2015/01/28 by Masaaki Fukasawa, Fukasawa, Masaaki · 3 citations
Economics, Econometrics and Finance · #60F05 #FOS: Economics and business #Financial Markets and Investment Strategies #Financial Risk and Volatility Modeling #Mathematical Finance (q-fin.MF) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1501.06980
openalex publication_date 2015/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Black-Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time-to-maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time-to-maturity tends to zero. For this purpose we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.