带有残差风险与交易成本的简单欧式期权定价

发布时间：2018-05-25 09:00

本文选题：期权定价 + 隐含波动率微笑现象　；参考：《华南理工大学》2013年硕士论文

【摘要】：近年来，随着我国金融市场的不断完善，衍生产品在我国的应用也有了极大的发展。在这种形势下，如何对期权合理定价显得尤为重要。在1973年，F.Black和M.S.Scholes在期权定价的研究上做出了开创性的贡献，得出了著名的Black-Scholes期权定价公式。然而，由于Black-Scholes公式是在一系列理想化假设下得出的，实际应用其定价期权时就会产生系统性的定价偏差。因而，得出与实际市场环境相一致的期权定价公式对金融风险的控制与管理显得尤为重要，其研究具有重要的理论价值与现实意义。特别需要指出的是，存在交易成本时，Black-Scholes公式中的连续时间交易假设不再成立。在现实世界，交易只能在离散时间进行。当连续时间期权定价公式用于离散时间场合时，就会产生定价误差。这是由于此时无风险证券及股票构成的投资组合在对冲期权风险时会产生规避误差（残差风险）。研究规避误差对期权定价的影响具有重要的理论与实际价值，这对准确定价期权、规避管理金融风险、解释隐含波动率的微笑现象具有重要的意义。假设股价S满足S=S_t=S_oe~(μt+σB) t∈[0,T],（1）其中S0, μ,σ为常数； μ为“期望回报率”，σ为波动率(常数)，(Bt)为标准Brown运动。在离散时间场合，在无交易成本假设下，本文首先考虑残差风险对基于上述股票价格模型（1）的欧式期权定价的影响。我们提出了期权定价的平均自融资极小方差规避规则，得到了带有残差风险的期权定价规避策略X1(t)（或称之为广义Delta规避策略）及含有残差风险的欧式看涨期权价格满足的偏微分方程和相应的期权定价公式。由上述期权定价公式，我们发现残差风险对期权定价有很大的影响，在实际期权定价时，，应予以考虑。其次，本文将带残差风险的Delta规避策略X1(t)用于Leland期权定价模型，得到了新的存在成比例交易费时的欧式看涨期权价格满足的偏微分方程及其相应的期权定价公式，该公式显示交易成本及残差风险在实际中对期权定价均有影响。最后，我们在不考虑交易成本的情况下，对带残差风险的欧式期权定价模型进行了数值分析，发现残差风险及交易费可以较好地解释隐含波动率的微笑现象。我们认为没有考虑残差风险可能是Black-Scholes公式中隐含波动率发生微笑现象的另一原因。
[Abstract]:In recent years, with the continuous improvement of China's financial market, the application of derivative products in China has been greatly developed. In this situation, how to reasonable pricing options is particularly important. In 1973, F. Black and M.S.Scholes made groundbreaking contributions to the study of option pricing, and obtained the famous Black-Scholes option pricing formula. However, since the Black-Scholes formula is derived under a series of idealized assumptions, systematic pricing deviations will occur when the pricing options are applied in practice. Therefore, it is very important to obtain the option pricing formula which is consistent with the actual market environment for the control and management of financial risk, and its research has important theoretical value and practical significance. It is especially important to point out that the hypothesis of continuous time trading in Black-Scholes formula is no longer true when transaction costs exist. In the real world, transactions can only take place at discrete times. When continuous time option pricing formula is used in discrete time situation, pricing error will occur. This is because the portfolio composed of risk-free securities and stocks will produce a hedge error (residual risk) when hedging the option risk. It is of great theoretical and practical value to study the influence of evading error on option pricing, which is of great significance to accurately price options, avoid managing financial risks and explain the smile phenomenon of implied volatility. Assuming that the stock price S satisfies the S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / S / Where S0, 渭, 蟽 are constants, 渭 is "expected rate of return", and 蟽 is standard Brown motion. In discrete time, under the assumption of no transaction cost, this paper first considers the influence of residual risk on the pricing of European options based on the above stock price model. We propose an average self-financing minimum variance avoidance rule for option pricing. In this paper, an option pricing evading strategy with residual risk is obtained, and the partial differential equation and the corresponding option pricing formula for the European call option with residual risk are obtained. From the above option pricing formula, we find that residual risk has a great influence on option pricing, which should be considered in the actual option pricing. Secondly, this paper applies the Delta evading strategy X1t with residual risk to the Leland option pricing model, and obtains a new partial differential equation and its corresponding option pricing formula for the price satisfaction of European call options with proportional transaction time. The formula shows that transaction cost and residual risk have influence on option pricing in practice. Finally, we analyze the European option pricing model with residual risk without considering transaction cost, and find that residual risk and transaction cost can explain the smile phenomenon of implied volatility. We think that the failure to consider the residual risk may be another reason for the smile phenomenon of implied volatility in the Black-Scholes formula.
【学位授予单位】：华南理工大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F830.91;F224

【共引文献】