Introduction

From the point of view of theoretical studies of world economics and finance, option pricing is a kernel and important problem to which economists pay due attention. In practice, exact option pricing is very important to monetary derivative markets and world economic market and plays an important part in the stabilization of the markets. In the studies of option pricing, there have been many significant results (Black and Scholes 1973, Merton 1976, Sharpe 1978, Whaley 1981, Gesk and Roll 1984), and approximation methods for American put option (MacMillan 1986, Stapleton and Subrahmanyam 1997).

But, Up to now, no exact analytic formula has ever been produced for the value of an American put option on a non-dividend-paying stock (Hull 2000). The Black-Scholes models and its extensions assume that the probability distribution of the stock price at any given future time is lognormal. This assumption is not perfect (Hull 2000). Here, we will try to present the new pricing formulas for options based on another probability distribution—the Partial Distribution and DF process (F. Dai 2000).

This paper proves, by empirical research, that the Partial Distribution can preferably describe the structure properties of prices of stocks and stock indices. Based on the Partial Distribution and Partial Process, we present here, for the first time, the concepts and expressions of DF process and DF structure, and put forward a new kind of analytic method of pricing options on a non-dividend-paying underlying. The DF structure models are able to price the call and put options exercised at any time, so it is applicable to pricing the American and European options, especially the American put options.

The option’s underlying assets discussed in this paper are the stock or stock indices on a non-dividend- paying. Here, we also offer the method of estimating the parameters in partial distribution. Finally, examples are given to compare the options priced by DF formulas and by Black-Scholes formulas. The examples show, as a whole, that the DF’ prices of options are closer to the trading prices of the United States options market than Black-Scholes’ prices.

Prof. Feng Dai, Prof. Zifu Qin