渐近展开方法在金融计量与金融工程中的应用

项目名称： 渐近展开方法在金融计量与金融工程中的应用

项目编号： No.11201009

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 李辰旭

作者单位： 北京大学

项目金额： 22万元

中文摘要： 当前，连续时间随机模型由于其丰富的理论背景，日益彰显其在诸多自然与社会科学量化研究领域不可替代的作用，然而此类模型在金融计量和统计推断领域中的发展仅从近十年开始, Ait-Sahalia (1999,2002,2008) 创新性地开辟了基于逼近的似然函数实施极大似然估计的重要方法，但正如历史上任何一个新理念的引入一样，该方法在具体的理论和应用中仍亟需改进和完善。在此课题中，我们深入本质，发展一系列全新的基于扩散（diffusion）和跳跃-扩散(jump-diffusion)过程转移密度的完全显式近似的金融计量（包括相关的统计推断）和金融工程方法。经过过去2年的努力探索以及与该领域著名学者的探讨和学习，我们将Malliavin-Watanabe随机分析理论应用于金融计量经济学和金融工程学的诸多问题中，例如，极大似然估计等依赖转移密度的统计推断问题和金融衍生品定价等相关课题。

中文关键词： 极大似然估计；扩散过程模型；转移密度；渐近展开；期权定价

英文摘要： Continuous-time models have demonstrated their indispensable roles in describing the laws in various random phenomenon in nature and science. However, their applications in financial econometrics and statistics started to attract significant research interests since merely about a decade ago. A milestone is the groundbreaking work of Ait-Sahalia (1999, 2002, 2008), which proposed to use approximated likelihood functions to perform maximum likelihood estimations for diffusion and jump-diffusion models. Such an idea provided a refresh perspective for the research in financial econometrics and even beyond. Motivated by some open questions arising from the literature, e.g., how to establish effective and fully closed-form asymptotic expansions for the likelihood functions (transition densities) of the so-called irreducible diffusions and jump-diffusions, we will introduce novel methods for performing approximate density (likelihood) based statistical inferences and apply the resulted econometric tools in theoretical and empirical analysis of derivative pricing models. Meanwhile, by developing similar quantitative methods, we will also investigate a broad range of issues in financial engineering, e.g., derivatives pricing and risk management.

英文关键词： maximum-likelihood estimation；diffusion model；transition density；asymptotic expansion；option pricing