欧式篮子期权定价研究综述和数值分析

发布时间：2018-01-07 05:28

本文关键词：欧式篮子期权定价研究综述和数值分析　出处：《清华大学》2014年硕士论文　论文类型：学位论文

【摘要】：由于交易市场的多样性、客户需求的差异性、金融衍生品市场的不断完善和金融理论研究的发展，金融机构会设计各式各样的新型期权。篮子期权是新型期权中的一种，它是多资产期权，通常在多种外国货币的交易中使用，利用它来套期保值。多个标的资产价格的加权平均决定篮子期权的到期收益。通常，由投资组合理论可知，一篮子标的资产的波动率相对比较小，这就导致篮子期权的价格要比单个标的资产期权价格的总和小，在费用上效率更高。已有许多学者基于不同的市场假设，，建立不同模型，选用不同方法，对篮子期权定价进行了探索和研究。基于篮子期权定价的理论意义和现实意义，本文将对篮子期权的定价模型、定价方法以及定价公式进行综述分析，以便在实际交易中更好地进行应用。一方面，在Black-Scholes模型下，对欧式篮子期权定价研究进行综述分析。对于几何平均篮子期权，基于几何平均篮子期权定价可以转化为一维问题，通过引进组合自变量直接求解多资产Black-Scholes方程，得到几何平均篮子期权定价公式；对于算术平均篮子期权，对国内外相关研究文献进行整理，列出五种不同的解析近似定价公式，并就部分公式作一些理论证明推导。另一方面，对Black-Scholes模型的假设进行放松，综述分析不同市场假设下的欧式篮子期权定价。在标的资产价格变化模式放松的基础上，基于不同的解析近似法，分别给出跳跃扩散模型和分数布朗运动下的欧式篮子期权定价公式；在对常数波动率假设放松的基础上，整理出Heston随机波动率模型下两个资产的篮子期权的近似定价公式；在对无违约风险假设放松的基础上，给出有违约风险的几何平均篮子定价模型和公式；在对无摩擦市场假设放松的基础上，利用无风险对冲原理推导出支付交易费用的篮子期权定价模型。本文还对Black-Scholes模型下的欧式看涨篮子期权定价进行数值模拟。基于大数定理，利用蒙特卡罗方法(Monte Carlo method)进行大量模拟，估计欧式算术平均看涨篮子期权价格，同时采用减小方差技术中的控制变量法来提高模拟效率，以及用低偏差Halton和Faure序列来缩减取样范围，即拟蒙特卡罗法(Quasi-MonteCarlo method)进行模拟。
[Abstract]:Because of the diversity of trading market, the difference of customer demand, the continuous improvement of financial derivatives market and the development of financial theory research. Financial institutions design a variety of new options. Basket options are one of the new types of options. They are multi-asset options and are usually used in transactions involving a variety of foreign currencies. It is used to hedge. The weighted average of multiple underlying asset prices determines the maturity return of basket options. Usually, from portfolio theory, the volatility of a basket of underlying assets is relatively small. As a result, the price of basket options is smaller than the sum of individual underlying asset options, and the cost efficiency is higher. Many scholars have established different models and different methods based on different market assumptions. Based on the theoretical and practical significance of basket option pricing, this paper will summarize and analyze the pricing model, pricing methods and pricing formulas of basket options. In order to be better used in actual transactions. On the one hand, under the Black-Scholes model, the research on European basket option pricing is reviewed and analyzed. Based on the geometric mean basket option pricing can be transformed into a one-dimensional problem, the geometric average basket option pricing formula is obtained by introducing portfolio independent variables to solve the multi-asset Black-Scholes equation directly. For arithmetic average basket option, the relevant literatures at home and abroad are sorted out, five kinds of analytical approximate pricing formulas are listed, and some formulas are proved theoretically. On the other hand, the hypothesis of Black-Scholes model is relaxed, and the European basket option pricing under different market assumptions is summarized and analyzed. Based on different analytical approximations, the jump diffusion model and the pricing formula of European basket options under fractional Brownian motion are given respectively. On the basis of relaxing the assumption of constant volatility, the approximate pricing formula of basket options for two assets under Heston stochastic volatility model is put forward. On the basis of loosening the assumption of non-default risk, the geometric average basket pricing model and formula with default risk are given. On the basis of loosening the assumption of frictionless market, a basket option pricing model for payment of transaction costs is derived by using the risk-free hedging principle. In this paper, the pricing of European call basket options under Black-Scholes model is numerically simulated, based on the theorem of large numbers. Monte Carlo method is used to carry out a large number of simulations to estimate the price of European arithmetic average call basket option. At the same time, the control variable method in the variance reduction technique is used to improve the simulation efficiency, and the low deviation Halton and Faure sequences are used to reduce the sampling range. Quasi-Monte Carlo method is used to simulate.
【学位授予单位】：清华大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：F830.9;F224

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