Collaborative Research: Small time behavior of multiscale diffusions motivated by stochastic volatility models

公认的大偏差理论描述了扩散过程在短时间内从状态空间的一点移动到另一部分的小概率。这在金融数学的背景下特别有趣，因为人们对股票价格在短时间内的行为感兴趣。例如，这将与短期期权价格相关，即短期内达到违约水平的可能性。多因素随机波动率模型在过去几年中得到了深入的研究，它们被发现非常有用地描述了隐含波动率的微笑/偏斜。在短期限内，出现在大偏差估计中的利率函数通常不会以封闭的形式给出，而是取决于模型的一些细节，如波动率。另一方面，已有研究表明，多尺度随机波动率模型及其渐近性对于获得期权价格和套期保值策略的精确近似是非常有用的，并且只需要几个组参数就可以有效地计算出来，易于对数据进行校准。因此，研究波动率时间尺度分离制度下的利率函数的渐近展开是很自然的。特别是，与随机波动率因子的均值回归时间相比，到期时间较小但较大的制度变得非常有趣和具有挑战性。该奖项支持的研究是关于推导并充分证明这种渐近展开的合理性。该问题将被表示为扩散过程的某些泛函的指数矩的对数渐近，进而与Hamilton-Jacobi-Bellman方程的齐化/平均理论相联系。这项研究的合作性质结合了多尺度随机波动率建模领域的专业知识，即大偏差理论领域，以及HJB方程的粘性解和均化/平均。该奖项将支持一项多学科的研究工作。它涉及金融数学领域中非常实际的建模问题，以及扩散过程的大偏差理论和非线性偏微分方程齐化的理论收敛结果。预计关于小额违约概率的结果将非常有助于理解信贷市场的违约机制。因此，这项研究非常及时。UCSB的高级PI积极参与组织会议、研讨会和特别会议，以促进学术研究人员和从业人员之间的互动。这项研究将在UCSB新成立的金融数学和统计研究中心(CRFMS)的活动中得到充分展示。该合作项目还将作为两个机构--南加州大学和密歇根大学--的研究生和博士后研究员的培训工具。

The well-established theory of large deviations describes the small probabilities that a diffusion process moves from one point to another part of the state space in a short time. This is particularly interesting in the context of financial mathematics in which one is interested in the behavior of the stock price in a short time. For instance, this will be relevant to option prices at short maturities, or probabilities to reach a default level in a short time. Multi-factor stochastic volatility models have been studied intensively in the past years, and they have been found very useful to describe the observed smile/skew of implied volatilities. At short maturities the rate functions appearing in the large deviation estimates will usually not be given in closed form, and will depend on some details of the models like volatility of volatility. On the other hand it has been shown that multiscale stochastic volatility models and their asymptotics are very useful to obtain accurate approximations of option prices and hedging strategies which can be efficiently computed with only a few group parameters easy to calibrate to data. It is then natural to study asymptotic expansions of the rate function in the regime of separation of volatility time scales. In particular, the regime in which the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor turns out to be very interesting and challenging. The research supported by this award is about deriving and fully justifying such asymptotic expansion. The problem will be formulated as logarithmic asymptotic for exponential moments of certain functionals of diffusion processes, which in turn is connected with homogenization/averaging theory for Hamilton-Jacobi-Bellman equations. The collaborative nature of the research combines expertise in the area of multiscale stochastic volatility modeling, in the areas of large deviation theory, and viscosity solution and homogenization/averaging for HJB equations. This award will support a multidisciplinary research effort. It concerns very practical modeling issues in the area of financial mathematics as well as theoretical convergence results in the theories of large deviations for diffusion processes and homogenization of nonlinear partial differential equations. It is expected that the results obtained on small default probabilities will be very helpful in understanding default mechanisms in credit markets. This research therefore is very timely. The senior PI at UCSB is actively involved in organizing conferences, workshops and special sessions which promote interaction between academic researchers and practitioners. The research will be given full exposure in the activities of the newly created Center for Research in Financial Mathematics and Statistics (CRFMS) at UCSB. The collaborative project will also serve as a training tool for graduate students and postdoctoral fellows at both institutions, UCSB and KU.