SIAM Journal on Financial Mathematics, Volume 11, Issue 4, Page 1137-1167, January 2020.
Anomalous diffusions arise as scaling limits of continuous-time random walks whose innovation times are distributed according to a power law. The impact of a nonexponential waiting time does not vanish with time and leads to different distribution spread rates compared to standard models. In financial modeling this has been used to accommodate random trade duration in the tick-by-tick price process. We show here that anomalous diffusions are able to reproduce the market behavior of the implied volatility more consistently than the usual Lévy or stochastic volatility models. Two distinct classes of underlying asset models are analyzed: one with independent price innovations and waiting times, and one allowing dependence between these two components. These models capture the well-known paradigm according to which shorter trade duration is associated with higher return impact of individual trades. We fully describe these processes in a semimartingale setting leading to no-arbitrage pricing formulas, study their statistical properties, and in particular observe that skewness and kurtosis of asset returns do not tend to zero as time goes by. We finally characterize the large-maturity asymptotics of call option prices, and find that the convergence rate to the spot price is slower than in standard Lévy regimes, which in turn yields a declining implied volatility term structure and a slower time decay of the skew.

中文翻译：

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期权价格的异常扩散：连接交易时间和波动期限结构

SIAM金融数学杂志，第11卷，第4期，第1137-1167页，2020年1月。
异常扩散是连续时间随机游动的缩放极限产生的，创新时间根据幂定律分布。与标准模型相比，非指数等待时间的影响不会随时间消失，并且会导致不同的分布传播率。在财务建模中，已使用它在逐笔价格过程中适应随机交易时间。我们在这里表明，与通常的Lévy或随机波动率模型相比，异常扩散能够更一致地重现隐含波动率的市场行为。分析了两类不同的基础资产模型：一类具有独立的价格创新和等待时间，一类允许这两个组件之间的依赖。这些模型捕获了众所周知的范例，根据该范例，较短的交易时间与较高的单个交易的收益影响相关联。我们在导致无套利定价公式的半市场环境中充分描述了这些过程，研究了它们的统计属性，特别是观察到随着时间的流逝，资产收益率的偏度和峰度不会趋于零。我们最终描述了看涨期权价格的大型成熟渐近性，并发现与现货价格的收敛速度比标准Lévy体制慢，这反过来又导致隐含波动率期限结构的下降和偏斜的时间衰减变慢。特别要注意的是，随着时间的流逝，资产收益率的偏度和峰度不会趋于零。我们最终描述了看涨期权价格的大型成熟渐近性，并发现与现货价格的收敛速度比标准Lévy体制慢，这反过来又导致隐含波动率期限结构的下降和偏斜的时间衰减变慢。特别要注意的是，随着时间的流逝，资产收益率的偏度和峰度不会趋于零。我们最终描述了看涨期权价格的大型成熟渐近性，并发现与现货价格的收敛速度比标准Lévy体制慢，这反过来又导致隐含波动率期限结构的下降和偏斜的时间衰减变慢。