Exercise: Newton's Method for Implied Vol — Convergence and Edge Cases

Prerequisites: Implied Volatility, Derivation of the Black-Scholes Formula

Problem

1. Convergence. Consider the Newton iteration

Given that $C_{\text{BS}}(\sigma)$ is strictly increasing and strictly convex for $\sigma < \sigma^*$ (where $\sigma^*$ satisfies $d_1(\sigma^*) = 0$) and strictly concave for $\sigma > \sigma^*$, argue from first principles why Newton's method converges for any starting guess $\sigma^{(0)} > 0$ when the market price lies in the arbitrage-free band.

1. Edge case — deep OTM. Consider a $120$-strike call with $S_0 = 100$, $T = 0.25$, $r = 0$, market price $C_{\text{market}} = 0.005$ (very cheap). Compute the vega at $\sigma = 0.15$. Why might Newton's method converge slowly here? What is the practical fix?
2. Edge case — price outside arbitrage bounds. For a $95$-strike call with $S_0 = 100$, $T = 0.25$, $r = 0$, the arbitrage-free band is $\max(100 - 95, 0) = 5 < C_{\text{market}} < 100$. Suppose a stale data feed reports $C_{\text{market}} = 4.50$ (below the lower bound). Describe what Newton's method does. How should a production implementation handle this?
3. Starting point. Manaster and Koehler (1982) suggest initialising with

Apply this heuristic to the call of Part 2. How does it compare to a naive start of $\sigma^{(0)} = 0.2$?

Hint

For part 1, use the fact that a convex, monotone function's Newton iterates from the right of the root decrease monotonically to the root. The concave regime flips the direction. Combining both cases gives global convergence for the bijective $C_{\text{BS}}$.