Brownian Bridge™

1. The BS vega is

\mathcal{V}=Se^{-qT}\sqrt{T}\,\phi(d_1)

. Why does plain Newton-Raphson misbehave for deep OTM options where

d_1\gg 1

The objective

f(\sigma)

acquires several roots deep OTMThe BS price becomes non-monotone in

\sigma

for deep OTM strikesVega is exponentially small, so the step

-f/\mathcal{V}

overshoots wildlyThe Brenner-Subrahmanyam initial guess is undefined deep OTM

2. Brent needs a bracket

[\sigma_{\mathrm{lo}},\sigma_{\mathrm{hi}}]

with

f(\sigma_{\mathrm{lo}})\,f(\sigma_{\mathrm{hi}})<0

. A robust universal bracket for a valid European-call quote is:

[10^{-8},\,10.0]

—

f

is monotone and the no-arbitrage bounds place the root inside

[0,\,1]

— implied vols above 100% never occur in practice

[\sigma_{\mathrm{ATM}}-0.5,\ \sigma_{\mathrm{ATM}}+0.5]

— centred on ATM volNo fixed bracket exists; it must be found by trial and error

3. Near the root, Newton-Raphson converges quadratically. In terms of correct decimal digits, this means the digit count:

Grows linearly with the iteration numberGrows by a fixed additive constant each iterationDepends on the initial guess but not on

f

Roughly doubles each iteration

4. Because the BS call price is strictly increasing in

\sigma

at every strike, the implied-vol root-finding problem has a solution for any positive market price.

TrueFalse

5. In a hybrid Newton/Brent solver the bracket must stay valid after each iterate

\sigma_{\mathrm{new}}

. The correct update is:

Replace the bracket midpoint regardless of the sign of

f

Replace the endpoint whose

f

-value shares the sign of

f(\sigma_{\mathrm{new}})

Reset the bracket to

[10^{-8},10]

whenever Newton divergesAlways overwrite

\sigma_{\mathrm{hi}}

with

\sigma_{\mathrm{new}}

6. The Brenner-Subrahmanyam guess

\sigma_0\approx\sqrt{2\pi/T}\,C_{\mathrm{mkt}}/(Se^{-qT})

comes from which ATM approximation?

Linearising the ATM price as

C\approx Se^{-qT}\sigma\sqrt{T/(2\pi)}

The Bachelier (normal-model) ATM option price formulaDifferentiating the exact BS ATM price with respect to

\sigma

Put-call parity applied to the at-the-money straddle position

Quiz: Newton-Raphson and Brent for Implied Volatility