Exercise: Computing Implied Vol by Hand (ATM approximation)

Prerequisites: Implied Volatility, Derivation of the Black-Scholes Formula

Problem

The Brenner-Subrahmanyam (1988) approximation for the at-the-money (ATM) call price is:

1. Derive this approximation by evaluating $C_{\text{BS}}(S_0, K = S_0, T, r = 0, \sigma) = S_0(\Phi(d_1) - \Phi(d_2))$ at the ATM strike, then Taylor-expanding around $\sigma\sqrt{T} = 0$ to leading order.
2. Invert the approximation to get $\sigma_{\text{imp}} \approx (C_{\text{market}}/S_0)\sqrt{2\pi/T}$.
3. Apply the approximation to the following trades:
   • ATM call, $S_0 = 100$, $T = 0.25$ years, $r = 0$, $C_{\text{market}} = 4.00$. Estimate $\sigma_{\text{imp}}$.
   • ATM call, $S_0 = 50$, $T = 1.0$ years, $r = 0$, $C_{\text{market}} = 3.00$. Estimate $\sigma_{\text{imp}}$.
4. For the second trade, the exact implied vol (from numerical inversion) is $\sigma_{\text{imp}} \approx 0.1508$. Compare to your approximation and comment on the accuracy.

Hint

For part 1, at $K = S_0$ and $r = 0$, note that $d_1 = \sigma\sqrt{T}/2$ and $d_2 = -\sigma\sqrt{T}/2$ — symmetric around $0$. Use $\Phi(x) - \Phi(-x) = 2\Phi(x) - 1 \approx 2\phi(0)x = \sqrt{2/\pi}\, x$ for small $x$.