混合布朗运动下的欧式脆弱期权定价研究

发布时间：2018-03-31 13:21

本文选题：混合分数布朗运动　切入点：跳扩散过程　出处：《中国矿业大学》2017年硕士论文

【摘要】：随着近年来中国金融市场的不断开放、发展,国内的期权市场有了较大的实际意义上突破。伴着2015年新年的脚步,上证50ETF期权登录上海证券交易所。这意味着起步24年后,中国内地股市迎来了“期权时代”。在此之后,一些其他种类的期权将会陆续进入金融市场,交易规模也将不断增大。随之也会出现场外市场,那么当期权在场外市场进行交易时,由于没有类似清算所之类的监督机构来监管期权的空头方在到期时履行相应的义务,就会导致期权的多头方要同时承受市场风险和信用风险,而这必定会导致交易过程中违约的出现。到目前为止,关于市场信用违约风险的理论研究已经相对成熟,但是对于在存在违约风险的市场定价方面的研究就相对较少,考虑到的条件也较少,不能很好的描述期权价格的改变。因此本文考虑将跳扩散过程,随机利率和混合布朗运动引入,考虑简约模型与结构模型相结合,在以往的基础上对定价公式进行完善,增强定价公式的准确性。本文主要研究了以下几个问题:(1)由于真实市场中股票价格不能完全符合几何布朗运动,金融资产收益的分布具有“尖峰厚尾”的特征,且股价变化也不是随机游走,而是不同时间呈现不同程度的长期相关性和自相似性。这些特征与标准的布朗运动存在一定的差距,而分数布朗运动正好具备自相似性和长期相关性,更加适合金融市场的特性。因此本文在假设股票价格服从几何、分数布朗运动的条件下对脆弱期权的定价进行研究。(2)本文引入了跳扩散过程,但一般的Girsanov定理不能运用在这种情况下,所以本文先研究了跳过程经测度变换后在新测度下的表达公式,在变换后则可以在新测度下应用Girsanov变换对期权进行定价。(3)在实际的金融市场中,随着场外交易的增多,利率和违约行为都具有较强的随机性。因此本文在随机利率和随机违约强度基础上,对混合布朗运动模型下的欧式脆弱期权定价进行了相关研究,并求出解析解。本文通过鞅测度变换得到了欧式脆弱期权定价的显式解。通过数值试验将本文的定价公式与经典Black-Scholes定价公式进行比较研究,结果表明本文的期权定价公式更加符合实际金融市场的特征。
[Abstract]:With the opening and development of China's financial market in recent years, the domestic options market has made a big breakthrough in practical terms. With the pace of the 2015 New year, Shanghai Stock Exchange 50ETF options entered the Shanghai Stock Exchange. This means 24 years after the start. China's mainland stock market is ushered in an "option era." after that, some other types of options will gradually enter the financial market, and the scale of transactions will continue to grow. There will also be an over-the-counter market. Well, when options are traded in the over-the-counter market, because there is no supervisory body such as clearing houses to monitor the options' short parties to fulfill their corresponding obligations at maturity, So far, the theoretical research on market credit default risk has been relatively mature, which will lead to both market risk and credit risk, which will inevitably lead to the occurrence of default in the course of trading. However, there are relatively few studies on the market pricing with default risk, and the conditions are also less, which can not describe the change of option price very well. Therefore, this paper considers the jump diffusion process. With the introduction of stochastic interest rate and mixed Brownian motion, considering the combination of the reduced model and the structural model, the pricing formula is improved on the basis of the past. To enhance the accuracy of pricing formulas, this paper mainly studies the following questions: 1) since stock prices in real markets do not fully conform to the geometric Brownian motion, the distribution of financial assets returns has the characteristics of "peak and thick tail". And the change of stock price is not random walk, but show different degree of long-term correlation and self-similarity at different time. These characteristics are different from the standard Brownian motion. The fractional Brownian motion has self-similarity and long-term correlation, which is more suitable for the characteristics of financial market. In this paper, we introduce the jump diffusion process, but the general Girsanov theorem can not be applied in this case. So this paper first studies the expression formula of the jump process under the new measure after the measure transformation. After the transformation, we can use the Girsanov transform to price options under the new measure.) in the actual financial market, with the increase of over-the-counter transactions, Both interest rate and default behavior have strong randomness. Therefore, on the basis of stochastic interest rate and stochastic default intensity, the pricing of European fragile options under mixed Brownian motion model is studied in this paper. In this paper, the explicit solution of European fragile option pricing is obtained by martingale measure transformation. The pricing formula of this paper is compared with the classical Black-Scholes pricing formula by numerical experiments. The results show that the option pricing formula is more in line with the characteristics of the actual financial market.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.6;F830.9

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