Financial time series typically exhibit strong fluctuations that cannot be described by a Gaussian distribution. Recent empirical studies of stock market indices examined whether the distribution of returns can be described by a (truncated) Levy-stable distribution with some index 0<α≤2. While the Levy distribution cannot be expressed in a closed form. In a study [ P. Gopikrishnan et al., Phys. Rev. E 60, 5305 (1999)] it was found that the tails of P(r) exhibit a power-law decay, with an exponent α ~ 3, thus deviating from the Levy distribution. Ofer Biham et al. (2001) studied the distribution of returns in a generic model that describes the dynamics of stock market indices. For the distributions P(r) generated by this model, we observe that the scaling of the central peak is consistent with a Levy distribution while the tails exhibit a power-law distribution with an exponent α>2, namely, beyond the range of Levy-stable distributions.
The results are in agreement with both empirical studies and reconcile the apparent disagreement between their results. 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 2 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) (F. Dai, 2001) 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 µ 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 another hand, there is no accurate analytic formula for American Puts option pricing now. The American put value must be at least its exercise value since it can be exercised immediately. Geske and Johnson (1984), however, obtained a solution as an n-fold compound option, using Geske’s (1979) compound option formula and the equivalent martingale/risk neutrality assumption. Bunch, D. S. and H. Johnson (1992) given a simple and numerically efficient valuation method for American puts option. Y.Tian (1993, 1999) given the binomial option pricing models. Up to now lots of numerical approximation procedures were proposed for pricing American put options. Because of various difficulties in calculating the price of American put options, however, intensive efforts are still needed for developing new accurate formula to this problem. Here, we shall present an accurate analytic formula for American Puts option pricing based on partial distribution.
Prof. Feng Dai
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