【作者】 李英华；

【导师】 李兴斯；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2013， 博士

【摘要】 具有投资、避险、操作灵活多样等特性的期权已成为金融衍生品的核心组成部分,是以对其研究也持续不断。由于期权定价涉及很多内在随机因素,外在创新变更内容,使得期权定价这一课题一直是学者们高度关注的研究领域。本文主要运用了信息熵及其优化原理和树图方法相结合,对期权的几个核心问题,如：构建期权定价模型、分析期权的套期保值参数(Greeks)、计算奇异期权定价和不完全市场期权定价等方面展开研究,具体研究内容及相关成果详见如下：1.构建最大熵树期权定价模型把金融市场看作一个信息系统,已知的标的资产的不完全信息作为约束条件,基于最大熵原理及凝聚函数法光滑收益函数,来构建最大熵树期权定价模型。数值算例印证了最大熵树模型的有效性。具体结论如下：(1)熵和矩信息密切相关,能在不完全信息下计算出无偏、意义明确的树图参数。(2)凝聚函数光滑收益函数,使模型的收敛阶为O(1/n)。2.考虑历史信息的期权定价模型若已知标的资产的历史信息,通过最小叉熵原理与控制敲定价在最后一层节点的位置相结合,得到光滑的最小叉熵树期权定价模型。具体结论如下：(1)该模型能更直观的体现最小叉熵树与历史信息的关联,避免了树图的收敛波动问题。(2)光滑收敛的模型有利于精度插值运算。3.套期保值参数(Greeks)分析Greeks从侧面反映了影响期权风险的因素,及时调整套期保值策略规避风险。具体结论如下：(1)计算了最大熵树模型的Delta收敛到B-S模型的Delta的收敛阶。(2)最大熵树期权定价模型的避险灵敏度不低于B-S模型的避险灵敏度。4.定价奇异期权的拆分熵树模型本章转化算术平均亚式期权为两资产期权,然后,运用熵树模型对两资产期权定价,通过数值算例分析,新方法计算简便准确。具体结论如下：(1)通过最大熵原理能得到无偏、合理的树图参数。(2)可以避免算术平均亚式期权关于界限函数的参数选取。(3)把算术平均亚式期权和两资产期权建立联系。5.效用最大化的不完全市场多叉树期权定价通过多叉树与效用函数最大化相结合所得的套期保值策略带来了求解不完全市场期权定价的新思路,对比算例表明新方法可行、有效。具体结论如下：(1)在计算期权标的资产财富增长速度的过程中,通过增值熵推导了效用函数；(2)由多叉树和效用函数的结合得到了多目标求解不完全市场的期权定价新模型。更多还原

【Abstract】 Option has become the center of financial derivatives due to its various functions and therefore studied extensively. Option pricing involves the internal stochastic factors and external innovations, which make it become a hot research topic in recent years.This thesis combines tree option price model with the entropy and relevant optimization principles and whereby analyzes several important problems of option, such as constructing the entropy tree models for option price, analyzing the Greeks of option, pricing Asian option as well as option price under the incomplete market. The main investigations and achievements are listed as follows:1. Constructing the entropy tree option price modelConsidering the financial market as the information system, the entropy tree option price model is obtained based on the maximum entropy principle and the aggregate function smoothing the payoff function of the option under the underlying assets information constrains. The essence for measuring uncertainty of the entropy is employed to derive the probability as such the entropy is introduced to the binomial tree option price model. The numerical examples are calculated to verify the effectiveness of the entropy tree model. The conclusions are in the following:(1) Entropy and moments are closely related, so it can compute the binomial tree unbiased, impartial parameters under the incomplete information.(2) To use the aggregate function to smooth the payoff function, the rate of the maximum entropy tree option price model is O (1/n).2. The option price mode considering the underlying assets history informationGiven the underlying assets history information, the smooth minimum cross-entropy tree option price model is obtained based on the minimum cross-entropy principle and controlling the position of the strike in the last layer nodes. The conclusions are as follows:(1) The smooth minimum cross-entropy tree directly embodies the function of the underlying assets history information and avoids the convergent oscillations of CRR model.(2) It will benefit the accuracy interpolation in the smooth minimum cross-entropy tree.3. Analyzing the Greeks of the entropy tree modelGreeks of the entropy tree model reflect the factors which influence the option price risk, so that it can promptly adjust the investment strategy and observe the sensitivity of the entropy tree. The conclusions are in the following:(1) It computes the Delta of the entropy tree, and proves that the Delta of the entropy tree converges to the Delta of the B-S model.(2) Sensitivity for avoiding risk of the entropy tree is not lower than that of the B-S model.4. The separation entropy tree for the exotic option priceThis thesis transforms the arithmetic average Asian options to two assets option, and applies the entropy tree to compute the two assets option. The numerical examples show the new model is accurate and easy to compute. The conclusions are in the following:(1) The parameters of the entropy tree are unbiased, impartial for the two assets option.(2) It can avoid the choice of path function parameters.(3) It builds the relation between the arithmetic average Asian options and the two assets option.5. m-nomial tree option price model based on utility function in the incomplete marketThe hedging with m-nomial tree and maximum utility function derives the option price in the incomplete market. The numerical examples show the new method is feasible and effective. The conclusions are in the following:(1) During analyzing the maximum growth rate of wealth, the utility function is obtained by the generalized entropy.(2) Combining m-nomial tree with maximum utility function, the multi-object option price model is derived in the incomplete market.更多还原