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

Abstract

Abstract. 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. IntroductionThe 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 a...