论文标题
时间消化的对数正态SABR模型的渐近学：隐含的波动率表面
Asymptotics of the time-discretized log-normal SABR model: The implied volatility surface
论文作者
论文摘要
我们为对数正态SABR模型提出了一种新颖的时间离散化，该模型是一种流行的随机波动率模型，广泛用于财务实践。我们的时间离散化是Euler-Maruyama计划的变体。我们在某个渐进状态下以大量时间步长的极限研究其渐近性能，其中包括有限成熟度，频率小和较大的初始波动率，并具有固定产物的频率和初始波动性。我们得出了几乎确定的限制，并且在大量时间步骤的限制下，对数资产价格的偏差很大。我们得出暗示的挥发率表面的确切表示，以在此制度中进行任意成熟和罢工。使用这种表示，我们获得了对较小的成熟度和极端打击的隐含波动率的分析膨胀，这些幅度以连续时间模型的领先顺序已知渐近结果繁殖。
We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in this regime. Using this representation we obtain analytical expansions of the implied volatility for small maturity and extreme strikes, which reproduce at leading order known asymptotic results for the continuous time model.
