Implied Standard Deviation For Black/Scholes Call - Newton 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 Newton-Ralphson method to derive the implied standard deviation (volatility).
To use Newton-Ralphson method, the first differential of the standard deviation with respect to the price (Black/Scholes) is required. In this case, we can use Vega (Kappa) the sensitivity of the call price to the implied standard deviation.
Regarding the initial value for the procedure, Brenner and Subrahmanyam (1988) came out with a value of C/(0.398*S*t0.5), where C is the call price, S is the stock price (spot price), and t is the life of the option. This is the value that we use in this Newton-Ralphson procedure.
The following figure shows the outcome from the Newton's 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. Each of the procedures takes only 3 iterations to converge. 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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