强路径依赖期权定价的偏积分-微分方程新型快速求解研究

Strongly path-dependent options can avoid effectively systemic risk which traditional risk management tools are incapable of preventing. Reasonable pricing for these options have significant impacts on financial risk management. Stochastic volatility jump-diffusion model can reproduce typical characteristics of asset return fluctuation in market very well. The strongly path-dependent options pricing under the stochastic volatility jump-diffusion model can be reduced to solving a two dimensional partial integro-differential equation with boundary conditions. The high dimensionality of the PIDE and the separate treatment of diffusion and integral components lead to the additional complexity，slow convergence and low accuracy of the existing numerical method. From the perspective of the asset prices distribution, this project will degrade the equation's dimensions by constructing an approximating Markov chain of the pricing model. Then, by Fourier space time-stepping method this project will convert the PIDE into a linear system of easily solvable ordinary differential equations in the frequency domain. This project will speed up the solution of the PIDE by fast Fourier transform algorithm. Moreover, this project will control effectively numerical errors using error bound minimization method. Based on the above obtained results, this project will avoids the problems produced by traditional numerical methods and fulfill fast and accurate pricing strongly path-dependent options and improve effectively the ability of hedging financial risk by those options. On one hand, this research will help to complete option pricing theory to some extent. On the other hand, the obtained algorithms which are a highly efficient method to solve partial integro-differential equation can be expected to be extensively applied to deal with some related problems encountered in the areas of natural science and social science.

强路径依赖期权能有效规避传统风险管理工具无法防范的系统性风险，其合理定价对金融风险管理具有重要影响。随机波动跳扩散模型能很好复制市场资产收益变化的典型特征，基于该模型的强路径依赖期权定价归结为具有边值条件的二维偏积分-微分方程的求解。定价方程的高维性以及对积分项和扩散项的分离处理导致现有定价算法过于复杂、收敛速度慢和精度低。本项目从资产价格分布角度，通过构建定价模型的近似马氏链使定价方程降维，通过傅里叶空间时步方法将偏积分-微分方程转化为频域上易求解的线性常微分方程组，利用快速傅里叶变换算法加速定价方程的求解，利用误差界最小化方法对数值误差加以有效控制，避免了传统数值方法出现的问题，实现强路径依赖期权的准确、快速定价，切实提高其对冲风险的能力。该研究不仅可以完善期权定价理论，所获算法作为一种有效的偏积分-微分方程求解方法，也将在解决自然科学和社会科学各领域的相关问题中得到广泛应用。

路径依赖期权能有效规避传统风险管理工具无法防范的系统性风险，其合理定价对金融风险管理具有重要影响。随机波动跳扩散模型能很好复制市场资产收益变化的典型特征，基于该模型的路径依赖期权定价归结为具有边值条件的二维偏(积分)微分方程(Partial （Integro）Differential Equation, PIDE（PDE）)的求解。定价方程的高维性以及对积分项和扩散项的分离处理导致现有定价算法过于复杂、收敛速度慢和精度低。本项目进行了如下三方面的研究：第一，美式、障碍、几何亚式等路径依赖期权定价PIDE（PDE）的降维；第二，美式、障碍、几何亚式、远期开始等路径依赖期权定价PIDE（PDE）高效求解数值方法的开发；第三，几何亚式等路径依赖期权期权数值定价误差控制的理论与实验研究。研究结果表明，基于扰动理论的分析近似方法对于定价障碍期权、几何亚式期权和回望期权快速、有效；基于傅里叶变换技术的级数展式方法对于定价具有欧式期权特征的路径依赖期权比如远期开始期权准确、快速；基于扰动理论的模型校正是一种简单有效的模型参数估计方法。研究结果不仅可以完善期权定价理论，所获数值方法作为有效的PIDE（PDE）求解方法，也将在解决自然科学和社会科学各领域的相关问题中得到广泛应用。