Numerical methods

Newton–Raphson and implied volatility

Inverts a price for a volatility in four or five steps, using vega as the derivative.

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}, \qquad \sigma_{n+1} = \sigma_n - \frac{C_{\text{BS}}(\sigma_n) - C_{\text{mkt}}}{\mathcal{V}(\sigma_n)}

Where

$\mathcal{V} = S_0\sqrt{T}\,\varphi(d_1)$: Vega, strictly positive for a live option.

Assumptions

Price is strictly increasing in $\sigma$ , so the root is unique and always bracketable.
Vega vanishes in the wings, so pure Newton breaks there.

Sanity check. Production code is Newton with a maintained bracket and a bisection fallback. Never Newton alone.

Where this is taught

Root finding, with implied volatility as the worked case · NUM · Root finding