Brenner-Subrahmanyam ATM Implied Volatility Approximation

Scenario

In rapid volatility market making, traders require an immediate mental rule-of-thumb to invert the Black-Scholes formula for at-the-money options without running numerical root-finding algorithms.

Your Task

1. Starting with the Black-Scholes pricing formula for an ATM European call option ($S_0 = K$) with interest rate $r = 0$, expand the cumulative normal distribution function $\Phi(x)$ in a first-order Taylor series around $x = 0$.
2. Derive the celebrated Brenner-Subrahmanyam approximation formula: $\sigma \approx \sqrt{\frac{2\pi}{T}} \frac{C_{\text{atm}}}{S_0} \approx 2.5066 \frac{C_{\text{atm}}}{S_0 \sqrt{T}}$
3. State the common practitioner rule-of-thumb for 1-month options ($T \approx 1/12$), and benchmark the analytical approximation against exact Black-Scholes implied volatility for $S_0 = 100, K = 100, T = 0.25, C_{\text{atm}} = 4.00$.

Constraints

• Exact ATM condition $S_0 = K$.
• Zero interest rate $r = 0$.

Examples For $S_0 = 100, T = 0.25, C = 4.00$, the approximation gives $\sigma \approx \frac{2.5066 \times 4.00}{100 \times 0.5} = \frac{10.0264}{50} = 20.05\%$.