随机波动率模型有股息情形下欧式期权的定价

发布时间：2018-01-10 16:12

本文关键词：随机波动率模型有股息情形下欧式期权的定价　出处：《清华大学》2014年硕士论文　论文类型：学位论文

【摘要】：这篇论文讨论的是随机波动率模型有股息情形下的欧式期权定价问题。在这篇论文中，我们严格推导了随机波动率模型有股息情形下的欧式期权价格所应满足的偏微分方程以及相应的概率表达式。随机波动率模型是假定风险资产回报率的标准差为一个随机变量的模型，在这篇论文中，我们假定风险资产回报率的标准差是一个随机微分方程的解，并且影响风险资产回报率的标准差的随机因素是与影响风险资产回报率的随机因素不同的随机因素。在经典的Black和Scholes模型下，风险资产回报率的标准差被假定为一个常数，因此此时市场是完备的，也就是说任何一个金融衍生品的价格都能通过持有无风险资产和该金融衍生品的标的资产获得。在Black和Scholes模型下，我们能够容易的获得欧式期权价格所应满足的偏微分方程以及相应的概率表达式。但是实际数据表明，风险资产回报率的标准差随时间不是一个常数。对于随机波动率模型，由于引入了新的影响风险资产回报率的标准差的随机因素，这使得随机波动率模型更符合实际，然而在随机波动率模型下，市场是非完备的，也就是说任何一个金融衍生品的价格都不能仅仅通过持有无风险资产和该金融衍生品的标的资产获得。正是由于这个区别，导致随机波动率模型下的期权定价问题变的更加复杂，因为我们不能够像经典的Black和Scholes模型那样，通过持有无风险资产和该金融衍生品的标的资产的方法，，获得期权价格的概率表达式。在这篇论文中，为解决这个问题，我们采用了先利用无风险套利的方法获得欧式期权价格所应满足的偏微分方程，之后再利用Feynman-Kac公式将得到的结果转化为相应的概率形式的方法，无风险套利的方法在随机波动率模型下仍然适用。此外本论文还考虑了风险资产为股票且派发股息的情形，并同样对欧式期权的价格的概率表达式进行了严格推导，与不派发股息情形唯一不同的是，此时我们将得到的股息投资到无风险资产上。在随机波动率模型有股息情形下，我们得到的欧式期权的价格表达式和无股息情形下十分的相似，这和经典的Black和Scholes模型下得到的结果一致。在文章的最后，我们简要的讨论了如何在实际中应用上面得到的结论给欧式期权定价，我们主要讨论模型的参数估计问题。
[Abstract]:This paper deals with the pricing of European options in the case of a stochastic volatility model with dividends. We strictly deduce the partial differential equation and the corresponding probabilistic expression for the price of European option in the case of dividend. The stochastic volatility model assumes that the standard deviation of return on risky assets is assumed to be. A model of random variables. In this paper, we assume that the standard deviation of return on risky assets is the solution of a stochastic differential equation. And the random factors that affect the standard deviation of return on risky assets are different from those affecting the rate of return of risky assets. Under the classical Black and Scholes models. The standard deviation of the return on risky assets is assumed to be a constant, so the market is complete. That is, the price of any financial derivative can be obtained by holding a risk-free asset and the underlying asset of the financial derivative. Under the Black and Scholes models. We can easily obtain the partial differential equation and the corresponding probabilistic expression of the European option price. But the actual data show that. The standard deviation of return on risky assets with time is not a constant. For the stochastic volatility model, a new random factor affecting the standard deviation of return on risky assets is introduced. This makes the stochastic volatility model more realistic. However, under the stochastic volatility model, the market is incomplete. That is to say, the price of any financial derivative cannot be obtained only by holding riskless assets and the underlying assets of the financial derivative. It is precisely because of this distinction. The problem of option pricing under stochastic volatility model becomes more complicated because we can not be like the classical Black and Scholes model. The probabilistic expression of the option price is obtained by holding the riskless asset and the underlying asset of the financial derivative. In this paper, to solve this problem. We use the risk-free arbitrage method to obtain the partial differential equation which the European option price should satisfy. Then we use the Feynman-Kac formula to convert the obtained results into the corresponding probability form. The risk-free arbitrage method is still applicable under the stochastic volatility model. In addition, this paper also considers the case that the risky asset is a stock and pays dividends. The probabilistic expression of the price of the European option is also strictly derived, which is only different from the case of no dividend payment. Under the stochastic volatility model with dividend, the price expression of the European option is very similar to that in the case of no dividend. At the end of this paper, we briefly discuss how to apply the above conclusions to European option pricing in practice. We mainly discuss the parameter estimation of the model.
【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】：F224;F830.91

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