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*t
0.5), 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.
