Implied Standard Deviation For Black/Scholes Call - Bisection Approach
The implied standard Deviation or implied volatility is the volatility value that would make the theoretical value (in this case the Black/Scholes Model) equals to the given market price. This volatility incorporates all sources of mispricing, including data errors, effects of the bid-ask spread and temporary imbalances in supply and demand. Nevertheless, implied volatility reflects the future aspect of the market (which is reflected on the market price). In this example, we will utilize the Bisection method to derive the implied standard deviation (volatility).

Unlike Newton-Ralphson procedure, Bisection method does not require the first differential of the standard deviation with respect to the price (Black/Scholes) as an input. However, it does require two initial values for the iteration.

Bisection searching method utilizes linear interpolation. It uses a minimum and a maximum starting numbers in the iteration process. The steps it takes to convert depends greatly on the starting numbers. In general, this method takes more iterations to convert compares to the Newton method.

The following figure shows the outcome from the Bisect procedure. Three different sets of data are tested. The implied standard deviation for each of the sets is displayed in the green section of each table. The first set of data takes 4 iterations to converge, where as the second and third set take 2 and 5 iterations, respectively, to converge. Bisect method does not converge as fast as the Newton-Ralphson method. Both the implied standard deviation and the steps are user-defined functions. The last table computes the call option price based on the implied standard deviation in table 1. The price comes out to be 10 - which matches with the call price uses for the formula.
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