The Black-Scholes formula expresses option value as a function of implied volatility, but markets quote options by price — not volatility. Recovering the implied volatility from an observed price requires solving a nonlinear equation: , which has no closed-form inversion. The standard solution is Newton's method: start with a guess, replace the curve by its tangent line at the guess, read off the tangent's root as the next guess, and iterate. Each step uses the formula
and the sequence typically converges to the true root with quadratic speed — roughly doubling the number of correct decimal places at each step. Newton's method is the engine inside most "solve for implied vol" routines, bond yield-to-maturity solvers, and nonlinear calibration algorithms in quantitative finance.
Learning objectives. By the end of this note you should be able to:
- derive the Newton-Raphson iteration from the linear approximation (tangent line) at the current guess;
- apply the iteration to approximate roots to a prescribed accuracy, stopping when successive iterates agree to the required decimal places;
- describe the failure modes (overshooting, cycling, diverging) and explain geometrically why they occur;
- formulate the implied-volatility root-finding problem and describe how Newton's method solves it.
In this lesson
- Geometric derivation
- Convergence and stopping criterion
- Worked examples
- Failure modes
- Newton's method in quantitative finance
Before you start
- The Derivative as a Limit: Rates of Change and the Derivative Function
- Related Rates, Linear Approximations, and Differentials
- Continuity, the Intermediate Value Theorem, and the Extreme Value Theorem
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