Standard Deviation For Black/Scholes Call - Bisection Approach
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.
* * Complete program (with open source
codes) available in Package Set 3 and the Combo Package.