We propose novel nonparametric estimators for stochastic volatility and the volatility of volatility. In doing so, we relax the assumption of a constant volatility of volatility and therefore, we allow the volatility of volatility to vary over time. Our methods are exceedingly simple and far...

We overcome the limitations of the previous literature in the European options pricing. In doing so, we provide a closed-form formula that doesn't require any numerical/computational methods. The formula is as simple as the classical Black-Scholes pricing formula. In addition, we simultaneously...

This is the first paper to provide a simple, explicit formula (that doesn’t requirenumerical/computational methods) under stochastic volatility. The formulais as simple as the classical Black-Scholes pricing formula. Furthermore,this paper modifies the Black-Scholes model to make it consistent...

We provide explicit, simple price formulas for the Europeanoptions under stochastic volatility and stochastic interest rate. The formulasare as simple as the classical Black-Scholes formula. Moreover, the formulasdo not require the normality of the returns. We do not need to know thedistribution...

We provide explicit, simple price formulas for European options under stochastic volatility. The formulas are as simple as the classical Black-Scholes formula. We also explicitly include the volatility of volatility in the price formula