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*t
0.5),
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.
