Option Pricing Models Based on Levy Process and Entropy, and Applications

研究了不完全资产市场中的期权定价问题。在以往的研究中，我们构建了以几何Levy过程为基础资产价格过程，以最小熵鞅测度为等价鞅测度的[GLP & MEMM]定价模型的基本框架。因此，本文首先研究了该模型的基本性质，建立了将该模型应用于实际期权定价问题的方法，然后通过计算机模拟的方法研究了该模型的性质，并得出了该模型具有实现波动微笑的可能性的结论。其次，对期权的市场价格进行了实证分析，得到了在标的资产过程非常快速的情况下，该模型非常适合于市场上的期权价格。特别是我们可以看到，[几何稳定过程和MEMM]模型是非常可取的，因为这个模型有可能适合市场上非常广泛的期权价格数据。

We have studied the option pricing problems in the incomplete asset market. In the previous research we have constructed the basic frameworks of the [GLP ＆ MEMM] pricing models, in which the geometric Levy processes are adopted as the underlying asset price processes and the MEMM (=minimal entropy martingale measure) is adopted as the equivalent martingale measure. So in this research we have investigated the fundamental properties of this model and have established the method how to apply this model to the practical option pricing problems.After that we have investigated the properties of this model by the computer simulation methods, and we have obtained a good result that this model has the possibility to realize the volatility smile/skew properties of the option prices in the markets.Next we have made empirical analysis on the market prices of the options, and we have obtained such a result that this model fits very well to the option prices in the market in the case that the underlying asset process moves very rapidly. Especially we could have seen that the [Geometric Stable Process ＆ MEMM] model is very desirable in the sense that this model has the possibility to fit to very wide class of the option price data in the market.