"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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=592918
• 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.

本研究涉及蒙特卡罗方法在金融学和统计学问题中的应用。我们经常希望估计一个随机过程的函数的期望值，而随机过程的分布Q是难以或难以模拟的。我们研究了“随机重要性抽样”的使用。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对观测值进行加权，这弥补了使用错误分布进行模拟的事实。估计量是随机权值W的加权平均值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，权值为0的观测值被丢弃。有许多潜在的应用，因为对于随机过程，可能性（“Radon Nikodym导数”）很难精确计算，但它们的无偏估计是可处理的。例如，如果过程是扩散过程或跳跃扩散过程，评估似然需要评估过程的积分，即在一个区间内的每个时间点模拟它。然而，获得一个无偏估计量是很容易的，它只在有限多个点上对过程进行采样。使用这种估计器的模拟将用于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抽样权值。我们将这种方法应用于随机波动模型的分析，这对金融中的不拟合布莱克-斯科尔斯模型提供了重要的改进。我们将考虑其他已知对数现货价格特征函数的问题，如带或不带随机利率的仿射随机波动模型、时变Levy过程和其他复杂模型。使用基于模拟的随机重要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，因为它允许在简单测度p下进行代入。这项工作还将有助于分析统计学和生物统计学中连续时间纵向模型中更复杂的缺失数据问题。