Computation and stability of the implied volatility

We investigate a problem connected with the evaluation of the implied volatility and it stability in a Black and Scholes model for a call option. Black and Scholes in [1] have derived a model for the equilibrium price of a European Stocks purchase option. According to the B-S model equilibrium option prices are a function of the time at maturity of the option, the exercised price, the current price of the underlying stocks, the risk free rate of interest, the instantaneous variance of the stocks rate of return and the exponential transaction cost. Among the six variables only the first and the last can be directly observed; the instantaneous variance can only be estimated. Recently several authors [2],[3],[4] for example rather than estimating the volatility from the past data have attempted to employ the Black-Scholes option pricing formula to derive an "implied volatility". An implied volatility is the value of the instantaneous volatility of the stock's return which when employed in the Black-Scholes formula, results in a model price equal to the market price. In this note necessary and sufficient condition for existence of positive implied volatility is given along with an algorithm which converges monotonely and uniformly to the unique implied volatility when it exists, also we have solve the transandental equation to find a good approximation of the starting value of the Newton Raphson procedure.