SABR型过程的泛函解析（Ir-）正则性
Functional Analytic (Ir-)Regularity Properties of SABR-type Processes
摘要：
SABR模型是利率市场中的一种基准随机波动率模型，在过去十年中备受关注。它的流行源于一个易于处理的渐近展开的隐含波动率，来自热核方法。随着市场利率降至历史低位，这种扩张似乎产生了不一致的价格。由于该模型是深深嵌入在市场实践中，替代定价方法SABR已解决了许多方法，在最近几年。所有标准的期权定价方法都对基础模型做出了一定的规律性假设，但对于SABR，这些假设很少得到满足。在这里，我们检查正则性的模型从这个角度来看，一些（渐近和数值）期权定价方法。特别是，我们强调微妙的退化的SABR模型（和相关的过程）的起源，认为目前使用的流行的热核方法和所有相关的方法（子）黎曼几何不适合SABR型过程，当利率接近零。我们描述了一个更一般的半群框架，它允许通过对称狄利克雷形式导出一个合适的几何SABR型过程（在某些参数制度）。此外，我们推导出正则性（Feller性质和强连续性）的流行的数值方案SABR半群的适用性所必需的，并确定合适的Banach空间和希尔伯特空间。最后，我们评论的短时间和大时间的SABR型过程超出热核框架的渐近行为。
英文摘要：
The SABR model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in market practice, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR these are rarely satisfied. We examine here regularity properties of the model from this perspective with view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller- properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups, and identify suitable Banach- and Hilbert spaces for these. Finally, we comment on the short time and large time asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.
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