Salvatore Micciche et al. (2002) investigated the historical volatility of the 100 most capitalized stocks traded in US equity markets. An empirical probability density function (pdf) of volatility is obtained and compared with the theoretical predictions of a lognormal model and of the Hull and White model. The lognormal model well describes the pdf in the region of low values of volatility whereas the Hull and White model better approximates the empirical pdf for large values of volatility.
Both models fail in describing the empirical pdf over a moderately large volatility range. Since 90’s of 20th century, the U.S. stock market soar first and slump later, and the fluctuation is violent, and the volatility is large. At the same time, we can use the Levy model. But, this does not mean that the U.S. stock market or other market will be in the violent fluctuation forever. Even if the stock market is generally in the violent fluctuation, the stock price is also in the low values of volatility at some periods of time.
In fact, we need to use the most proper model to analyze the probability distribution of stocks price in case the stock price behavior is in the region of low values of volatility. In this case, people accustom to the use the lognormal model. However, when a company collapses, the price of its stock will be the zero. The lognormal model can’t describe the possibility of zero price of a stock. The partial distribution P( µ, σ2) can do this. So the partial distribution should be applied to describe the price distribution of commodities and stocks at the low values of volatility. When value of ยต is lower, partial distribution have a sharper peak than lognormal distribution. In addition, Levy distribution and the truncated Levy distribution is usually applied to describe the price behavior in symmetry. Because of the price is non-negative, the distribution of price is generally non-symmetry.
The non-symmetry can be reflected in lognormal or partial distribution. In this paper, the models of options pricing is not studied in the current mode. In fact, the new models of option pricing are presented based on partial process and DF structure -DF structure models.
Because the partial process can nicely describe the properties of construction of stock prices, the DF structure method of option pricing is practical, reasonable and authentic. This can be best explained from the results in the table 6-1 and table 6-2, which are calculated by DF structure formulas and are closer to the actual trading price of options on the whole. As for the DF structure models of option pricing, we still have the following elucidations:
● With the time, the distribution of stock prices will continuously change, and the DF structure models can be vary with the price distribution of underlying stock at any time, so it may price option more accurately than B-S models.
● The DF structure models can estimate the prices of stock options at the expiration date or any time before the expiration date. So the models can be used for pricing European option, and American option as well, particularly for pricing American put option. Now the exact formula for pricing American put option has not been obtained.
● By means of the optimal pricing method [20], we shall get the model of calculating optimal execute price, and know the optimal opportunity for any options.
● This paper has only discussed the pricing of the options on stock and stock index. If the underlying, such as spots, futures, and foreign exchange, follow the partial distribution, and have no dividends, the textual conclusion still holds.
The textual data source is in the websites:
http://finance.yahoo.com
http://www.stockstar.com.cn
We are very grateful to Professor Weixuan Xu of Chinese Academy of Sciences for paying attention to our researches.
Prof. Feng Dai, Prof. Zifu Qin
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Summary: Index