Time and Finance: Exploring Variance in the Black-Scholes Model

In their 1973 paper, The Pricing of Options and Corporate Liabilities, Fischer Black and Myron Scholes published mathematical methods they had devised with the goal of accurately pricing European options. When using the model to predict future options prices, all input variables in the model can be empirically viewed, and calculated, at present time except for the future volatility of the underlying security. Retrospectively analyzing the volatility implied by the Black-Scholes model using price history shows that this implied volatility is an inaccurate estimate of actual future volatility. This project sought to explore the relationship between the implied future volatility of a stock and the Black-Scholes model, and if future implied volatility can be better predicted by incorporating an alternate measure of variance, Allan Variance. Allan Variance was first shown by physicist David Allan in Allan\u27s Statistics of Atomic Frequency Standards to measure frequency stability in oscillations found in atomic clocks. While the preliminary work done in this paper does not suggest a simple way to incorporate Allan Variance into the Black-Scholes model, and measuring variance in financial securities in general, it is indeed worth further exploration