A martingale method for option pricing under a CEV-based fast-varying fractional stochastic volatility model

Modeling the volatility smile and skew has been an active area of research in mathematical finance. This article proposes a hybrid stochastic-local volatility model which is built on the local volatility term of the CEV model multiplied by a stochastic volatility term driven by a fast-varying fractional Ornstein-Uhlenbeck process. We find that the Hurst exponent of the implied volatility is less than 1/2 usually but it is larger than 1/2 during an immediate period of recovery from the COVID-19 pandemic. We use a martingale method to obtain option price and implied volatility formulas in the both short- and long-memory volatility cases. As a result, the existing CEV implied volatility can be complemented to reflect implied volatility patterns (skewed smiles) that arise in pricing short time-to-maturity options in equity markets by incorporating convexity into it and controlling the downward slope of it at-the-money. We verify that one additional parameter of the CEV-based fractional stochastic volatility model contributes to a better qualitative agreement with market data than the Black-Scholes-based fractional stochastic volatility model or the CEV-based non-fractional stochastic volatility model.