Stochastic Analysis of the Effect of Asset Prices to a Single Economic Investor

In this paper, we propose a single economic investor whose asset follows a geometric Brownian motion process. Our objective therefore is to obtain the fair price and the present market value of the asset with an infinitely horizon expected discounted investment output. We apply dynamic programming principle to derive the Hamilton Jacobi Bellman (HJB)-equation associated with the problem which is found to be equivalent to the famous Black-Scholes Model under no risk neutrality. In addition, for a complete market under equilibrium, we obtained the value of the present asset with risk neutrality and its fair price. Introduction The study that Asset Prices follow a simple diffusion process was first proposed in the historical work of Bachelier in 1900. His work is rather remarkable in that it addressed the problem of option pricing and his main aim was to actually provide a fair price for the European call option [3]. Bachelier assumed stock price dynamics with a Brownian motion without drift (resulting in a normal distribution for the stock prices), and no time-value of money. The formula provided may be used to valuate a European style call option [10]. Later on, [4] obtained the same results as Bachelier. As pointed out by [5] and [9], this approach allows negative realizations for both stock and option prices. Moreover, the option price may exceed the price of its underlying asset. According to [10], Black – Scholes Model is often regarded as either the end or the beginning of the option valuation history. Using two different approaches for the valuation of European style options, they present a general equilibrium solution that is a function of “observable” variables only, making therefore the model subject to direct empirical tests. The stock price dynamics is described by a geometric Brownian motion with drift as a more refined market model for which prices cannot be negative, the volatility can be viewed as the diffusion coefficient of this random walk [7]. The manifest characteristic of the final valuation formula is the parameters it does not depend on. The option price does not depend on the expected return rate of the stock or the risk preferences of the investors. It is not assumed that the investors agree on the expected return rate of the stock. It is expected that investors may have quite different estimates for current and future returns. However, the option price depends on the risk-free interest rate and on the variance of the return rate of the stock. To make simplier normally distributed financial asset returns, on assumption that its distribution is difficult, [11] concludes that for a distribution model to reproduce the properties of empirical evidence it must possess certain parameters as: a location parameter; a scale parameter; an asymmetry parameter and a parameter describing the decay of the distribution since financial returns distribution is difficult to determine since normally distributed financial asset returns is not supported by empirical evidence. We suppose the movement of an asset price is a stochastic process, thus the price paths of stocks and indices will all be modeled using this idea. The stochastic process followed by the underlying asset is considered in a continuous time. Our concern therefore is to construct the stochastic control problem to emerge from solving the infinitely horizon expected discounted investment output. While existence and closed form solution to the stochastic control problem Bulletin of Mathematical Sciences and Applications Submitted: 2016-03-11 ISSN: 2278-9634, Vol. 17, pp 33-39 Revised: 2016-07-18 doi:10.18052/www.scipress.com/BMSA.17.33 Accepted: 2016-08-26 © 2016 SciPress Ltd., Switzerland Online: 2016-11-01 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ follows from well-known theory, it is also found to be equivalent to the famous Black-Scholes Model under no risk neutrality, though the market is complete. Interestingly, our proposed method have been successfully used by many authors, see for examples [8, 13]. 1. Problem formulation We reformulate the works of [6] and [14] for a stock market with the following properties. Uncertainty is represented by a complete filtered probability space   P f f t ), ( , ,  and throughout the paper, we denote by   0  t t f the neutral filtration i.e.   t s s W f t    0 ); (  where (.)) (W is a standard 1-dimensional Brownian motion defined on this space with value in . R Consider an investor whose underlying risky asset (e.g. stock) follows the setup below. Let   t S be the unit price for the risky asset assume to follow a geometric Brownian motion process. Given the above assumptions, the dynamics of the risky asset is: 0 ) 0 ( ), ( ) ( ) (    S t dW dt t S t dS   (1.1) such that the solution of equation (1.1) in ito’s sense is given by: 0 , ) ( 2 1 exp ) 0 ( ) ( 2                  t t W t S t S    (1.1a) where  , are the drift and volatility parameters assume to be positive constants and ) (t W is a standard 1-dimensional Brownian motion [12]. Fair price of an asset for an infinite investment is considered in this paper. The Objective function as formulated in [6] of the investor is the infinite horizon expected discounted investment output given by:           0 , ) ( ), ( ), ( ), ( ) ( ), ( ), ( ), ( 0        t dt t k t S C t t k t S u t k t S P e E Sup t t k t S V t rt


Introduction
The study that Asset Prices follow a simple diffusion process was first proposed in the historical work of Bachelier in 1900. His work is rather remarkable in that it addressed the problem of option pricing and his main aim was to actually provide a fair price for the European call option [3]. Bachelier assumed stock price dynamics with a Brownian motion without drift (resulting in a normal distribution for the stock prices), and no time-value of money. The formula provided may be used to valuate a European style call option [10]. Later on, [4] obtained the same results as Bachelier. As pointed out by [5] and [9], this approach allows negative realizations for both stock and option prices. Moreover, the option price may exceed the price of its underlying asset.
According to [10], Black -Scholes Model is often regarded as either the end or the beginning of the option valuation history. Using two different approaches for the valuation of European style options, they present a general equilibrium solution that is a function of "observable" variables only, making therefore the model subject to direct empirical tests. The stock price dynamics is described by a geometric Brownian motion with drift as a more refined market model for which prices cannot be negative, the volatility can be viewed as the diffusion coefficient of this random walk [7]. The manifest characteristic of the final valuation formula is the parameters it does not depend on. The option price does not depend on the expected return rate of the stock or the risk preferences of the investors. It is not assumed that the investors agree on the expected return rate of the stock. It is expected that investors may have quite different estimates for current and future returns. However, the option price depends on the risk-free interest rate and on the variance of the return rate of the stock.
To make simplier normally distributed financial asset returns, on assumption that its distribution is difficult, [11] concludes that for a distribution model to reproduce the properties of empirical evidence it must possess certain parameters as: a location parameter; a scale parameter; an asymmetry parameter and a parameter describing the decay of the distribution since financial returns distribution is difficult to determine since normally distributed financial asset returns is not supported by empirical evidence.
We suppose the movement of an asset price is a stochastic process, thus the price paths of stocks and indices will all be modeled using this idea. The stochastic process followed by the underlying asset is considered in a continuous time. Our concern therefore is to construct the stochastic control problem to emerge from solving the infinitely horizon expected discounted investment output. While existence and closed form solution to the stochastic control problem follows from well-known theory, it is also found to be equivalent to the famous Black-Scholes Model under no risk neutrality, though the market is complete. Interestingly, our proposed method have been successfully used by many authors, see for examples [8,13].

Problem formulation
We reformulate the works of [6] and [14] for a stock market with the following properties.
Uncertainty is represented by a complete filtered probability space   and throughout the paper, we denote by   0 R Consider an investor whose underlying risky asset (e.g. stock) follows the setup below. Let   t S be the unit price for the risky asset assume to follow a geometric Brownian motion process. Given the above assumptions, the dynamics of the risky asset is: such that the solution of equation (1.1) in ito's sense is given by: where  , are the drift and volatility parameters assume to be positive constants and ) (t W is a standard 1-dimensional Brownian motion [12].
Fair price of an asset for an infinite investment is considered in this paper. The Objective function as formulated in [6] of the investor is the infinite horizon expected discounted investment output given by: given by is the capital and the stochastic control parameter to be controlled by the is the production rate and   is the cost of expansion in the form of liabilities and r is the discount rate term.
Using the dynamic programming principle, the Hamilton-Jacobi-Bellman (HJB) -equation corresponding to (1.2) is given by: Suppose that the value of the claim at time t is given by where(1.12) is the famous B-S Model.

Solution of the asset model and its unique price with no risk neutrality
The time-homogenous form of (1.6) is equivalent to (1.16) such that

Bulletin of Mathematical Sciences and Applications Vol. 17 35
Applying [2], the rate of change of investment output is with respect to S , then we have (2.2) rewritten as: By (2.3), equation (2.1) therefore reduces to ordinary differential equation of the form: be a complete filtered probability space and let ) (S V be the investment output of a single economic investor and let S be the stock price. Suppose that With the correlation coefficient given as  and the characteristics and particular solutions are therefore given as: Remark: We considered the problem on the assumption that the market is an incomplete market.
Therefore the drift parameter regarded as  is not equal to the discounted rate parameter regarded as r so that the fair price Sˆdoes not tend to zero.

Unique asset price under risk neutrality
In equation (1.12), the drift term  disappears. This follows from the idea of risk-neutral pricing by hedging with no possibility of arbitrage [1]. It will be shown that the drift on an underlying asset must be considered risk-neutral regardless of the pricing method used. Therefore, to fairly price asset, consider By inspection, equation (3.5) has been reduced to a parabolic type of equation with boundary conditions having eliminated the interest rate term. Hence, by separation of variables method let the solution be given as: