基于随机波动率的期权定价问题

This project aims to study the option pricing problems based on the stochastic volatilities derived from a class of financial derivatives and phenomena in real financial market, which describe interesting phenomena that the uncertain volatilities of risky assets stochastically fluctuate with time going by rather than be a constant. We will mainly consider three classes of stochastic volatility problems，the first is the American option pricing problems based on the stochastic volatility models driven by Brownian Motion and fractional Brownian Motion；The Second is the path-dependent option pricing problems based on stochastic volatility models driven by rough paths；The third is option pricing problems with stochastic volatility in the Markov regime switching market. Due to their practical background, these problems have been paid more and more attention from the scholars both in China and abroad. This project will establish the option pricing formulas and study the properties of option prices in the above stochastic volatility models，through stochastic analysis method，partial differential equation theory and singular perturbation method. Therefore the study of these problems can not only enrich the option pricing theories, but also provide some necessary theoretical basis and references in dealing with financial problems in reality.

本项目旨在研究基于随机波动率模型的期权定价问题，它来源于实际金融市场中广泛存在的一类金融衍生品及现象，并用于刻画风险资产随时间随机变化的不确定的非常数波动率。本项目主要讨论三类随机波动率问题：一是布朗运动、分数布朗运动驱动的随机波动率模型及美式期权定价问题；二是粗糙路径驱动的随机波动率模型及路径依赖期权定价问题；三是马尔科夫开关市场下的随机波动率模型及期权定价问题。这些问题具有鲜明的实际背景，所以倍受国内外学者的关注。本项目将通过随机分析、偏微分方程、奇异摄动分析等理论给出各类随机波动率模型下的期权定价公式并分析期权价格性质。对于这些问题的讨论不仅可以丰富期权定价理论而且将对处理实际金融问题提供必要的理论依据和参考。

本项目考虑了金融数学中一类广为关注的重要问题，即期权定价问题。我们采用随机波动率模型来描述风险资产在复杂金融市场环境下，随时间变化的不确定风险收益率，即非常数的波动率。我们得到了随机波动率模型下美式期权的期权定价方程以及期权价格的表达式。我们还得到了一类与期权定价相关的偏微分方程的爆破解结果。这些数学结果对期权定价的理论分析和实际应用具有一定的指导意义。