Estimating Option Implied Volatility with Interval Bisection
Summary
The document explains implied volatility as the volatility input that makes a model option price match an observed market price. It motivates volatility quotes as a way to compare options whose premiums are affected by different underlying prices, especially in delta-neutral trading. Because the Black–Scholes pricing formula has no analytical inverse for volatility, the article presents interval bisection as a numerical solution method.
Bisection requires a continuous pricing function and an initial interval whose endpoint prices bracket the target. It repeatedly evaluates the midpoint and keeps the half-interval on the appropriate side, stopping when the price error is within a tolerance. A C++ functor stores the option’s fixed inputs, while a function template makes the solver reusable for callable pricing functions. The document demonstrates the process for a European call and reports an example estimate. Bisection is straightforward and robust when properly bracketed, but the article notes it is less efficient than methods such as Newton–Raphson or Brent’s method; the example also depends on Black–Scholes assumptions and its pricing approximation.
Key ideas
- Implied volatility is the volatility input that aligns a model price with an observed option price.
- Interval bisection requires a continuous function and a bracket that encloses the target price.
- The method narrows the volatility interval by repeatedly testing its midpoint until the pricing error meets a tolerance.
- A C++ functor can hold fixed option parameters while a generic solver varies volatility.
- Bisection is easy to implement but less efficient than some alternative root-finding methods.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.