Implied Standard Deviation For Black/Scholes Call - Secant 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 Secant method to derive the implied standard deviation (volatility).
Unlike Newton-Ralphson precedure, Secant method does not require the first differential of the of the standard deviation with respect to the price (Black/Scholes) as an input. Nevertheless, it does require an initial value for the iteration just as any other numerical precedures.
In 1988, Brenner and Subrahmanyam came out with a value of C/(0.398*S*t0.5), where C is the call option price, S is the stock price (spot price), and t is the life of the option. This is the value that we will use for this Secant procedure.
The following figure shows the outcome from the Secant precedure. 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 3 iterations to converge, where as the second and third set take 4 and 6 iterations, respectively, to converge. This finding shows two things. First, the number of steps the precedure takes depends greatly on the parameter values used for the option. Second, secant 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 matchs with the call price uses for the formula.
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