Root finding, with implied volatility as the worked case

Every options system solves the same root-finding problem millions of times a day: given a price, find the volatility. It is the canonical numerical-methods interview question at a market-making firm, because the answer has to be both fast and impossible to break.

Aspect	Bisection	Newton–Raphson	Secant
Needs	A bracket where the sign changes	The derivative	Two starting points
Convergence	Linear — one bit per step	Quadratic	About 1.6 (superlinear)
Can it fail?	Never, given a valid bracket	Yes — can diverge or cycle	Yes, but less dramatically
Cost per step	One function evaluation	Function and derivative	One function evaluation

Table 1.1 · The three methods. Bisection halves the interval every step, so it needs about 50 iterations for double precision from a unit bracket. Newton typically needs four.

Equation 1.2

Newton–Raphson

Step to where the tangent line crosses zero. Near a simple root the error squares each iteration, which is why the digit count roughly doubles per step.

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

$f'(x_n)$: The derivative — for implied volatility this is vega, which you already compute.

Derivation 1.3

Inverting Black–Scholes

The problem is to find the $\sigma$ that reproduces an observed price.

$f(\sigma) = C_{\text{BS}}(\sigma) - C_{\text{market}}$
Define the residual whose root we want.
$f'(\sigma) = \mathcal{V} = S_0\sqrt{T}\,\varphi(d_1) > 0$
The derivative is vega, which is strictly positive for a live option.
$\sigma_{n+1} = \sigma_n - \frac{C_{\text{BS}}(\sigma_n) - C_{\text{market}}}{\mathcal{V}(\sigma_n)}$

\text{a unique root, since } C_{\text{BS}} \text{ is strictly increasing in } \sigma

Figure 1.4 · Quadratic against linear convergence. Newton squares the error each step, so the number of correct digits doubles: five iterations take you from one digit to sixteen. Bisection halves it, gaining a third of a digit a step — reliable, never fast, and the right fallback when the derivative misbehaves.

Why this particular inversion is well behaved. Price is strictly increasing in volatility, so the root is unique and bracketing is trivial: zero volatility gives the intrinsic value, and a large volatility gives nearly the spot. Vega is available for free from the same computation that produced the price. And near the money the relationship is close to linear, so the first Newton step from a sensible guess is usually within a fraction of a volatility point.

Root finding, with implied volatility as the worked case

After this lesson you should be able to

Newton–Raphson

Inverting Black–Scholes

The rest of this lesson is in Premium