The Theory Of Q-distributions And Its Application In Option Pricing

With the rapid development of modern information technology and the continuous improvement of the degree of globalization, the emergence of new types of derivative financial products has been overwhelming. Option, as an important member of derivative financial products, is a commonly used financial means hedged by investors. In real financial markets, the yield of the underlying assets doesn’t follow the Gaussian distribution but follow a distribution which has the characteristics of fat tail and leptokurtic. The famous standard B-S option pricing model is the most important and widely used pricing model, which is based on the assumption that the yield of underlying assets follows the Gaussian distribution. It will no doubt bring the deviation of the actual price and result in a volatility smile. Today, many scholars have studied the modified option pricing model, such as the stochastic volatility model, truncated Levy leap model, alpha stable distribution model, etc. I lowever, these methods are more complex and we can not obtain a closed form solution. The root cause of the deviation of standard B-S model is that the assumption which states the yield of underlying assets follows a Gaussian distribution. Then we expect to revise this hypothesis and deduce a generalized option pricing model. The q-Gaussian distribution can satisfy this expectation because it can describe the leptokurtic and fat tailed characteristics better.The current studies on the theories of q-distributions are not comprehensive, so this paper firstly further improve the statistical properties in order to improve its theoretical basis, which is deducing the statistical digital characteristics(expectation. variance and k-order moment) and the parameter estimations(moment estimation. maximum likelihood estimation and the q-maximum likelihood estimation) of some common used q-distributions. Then we introduce the theories of standard B-S model, the generalized B-S option pricing model based on the q-Gaussian distribution and the discrete option pricing models. Finally, we apply the data of actual financial index and the prices of underlying assets to conduct an empirical analysis. In these empirical examples, we first estimate the parameters of distributions and compare these estimation methods according to the mean square error method. Then for these option pricing models, we also compare their final consequences. The results show that the maximum likelihood estimation method compared with other two estimation methods is superior, so we choose this method to estimate the distribution parameters. According to the comparison results of such four option pricing models, they show that the generalized B-S model based on the q-Gaussian distribution is more accurate than standard B-S model and the other two discrete models also overestimate the option prices. Therefore, on the actual option pricing, the generalized B-S model based on the q-Gaussian distribution is more feasible and effective.