带双层反馈的非线性随机波动率期权定价模型研究

Nonlinear option pricing has become one of hot topics in quantitative finance in recent years; the option pricing model with feedback when considering the effect of hedgers on the underlying asset price, is one of such famous models. This proposal aims to rigorously develop, rather than arbitrarily suppose it to be, a series of stochastic volatility option pricing models with two-layer feedback, and to explore their practical values. To this end, this proposal, starting from the microstructure of real market which consists of fundamentalists, chartists, noise traders and risk hedgers, examines the two-layer feedback pricing mechanism of market; then taking security markets where the option market has already been open for short, medium and long terms for examples (which corresponds to common investor’s three states in perceiving the effect of hedgers: ignorance, sub-consciousness and consciousness, respectively), constructs the stochastic volatility stock pricing models with two-layer feedback, develops the associated non-linear option pricing equations and investigates three representative approaches to their solutions, i.e., Fourier transform based analytic approximation, perturbation analysis and high-order difference method. Finally, this proposal applies the newly proposed option pricing models to explain the Pinning effect of stock price, to predict the market volatility, and to explore both the temporary and lasting impacts of opening option market on the underlying market in China's current situation.

非线性期权定价是近年来数量金融领域的热门课题；考虑期权风险对冲交易影响的“带反馈的期权定价模型”，是一类重要的非线性期权定价模型。项目拟构建一系列数学严谨(而非武断设计)的“带双层反馈的随机波动率期权定价模型”、并发掘其应用价值。为此，项目从市场微观结构出发，考察含四类交易者(价值、技术、噪音与风险对冲交易者)的证券市场双层反馈定价机制；然后以个股期权市场开放初期、中期和后期的证券市场为例(对应于大众交易者对“期权风险对冲交易影响”的三种认知状态：懵懂，潜意识和完全意识)，构建联立的带双层反馈的随机波动率股票价格模型以及相应的非线性期权定价方程，并针对性地研究三种代表性求解方法(基于傅氏变换的解析逼近方法、扰动分析方法以及高阶差分格式方法)；最后应用带反馈的期权定价模型，解释股价锚定期权敲定价现象、预测市场波动率、科学探索我国当前市场状态下推出个股期权可能对股市造成的短期和中长期的影响。

考虑投资者微观结构与期权风险对冲交易影响的带反馈的期权定价模型，日渐引起学界和实务界的重视。项目首先针对含基本面分析者，趋势交易者与噪声交易者的证券市场，推导出带第一层反馈的股票价格随机波动率模型。然后，研究含个股期权的证券市场上的资产定价模型，将期权交易策略对标的资产价格产生的影响融入到带第一层反馈的随机波动率模型中，构建了带双层反馈的随机波动率资产价格模型，并推导出期权定价偏微分方程；最后，着重研究了四种期权定价方法：基于傅里叶变换的期权定价解析逼近方法；基于扰动分析的期权数值定价方法；近似精确的Monte Carlo数值方法以及基于深度学习网络的期权定价方法。.研究表明：（1）带反馈的随机波动率模型能更灵活、准确地刻画不同成熟度市场中的超峰、肥尾、有偏以及波动聚集等现象；（2）在投机盛行的证券市场，不考虑期权反馈效应的随机波动率模型会低估期权的隐含波动率；（3）非理性投资者的期权投机策略，会加剧标的资产价格波动；我国境内证券市场不宜过早全面开放个股期权，逐步试点方案较为稳妥。（4）基于扰动分析与傅里叶变换相结合的期权定价方法，能破解非线性期权定价模型难以获得解析式的瓶颈，可推导出简洁、准确、高效的期权拟闭型定价公式；基于深度学习算法的期权定价方法，显著优于传统的基于“蛮力求均值”的蒙特卡洛方法；而近似精确的Monte Carlo方法虽然提升了定价精度，但是以牺牲计算效率为代价。.研究价值：（1）本研究由微观结构导引出宏观模型，为金融数学模型家族引进更“接地气”的成员。（2）研究工作对即将开放个股期权的我国证券市场的稳健运行有重要实际指导意义。（3）有望为期权品种提供一种高性能、应用级的定价和风险量化管理工具。.项目取得成果：1）出版专著一部（科学出版社）；2）发表（含收录）学术论文10篇，其中：4篇为SCI/SSCI检索，4篇为EI检索；3）培养硕士研究生6名。