Fractional Option Pricing Model And Its Application In Evaluating The Value Of A Firm

Option pricing is a fundamental problem in option transactions. So far, most researches in option pricing, including Black-Scholes formula, have been conducted in the approach of the linear classic financial theories.However, the linear methodlogy of CMT has limitations inherently as they are invalid to capture complicated"patern"in stock price. So,a new research trend ,from the point of nonlinearity and evolution instead of in a linear view, emerges. This paper studies option pricing with assumptions that the stock price fluctuation follows a fractional geometric Brownian motion and the capital market has fractal characteristics. It is therefore of great importance in theory and practice.Firstly, this paper researches in the behavior of the stock price. It shows that traditional finance theory based on the assumptions of normal return distribution, random walk, and independence cannot accurately characterize the price behavior; while with the hypothesis of fractal capital market, non-normality, fractional Brownian motion, and the long-term memory of the financial time series, the behavior of the actual stock price can be characterized well. The actual financial time series follow a skewed random walk; have fractals and exhibit long-term memory. Based on the fractals of the capital market, this paper derives a fractional geometric Brownian motion model for the stock price. This paper constructs a formula for option pricing on the basis of the fractional Brownian motion model. On the one hand, this paper draws on the latest researches in this field: the explicit formula for European option pricing—the fractional Brownian motion formula for option pricing; on the other hand, the analytical solution is derived through the Monte Carlo simulation. Compared to the traditional Black-Scholes formula, the fractional Brownian motion formula for option pricing is not only affected by the expiration time, but also dependent upon the memory exponent H .Finally, this paper explores the non-independence,the non-linearity, and the fractal of a strategic investment of a firm, and applies the fractional Brownian motion formula for option pricing to evaluate the uncertainties of a strategic investment scientifically.