A call is quoted at 5.50. Your model at 20% volatility gives 5.00 with a vega of 0.39 per volatility point. Estimate the implied volatility with one Newton step, and say when that step is a poor estimate.

A call is quoted at 5.50. Your model at 20% volatility gives 5.00 with a vega of 0.39 per volatility point. Estimate the implied volatility with one Newton step, and say when that step is a poor estimate.

Approach: Treat the price as a function of volatility and take one Newton step, dividing the price error by vega. Then ask where that function stops being close to linear.

21.28%. One Newton step is sigma_new = sigma_old + (market - model)/vega = 20 + 0.50/0.39 = 21.28%. The step is exact only if the price is linear in volatility over that range, which holds well at the money where volga is near zero, so a single iteration usually lands within a hundredth of a point on an at-the-money option. It degrades badly for deep out-of-the-money strikes, where vega is tiny and volga is large: the price is convex in volatility there, so the step overshoots, and dividing by a vega near zero amplifies any rounding in the quoted price into several volatility points of error. The practical fix is to seed the search with a closed-form approximation, cap the step size, and fall back to a bisection between bounds when vega falls below a threshold.

Follow-up: The quoted price sits below the intrinsic value bound by half a tick. What should the implied volatility routine return?

Key concepts: implied volatility, Newton method, vega, volga.