Next we'll use a method for solving for implied volatility for European options. This can be done by numerically solving for the root using the bisection method.
To be able to understand why we use the bisection solver to find the root of the Black-Scholes equation, we need some tools. First we recapture the definition of the call and put price as a function of the estimated volatility and a set of parameters (denoted):
To extract the implied volatility, we need an inverse function of the Black-Scholes formula. Unfortunately, there is no analytical inverse of that function. Instead, we can say that the Black-Scholes formula, with the implied volatility minus the current market price of that option, has a call option in this case of zero. Following 
is the current market price for the call option studied in this section:
This enables us to use a numerical root solver to find the implied volatility. Following is an implementation of the bisection solver in F#. We...