摘要
2008年金融危机后,期权合约占全球衍生品市场总交易量的比重越来越大,投资者越来越多地使用期权进行风险对冲或套利。在2014年,我国的期货和证券交易所也将进行期权合约的正式交易,成为中国期权交易元年。在近些年,理论界对期权定价的研究日新月异,以Levy模型为代表的非正态期权定价方法发展迅速。在这些背景下,回顾我国权证市场历史,引入最新的期权定价模型和定价技术,不仅能帮助我们反思中国金融衍生品交易和监管,为未来期权市场的平稳运行提供帮助,也能从理论上探索适合我国金融市场特殊环境的期权定价模型。
以Black-Scholes-Merton模型为代表的传统期权定价模型普遍建立在金融资产收益率的正态分布假设下,但金融数据具有强烈的非正态特征,大量研究证实,使用Levy族随机过程对传统模型进行修正能提升定价精度。Levy过程是具有独立增量、平稳增量和随机连续特征的分布函数的统称,广泛应用于金融、医学、物理等领域,可以准确表现金融数据高阶统计特征,特别是资产的“跳跃特征”和“非对称特征”。虽然Levy过程具有这些优点,但与正态随机模型相比,Levy随机模型结构较复杂,模型参数估计、风险中性测度转换和随机数模拟都更困难,也不适合解决路径依赖期权的定价问题,并且Levy随机数生成算法的运行效率较低,这些都是Levy分布下蒙特卡洛期权模拟定价的难点,也是实际操作中必须解决的重点。
针对以上几个问题,本文用Levy随机过程对传统期权定价模型和方法进行修正与拓展,同时针对中国证券市场的欧式权证、美式权证、百慕大式权证的历史数据进行全面的实证分析,具体内容及相关成果如下：
1.在蒙特卡洛模拟定价的框架下,建立了多种Levy期权随机模型,针对这些模型的结构给出了参数估计和风险中性调整的方法,最后使用这些模型对大陆权证市场数据进行定价,用定价结果对Levy期权定价模型进行检验,也对大陆权证市场的有效性进行分析。
2.期权价格对标的资产的波动很敏感,考虑到金融资产波动率的时变特征,使用有偏GARCH模型对基础资产进行建模,同时引入Levy随机过程对模型的“新息项”进行模拟。完成Levy-GARCH模型风险中性测度转换问题后,我们用多种Levy随机过程和多种GARCH模型进行交叉建模与实证,使用大陆欧式权证交易数据对Levy-GARCH模型进行定价,验证这一模型是否能准确描述历史数据波动率的时变特征,并证实模拟收益率能否准确反映真实数据的分布特征。
3.针对美式期权定价的路径依赖问题,用Levy随机模型模拟基础资产的价格路径,用美式期权最小二乘法进行预期现金流的逐期迭代,最终用蒙特卡洛模拟的方法得到期权价格。在实证环节中,引入我国百慕大权证和美式权证数据,证实这一方法能否有效预测现金流贴现值,从而检验这一方法的准确度和定价效率。
4.借鉴方差减少技术的思路,从Levy随机数的时变布朗运动生成算法出发,同时生成两组高度相关的随机路径,用控制变量法建立期权模拟定价模型,并用拟蒙特卡洛模拟技术进行算法优化,形成时变布朗算法下的Levy过程方差减少技术,之后使用欧式权证数据进行模拟定价,验证这一方法是否能降低随机路径间的方差、能否加快模拟定价的收敛速度、能否提升整体定价效率。
5.综合我国沪深两市中欧式、美式和百慕式权证在多种模型下的定价结果,计算权证市场价格和模型理论价格的偏离程度,综合考虑定价误差的统计特征并与香港权证市场做对比分析,验证我国权证市场是否存在过分投机现象,分析市场是否缺乏有效性。1
After the2008financial crisis, the global derivatives trading volume in options proportion is growing, more and more investors build portfolios using options to hedge or arbitrage, our futures and stock options will soon open. Theoretical research of options is also changing, option pricing models under Levy processes developed rapidly.In this context, a review of the China's warrants market and the introduction of option pricing models can not only help us to reflect Chinese financial derivatives market regulation, but also to explore the option pricing theory for China's financial market environment.
Black-Scholes model and other traditional option pricing models are usually built on the assumption of normal distribution of financial assets under yields, and the strong non-normal characteristics of financial data have been confirmed, thus Levy stochastic processes can correct the traditional model by improving pricing accuracy. Levy processes are collectively referred to the distribution function with independent increments, steady incremental and stochastically continuous characteristics, it is widely used in finance, medicine, physics and other fields. Levy can be expressed in higher-order statistical features especially assets "jump feature" and "asymmetric characteristics". However, compared with the normal random model, Structures of Levy stochastic models are more complex, Levy model parameter estimation, and risk-neutral measure conversion and random number generation are more difficult, is not suitable to solve the path-dependent option pricing problems. But random number generating algorithm is a core issue for Monte Carlo simulation.
In this paper, Levy stochastic process with several traditional option pricing models for correction and expansion are discussed, while a comprehensive empirical research on China's mainland European Style Warrants, American and Bermudan Warrants. The specific content and related results are as follows:
1In the framework of Monte Carlo simulation pricing, we established multi-Levy process option pricing models, the structural model for the given parameter estimation and risk-neutral adjustment method are discussed, the last part of this chapter is an empirical analysis of China warrants trading data in order to prove the validate of Levy models.
2Option pricing is sensitivity to assets volatility, taking into account the time-varying characteristics of the financial assets, we use biased GARCH model to model the underlying assets while introducing Levy stochastic process to model the "innovations". China warrants trading data are used to do empirical verify for these models It is proved the Levy-GARCH models can describe the historical volatility of the time-varying data characteristics.
3For the path-dependent American option pricing problem, we use Levy-GARCH models to simulate price path of underlying assets, based on American option lest square Monte Carlo method we calculate expected cash flows iteratively, then with the Monte Carlo simulation method we obtained the option pricing result. The empirical research of American Style China's Warrants and Bermuda data confirmed the effectiveness of this approach.
4We introduced a variance reduction technology modified with change Brownian motion Levy random number generation algorithm. First we generate two highly correlated random path, one normal distribution path and one Levy path, then we used random path to simulate the control variable. Under the framework of control variable algorithm, we introduced this "variance reduction method under Levy processes". Finally, we use this method to simulate European China's warrants, the results verified analog path can reduce the variance between samples, this method can accelerate the convergence speed of Monte-Carlo simulation.
引文
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