The Calculation of Implied Variances from the Black-Scholes Model: A Note

Black and Scholes (1973) derived a model for the equilibrium price of a European Stock Purchase option. According to this model, equilibrium option prices are a function of the time to maturity of the option, the exercise price, the current price of the underlying stock, the risk-free rate of interest, and the instantaneous variance of the stock's rate of return. Only the first 4 of these variables can be observed; the instantaneous variance of the stock's return can only be estimated. Recently several authors have tried to use the Black-Scholes formula to derive an implied variance. This implied variance cannot be calculated explicitly, and researchers have used numerical methods such as the Newton-Raphson method. The necessary and sufficient conditions for the existence of a positive implied variance are given. An algorithm is presented which converges monotonically and quadratically to the implied variance when it exists. The algorithm provides a starting value for the first iteration of the Newton-Raphson procedure.