.7) as: be a delta-hedged-portfolio by longing a contingent, and shorting a delta unit of the underlying asset such that: ) ) w (3.10) in order to make the value of the portfolio riskless, where r is a riskless rate, say bank account.

Comparison of the models
In this subsection, we shall compare the basic features of the two models presented above. (3.14)

Remark 3.2:
It is obvious that is not a constant function but a function of the underlying asset price t S .

3.3: Special Cases of the GSGDM
In this subsection, we present the B-S and the square root models as special cases of the GSGDM.

3.3a: B-S Model and GSGDM
In (