Newton-Raphson and Brent for Implied Volatility

Setup

Market Context

Implied volatility $\sigma^*(K, T)$ is the unique value of $\sigma$ such that the Black-Scholes price matches the observed market price of an option with strike $K$ and expiry $T$. It is the standard language in which option markets quote prices: traders say "I buy the 1-month 25-delta put at 18 vol", not "at price €3.47".

Computing implied vol is therefore an inversion problem, performed thousands of times per second in a live trading system. Correctness and speed both matter. A slow inverter causes pricing latency; a numerically fragile one produces garbage implied vols for deep OTM strikes, polluting the vol surface.

Conventions

Throughout this module:

• $C_{\mathrm{BS}}(\sigma; S, K, r, q, T)$: Black-Scholes call price, where $S$ is spot, $K$ is strike, $r$ is the continuously compounded risk-free rate, $q$ is the continuous dividend yield, $T$ is time to expiry in years (Act/365 unless stated).
• Volatility $\sigma$ is annualised, in decimal ($0.20 = 20\%$).
• $F = S e^{(r-q)T}$: the forward price.
• $d_1 = \frac{\ln(F/K) + \frac{1}{2}\sigma^2 T}{\sigma\sqrt{T}}$, $\quad d_2 = d_1 - \sigma\sqrt{T}$.

The Black-Scholes call price:

$C_{\mathrm{BS}}(\sigma) = e^{-rT}\bigl[F \Phi(d_1) - K \Phi(d_2)\bigr].$

Assumptions before use:

1. Constant volatility $\sigma$ over $[0, T]$.
2. Lognormal spot dynamics under the risk-neutral measure.
3. No early exercise (European option).
4. Liquid, arbitrage-free market quote $C_{\mathrm{mkt}}$ with $C_{\mathrm{mkt}} \in \bigl((F - K)^+\, e^{-rT},\; S\, e^{-qT}\bigr)$ — the no-arbitrage bounds.

Theory: Implied Vol as a Root-Finding Problem

Define the objective function:

$f(\sigma) = C_{\mathrm{BS}}(\sigma) - C_{\mathrm{mkt}}.$

We seek $\sigma^* > 0$ such that $f(\sigma^*) = 0$. This function has the following properties that govern algorithm design:

1. Monotonicity. For $\sigma > 0$, $C_{\mathrm{BS}}$ is strictly increasing in $\sigma$: $f'(\sigma) = \mathcal{V} > 0$, where $\mathcal{V} = S e^{-qT} \sqrt{T}\, \phi(d_1)$ is the Black-Scholes vega and $\phi = \Phi'$ is the standard normal density. Monotonicity guarantees at most one root.
2. Boundary behaviour. $C_{\mathrm{BS}}(0) = e^{-rT}(F - K)^+$ (intrinsic value) and $C_{\mathrm{BS}}(\infty) = S e^{-qT}$ (full asset price). For $C_{\mathrm{mkt}}$ in the arbitrage-free interior, a root exists.
3. Vega degeneracy. At extreme strikes ($|d_1| \to \infty$), $\phi(d_1) \to 0$, so $\mathcal{V} \to 0$. Newton-Raphson $\sigma_{n+1} = \sigma_n - f(\sigma_n)/f'(\sigma_n)$ divides by a near-zero vega and can produce catastrophic oscillations or divergence.

Newton-Raphson

Algorithm

Starting from an initial guess $\sigma_0$, Newton-Raphson iterates:

$\sigma_{n+1} = \sigma_n - \frac{f(\sigma_n)}{f'(\sigma_n)} = \sigma_n - \frac{C_{\mathrm{BS}}(\sigma_n) - C_{\mathrm{mkt}}}{\mathcal{V}(\sigma_n)}.$

Convergence rate. Near a simple root, Newton-Raphson converges quadratically: if $|\sigma_n - \sigma^*| = \varepsilon$, then $|\sigma_{n+1} - \sigma^*| = O(\varepsilon^2)$. The number of correct decimal digits doubles per iteration. For a good initial guess, 3–5 iterations suffice.

Formal proof. By Taylor expansion around $\sigma^*$:

$f(\sigma_n) = f(\sigma^*) + f'(\sigma^*)(\sigma_n - \sigma^*) + \tfrac{1}{2}f''(\xi)(\sigma_n - \sigma^*)^2 = f'(\sigma^*)(\sigma_n - \sigma^*) + O((\sigma_n - \sigma^*)^2),$

since $f(\sigma^*) = 0$. Thus:

$\sigma_{n+1} - \sigma^* = \sigma_n - \sigma^* - \frac{f(\sigma_n)}{f'(\sigma_n)} = -\frac{f''(\xi)}{2f'(\sigma_n)}(\sigma_n - \sigma^*)^2 + O((\sigma_n - \sigma^*)^3).$

The error is bounded by $|e_{n+1}| \leq C |e_n|^2$ where $C = |f''(\sigma^*)| / (2|f'(\sigma^*)|)$ is the asymptotic constant.

Initial Guess

A poor initial guess causes slow convergence or divergence. The Brenner-Subrahmanyam (1988) approximation for near-ATM options provides a first-order starting point:

$\sigma_0 \approx \sqrt{\frac{2\pi}{T}} \cdot \frac{C_{\mathrm{mkt}}}{S e^{-qT}}.$

This is a linearisation of the ATM Black-Scholes formula $C_{\mathrm{ATM}} \approx S e^{-qT} \sigma \sqrt{T/(2\pi)}$. For skewed strikes, a better guess uses the Corrado-Miller approximation.

Failure Modes

Deep OTM puts and calls. When $|d_1| > 4$, the vega is below $10^{-8}$ per unit notional. Newton-Raphson steps become enormous and overshoot wildly. The algorithm must detect this case and switch strategies.

Near-zero time to expiry. As $T \to 0$, the whole vol surface compresses. Vega scales as $\sqrt{T} \to 0$. Newton-Raphson becomes unreliable for $T < 0.001$ years (less than one trading day).

Near-intrinsic options. If $C_{\mathrm{mkt}} \approx (F - K)^+ e^{-rT}$ (i.e., the option is priced at near-intrinsic), the root is near $\sigma = 0$ and both $f(\sigma)$ and $\mathcal{V}(\sigma)$ are near zero simultaneously. l'Hôpital's rule applies in the limit but the numerical ratio is unstable.

Brent's Method

Brent's method (1973) combines three strategies: bisection, secant, and inverse quadratic interpolation (IQI). It is guaranteed to converge to machine precision within $O(\log_2(1/\varepsilon))$ function evaluations, making it the default fallback.

Bracketing

Brent requires a bracket $[\sigma_{\mathrm{lo}}, \sigma_{\mathrm{hi}}]$ with $f(\sigma_{\mathrm{lo}}) \cdot f(\sigma_{\mathrm{hi}}) < 0$. Since $f$ is monotone and bounded:

$\sigma_{\mathrm{lo}} = 10^{-8} \quad (\text{near zero but finite}), \qquad \sigma_{\mathrm{hi}} = 10.0 \quad (1000\%\ \text{vol}).$

This bracket is valid for any arbitrage-free market quote. Initial $f(\sigma_{\mathrm{lo}}) < 0$ (intrinsic value is below the market) and $f(\sigma_{\mathrm{hi}}) > 0$ (deep-ITM call at 1000 vol recovers full asset value), so the sign condition is satisfied.

Algorithm Sketch

Maintain three points $a$, $b$, $c$ with $f(a) \cdot f(b) \leq 0$, $|f(b)| \leq |f(a)|$. At each step:

1. IQI if the last two steps used secant and the three points are distinct: fit a quadratic through $(a, f(a))$, $(b, f(b))$, $(c, f(c))$ and take the root.
2. Secant if the last step was not secant: use the linear interpolant through $(a, f(a))$, $(b, f(b))$.
3. Bisection if the candidate from steps 1/2 lies outside $(b, (a+b)/2)$ or convergence is slow: take the midpoint.