On valuing and hedging European options when volatility is estimated directly

We quantify the effects on contingent claim valuation of using an estimator for the unknown volatility σ of a geometric Brownian motion (GBM) process. The theme of the paper is to show what difficulties can arise when failing to account for estimation risk. Our narrative uses a direct estimator of volatility based on the sample standard deviation of increments of the underlying Brownian motion. After replacing the direct estimator into the GBM, we derive the resulting distribution function of the approximated GBM for any time point. This allows us to present post-estimation distributions and valuation formulae for an assortment of European contingent claims that are in accord with many of the basic properties of the underlying risk-neutral process, and yet better reflect the additional uncertainties and risks that exist in the Black–Scholes–Merton paradigm.

► We quantify the effects on option valuation of using an estimator for the unknown volatility of a geometric Brownian motion (GBM) process.
► We derive the distribution function of the approximated GBM at any time.
► We present post-estimation properties for an assortment of European options.
► We provide exact post-estimation GBM recipes for pricing and hedging.
► We characterize the difference between risk and uncertainty within our paradigm.