Implied Volatility - Solving The Inverse Pricing Model Function

Solving The Inverse Pricing Model Function

In general, a pricing model function, f, does not have a closed-form solution for its inverse, g. Instead, a root finding technique is used to solve the equation:

While there are many techniques for finding roots, two of the most commonly used are Newton's method and Brent's method. Because options prices can move very quickly, it is often important to use the most efficient method when calculating implied volatilities.

Newton's method provides rapid convergence; however, it requires the first partial derivative of the option's theoretical value with respect to volatility; i.e., which is also known as vega (see The Greeks). If the pricing model function yields a closed-form solution for vega, which is the case for Black–Scholes model, then Newton's method can be more efficient. However, for most practical pricing models, such as a binomial model, this is not the case and vega must be derived numerically. When forced to solve for vega numerically, it usually turns out that Brent's method is more efficient as a root-finding technique.