"Monte Carlo Methods in Finance, Statistics and Biostatistics"

“金融、统计和生物统计学中的蒙特卡罗方法”

• 批准号: 8335-2012
• 负责人: McLeish, Don
• 金额: $ 1.53万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=521117
• 关键词: Monte; Carlo; Methods; Finance; Statistics

This research concerns the applications of Monte Carlo methods to problems in Finance and Statistics. We frequently wish to estimate the expected value of a function of a stochastic process, whose distribution Q is intractible or difficult to simulate from. We investigate the use of "randomized importance sampling". We simulate from an alternate simpler distribution P and then weight the observations using (random) importance sampling (IS) weights W, which compensate for the fact that the wrong distribution was used to conduct the simulation. The estimator is a weighted average with random weights W. W estimates the relative likelihood of the observations under the two measures and can be chosen to take only integer values or rescaled, observations with weight 0 discarded. There are emany potential applications since for stochastic processes, likelihoods ("Radon Nikodym derivatives") are difficult to compute exactly, but unbiased estimators of them are tractable. For example if the process is a diffusion or jump-diffusion process, evaluating the likelihood requires evaluating an integral of the process, i.e. simulating it at every time point in an interval. However it is easy to obtain an unbiased estimator which samples the process only at finitely many points. Simulations using such estimators will be used to assess risk or evalute option prices. We can also obtain our sampling weights using only the characteristic function of a distribution. We apply this methodology to the analysis of stochastic volatilitiy models, which provide a critical improvement over the ill-fitting Black-Scholes model in finance. We will consider other problems in which the characteristic function of the log-spot price is known, such as affine stochastic volatility models with and without stochastic interest rates, time-changed Levy processes, and other complex models. Using simulation-based randomized importance sample weights permits adoption of broader classes of models, or accommodating incomplete observed data, since it allows imputation to be carried out under the simple measure P. This work will also facilitate the analysis of more complex missing data problems in continuous-time longitudinal models in statistics and biostatistics.

本研究是关于蒙地卡罗方法在金融与统计问题上的应用。我们经常希望估计随机过程的函数的期望值，其分布Q是棘手的或难以模拟。 我们调查使用“随机重要性抽样”。我们从另一个更简单的分布P进行模拟，然后使用（随机）重要性抽样（IS）权重W对观察值进行加权，这弥补了使用错误分布进行模拟的事实。估计量是具有随机权重W的加权平均。 W估计两个度量下观测值的相对似然，可以选择只取整数值或重新缩放，丢弃权重为0的观测值。 随机过程的似然估计（Radon Nikodym导数）很难精确计算，但其无偏估计是易于处理的，因此有许多潜在的应用。 例如，如果过程是扩散或跳跃扩散过程，则评估似然性需要评估过程的积分，即在间隔中的每个时间点对其进行模拟。然而，很容易得到一个无偏估计，它只在许多点上对过程进行采样。模拟使用这种估计将被用来评估风险或评估期权价格。我们也可以只使用分布的特征函数来获得我们的抽样权重。 我们将这种方法应用于随机波动率模型的分析，这提供了一个关键的改进，在金融中的拟合不佳的Black-Scholes模型。我们将考虑其他的问题，其中的特征函数的日志现货价格是已知的，如仿射随机波动率模型和不随机利率，时变Levy过程，和其他复杂的模型。 使用基于模拟的随机重要性样本权重允许采用更广泛的模型类别，或容纳不完整的观测数据，因为它允许插补进行简单的措施P.这项工作也将有利于分析更复杂的缺失数据问题，在连续时间纵向模型的统计和生物统计。