Exercise: Newton's Method for Implied Volatility Using Vega

Prerequisites: Vega, Implied Volatility

Problem

Implied volatility is defined implicitly: given an observed market price $C^{\text{mkt}}$, find $\sigma$ such that $C_{\text{BS}}(S, K, T, r, \sigma) = C^{\text{mkt}}$. This is a root-finding problem.

Newton's method iterates:

using vega as the Jacobian.

1. Implement Newton's method for IV. Target tolerance $10^{-6}$ on the price residual. Start with $\sigma_0 = 0.3$.

2. Test on $C^{\text{mkt}} = 10$, $S = 100, K = 100, T = 1.0, r = 0.05, q = 0$. Report the converged IV and number of iterations.

3. Convergence analysis. Newton's method converges quadratically near the root. Verify this by reporting $|\sigma_n - \sigma^*|$ at each step.
4. Failure modes. Try applying Newton's method with a pathological starting point ($\sigma_0 = 0.01$, very low). What happens? Why?
5. Brenner-Subrahmanyam approximation. As a warm-start, use the approximation $\sigma_0 \approx \sqrt{2\pi/T}\cdot C^{\text{mkt}}/S$ (for ATM options). Compute $\sigma_0$ for our example and see how many iterations Newton needs with this warm-start.

Hint

For numerical stability, cap updates: if $|\Delta\sigma| > 0.5$, truncate to $0.5$ to avoid overshooting. Newton's method can struggle in the low-vega region (very-OTM or very-ITM options); for our ATM test case, it should work cleanly.