Solution: Computing Implied Vol by Hand (ATM approximation)

Exercise: Computing Implied Vol by Hand (ATM approximation)

Part 1: deriving the approximation

Set $K = S_0$ , $r = 0$ . Then $\ln(S_0/K) = 0$ , $(r \pm \sigma^2/2)T = \pm\sigma^2 T/2$ , and

d_1 = \frac{\sigma^2 T/2}{\sigma\sqrt{T}} = \frac{\sigma\sqrt{T}}{2}, \qquad d_2 = -\frac{\sigma\sqrt{T}}{2}

Symmetric around $0$ . Let $x = \sigma\sqrt{T}/2$ . The Black-Scholes call price is

C_{\text{BS}} = S_0(\Phi(x) - \Phi(-x)) = S_0(2\Phi(x) - 1)

Taylor-expand $\Phi(x)$ around $0$ :

\Phi(x) = \frac{1}{2} + \phi(0)x + O(x^3) = \frac{1}{2} + \frac{x}{\sqrt{2\pi}} + O(x^3)

where $\phi(0) = 1/\sqrt{2\pi}$ . Hence

2\Phi(x) - 1 = \frac{2x}{\sqrt{2\pi}} + O(x^3) = x\sqrt{\frac{2}{\pi}} + O(x^3)

Substituting $x = \sigma\sqrt{T}/2$ :

C_{\text{BS}} \approx S_0 \cdot \frac{\sigma\sqrt{T}}{2} \cdot \sqrt{\frac{2}{\pi}} = S_0 \cdot \sigma\sqrt{\frac{T}{2\pi}}

which is the Brenner-Subrahmanyam formula. $\square$

Part 2: inversion

Rearrange for $\sigma$ :

\sigma_{\text{imp}} \approx \frac{C_{\text{market}}}{S_0}\,\sqrt{\frac{2\pi}{T}}

Part 3: applications

Trade 1.

S_0 = 100

T = 0.25

C_{\text{market}} = 4

\sigma_{\text{imp}} \approx \frac{4}{100}\sqrt{\frac{2\pi}{0.25}} = 0.04 \cdot \sqrt{25.133} \approx 0.04 \cdot 5.013 \approx 0.2005

Estimated implied vol $\approx 20\%$ .

Trade 2.

S_0 = 50

T = 1

C_{\text{market}} = 3

\sigma_{\text{imp}} \approx \frac{3}{50}\sqrt{2\pi} = 0.06 \cdot 2.5066 \approx 0.1504

Estimated implied vol $\approx 15.04\%$ .

Part 4: accuracy comparison

Exact value for Trade 2:

\sigma_{\text{imp}} \approx 0.1508

. Our approximation gave

0.1504

. Error of about 0.04 vol points — a

0.3\%

relative error.

The approximation is excellent for ATM options when $\sigma\sqrt{T}$ is small. The next-order correction involves an $x^3$ term, producing relative error on the order of $(\sigma\sqrt{T}/2)^2 / 6$ . For Trade 2 with $\sigma\sqrt{T} \approx 0.15$ , this is $\approx (0.075)^2/6 \approx 10^{-3}$ — matching the observed error magnitude. For deep-ITM or deep-OTM options the approximation fails because $d_1, d_2$ are no longer symmetric around zero and the Taylor expansion is invalid.

Takeaways

Brenner-Subrahmanyam is the quant's mental math. For ATM liquid options, $\sigma_{\text{imp}} \approx (C/S_0)\sqrt{2\pi/T}$ gives a vol estimate within $\sim 1\%$ of the exact value. Handy for sanity-checking a Newton-solver output or for estimating vol without a machine.
The derivation is a first-order Taylor expansion. The approximation is valid for small $\sigma\sqrt{T}$ (short-dated, low-vol). The error grows cubically in $\sigma\sqrt{T}$ , so for year-long deep OTM options the error becomes material.
Vega at ATM $\approx S_0\sqrt{T/(2\pi)}$ . The constant $\sqrt{2\pi}$ in the formula is exactly $1/\phi(0)$ , where $\phi$ is the standard-normal density. Every ATM Black-Scholes calculation traces back to this single constant.