Parameter estimation error: a cautionary tale in computational finance

We quantify the effects on contingent claim valuation of using an estimator for the volatility of a geometric Brownian motion (GBM) process. That is, we show what difficulties can arise when failing to account for estimation risk. Our working problem uses a direct estimator of volatility based on the sample standard deviation of increments from the underlying Brownian motion. After substituting into the GBM the direct volatility estimator for the true, but unknown, value of the parameter sigma, we derive the resulting marginal distribution of the approximated GBM. This allows us to derive post-estimation distributions and valuation formulae for an assortment of European contingent claims that are in accord with the basic properties of the underlying risk-neutral process. Next we extend our work to the contingent claim sensitivities associated with an assortment of European option portfolios that are based on the direct estimator of the volatility of the GBM process. Our approach to the option sensitivities - the Greeks - uses the likelihood function technique. This allows us to obtain computable results for the technically more-complicated formulae associated with our post-estimation process. We discuss an assortment of difficulties that can ensue when failing to account for estimation risk in valuation and hedging formulae.