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 simpler than the existing ones. Using intraday prices for the Standard & Poor’s 500 equity index, the estimates revealed strong evidence that both volatility and the volatility of volatility are stochastic. We also proceeded in a Monte Carlo simulation analysis and found that the estimates were reasonably accurate. Such evidence implies that the stochastic volatility models proposed in the literature with constant volatility of volatility may fail to approximate the discrete-time short rate dynamics.

Summary: We introduce a method that eliminates the specification error and spurious relationships in regression. In addition, we introduce a test of strong causality. Furthermore, hypothesis testing (inference) becomes almost unneeded. Moreover, this method virtually resolves error problems such as heteroscedasticity, autocorrelation, non-stationarity and endogeneity.

We provide simple formulas for pricing both the European and American options.