一类局部随机波动率模型的期权定价研究

发布时间：2018-04-27 23:21

本文选题：期权定价 + 隐含波动率　；参考：《浙江大学》2012年博士论文

【摘要】：Black-Scholes基于无套利原理推导出著名的Black-Scholes期权定价公式。该公式的问世的确引起了金融经济学与金融业的革命。然而Black-Scholes模型也有自已的不足,它的常波动率假设使得模型的隐含波动率曲面独立于敲定价格与到期时间,且不随时间变化。但实际上,市场隐含波动率曲面依赖于期权敲定价格K和到期时间T以及当前时间t,体现以下特性： (1)对相同的到期时间T,·体现出波动率微笑(smile)特性。 (2)隐含波动率曲面随时间t而动态变化的。这些市场特性与期权风险对冲紧密相连,因此在建模中体现这些特性非常重要。而Black-Scholes模型在这方面的的欠缺,引起了业内对它的推广。目前主要有两类模型来体现波动率特性：一类是随机波动率模型,它直接用新的随机过程对波动率建模；另一类是局部波动率模型,它的波动率建模为基础资产和时间的确定函数。我们认为局部波动率刻画了内部因素,随机波动率刻画了外部因素,它们都部分地刻画了波动率。如果把波动率函数建模为局部和随机波动率两部分的加权和形式,这样会更加全面。于是我们提出了一类加权和局部随机波动率模型,并以SSCEV模型为例来进行研究。我们选用J-P Fouque的奇异扰动方法,推导出欧式期权定价公式的一阶近似表达式,并给出了该近似公式的误差估计。为了计算的方便,我们把近似公式的一阶项和零阶项分别关于指数参数渐进展开,得到渐进解析公式。实证分析结果显示,SSCEV模型对到期时间大于90天的欧式期权数据拟合效果是比较好的。且对不同的到期时间,模型预测的波动率曲线动态方向与市场观测的相一致。这克服了局部波动率的这方面的问题,使得模型的对冲更加稳定。另外,基于Jim Gatheral的工作,我们对Pierre Henry-Labordere(2008)的近似结果进行改进,给出局部波动率模型下隐含波动率的一种新的近似公式,并巧妙的利用Taylor级数展开,推导出近似公式的简单级数形式。数值结果显示,在时间齐次与非齐次的CEV模型下,我们的近似公式均显示出较高的近似精度。且在计算上也要简单许多,因此它可以作为CEV模型精确解的良好替代。在时间非齐次的CEV模型情形,与Pierre Henry-Labordere(2008)的近似结果进行比较得出：我们基于它的修正不仅在精度上提高很多,而且有更好的稳定性.这说明了改进效果显著。
[Abstract]:Black-Scholes derives the famous Black-Scholes option pricing formula based on the arbitrage free principle. The emergence of the formula did lead to a revolution in financial economics and finance. However, the Black-Scholes model also has its own shortcomings. Its constant volatility hypothesis makes the implicit volatility surface of the model independent of the fixed price and maturity time, and does not change with time. In practice, however, the implied volatility surface of the market depends on the option strike price K and the maturity time T, as well as the current time t, reflecting the following characteristics: 1) for the same expiration time T, the volatility smile is reflected in the characteristic of mileage. 2) the implicit volatility surfaces change dynamically with time t. These market characteristics are closely related to option risk hedging, so it is very important to embody these characteristics in modeling. And the lack of Black-Scholes model in this respect, has caused the promotion of it in the industry. At present, there are mainly two kinds of models to reflect volatility characteristics: one is stochastic volatility model, which directly uses new stochastic processes to model volatility, the other is local volatility model. Its volatility is modeled as a deterministic function of the underlying asset and time. We consider that local volatility characterizes internal factors, random volatility characterizes external factors, and they all characterize volatility partially. If the volatility function is modeled as the weighted sum of local and random volatility, it will be more comprehensive. So we propose a class of weighted and local stochastic volatility models, and take the SSCEV model as an example to study. Using J-P Fouque's singular perturbation method, we derive the first order approximate expression of European option pricing formula, and give the error estimate of the approximate formula. For the convenience of calculation, we develop the first order term and zero order term of the approximate formula with respect to the exponential parameters respectively, and obtain the asymptotic analytic formula. The empirical results show that the SSCEV model has a good effect on European option data whose expiration time is longer than 90 days. For different expiration time, the dynamic direction of volatility curve predicted by the model is consistent with that of market observation. This overcomes the problem of local volatility and makes the hedging of the model more stable. In addition, based on the work of Jim Gatheral, we improve the approximate results of Pierre Henry-Labordereen 2008, and give a new approximate formula of implicit volatility under local volatility model, and use Taylor series to expand it cleverly. The simple series form of approximate formula is deduced. Numerical results show that our approximate formulas show high approximation accuracy under the time-homogeneous and non-homogeneous CEV models. And it is much simpler in calculation, so it can be used as a good substitute for the exact solution of CEV model. In the case of time-inhomogeneous CEV model, the comparison with the approximate result of Pierre Henry-Labordereen 2008) shows that our correction based on it not only improves the accuracy but also has better stability. This shows that the effect of improvement is remarkable.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2012
【分类号】：F224;F830

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