Financial Modeling
Brenner-Subrahmanyam ATM Implied Volatility & Straddle Breakeven Formula
Extract At-The-Money implied volatility directly without Goal Seek using Brenner and Subrahmanyam’s (1988) SQRT(2*PI()/T) * (C_ATM / S) approximation, plus straddle breakevens and daily 1-SD move.
Why SQRT(2*PI()/T) * (C/S) inverts Black-Scholes analytically at the money
Black-Scholes has no general closed-form inverse for arbitrary strikes, tempting analysts to run manual Goal Seek macros cell by cell. However, when strike K equals discounted forward price (zero-carry ATM where S = K), first-order Taylor expansion of N(0.5 * sigma * SQRT(T)) - N(-0.5 * sigma * SQRT(T)) around zero yields Menachem Brenner and Marti Subrahmanyam’s (1988) famous linear identity: C_ATM / S = sigma * SQRT(T / (2 * PI())), which inverts directly to sigma_ATM = SQRT(2 * PI() / T) * (C_ATM / S).
Notice where floor traders’ mental shortcut “IV = 2.5 * (Call% / SQRT(T))” comes from: SQRT(2 * PI()) = 2.506628! On a 100.00 underlying with a 3-month (T = 0.25 yr) ATM call trading at 4.00 (and straddle 8.00), Brenner-Subrahmanyam gives 2.506628 * (4 / 100) / 0.5 = 20.05% annualized implied volatility and an implied daily 1-SD move of 100 * (0.2005 / SQRT(252)) = 1.26.
| Options Desk Metric (S=100, C_ATM=4.00, T=0.25y) | Formula | Result |
|---|---|---|
| Brenner-Subrahmanyam ATM IV | SQRT(2 * PI() / T) * (C_ATM / S) | 0.2005 (20.05% annualized) |
| Implied 1-Day 1-SD Move | S * (sigma_ATM / SQRT(252)) | 1.26 points |
| ATM Straddle Upper / Lower Breakeven | S + 2*C_ATM / S - 2*C_ATM | 108.00 / 92.00 (+/-8.00%) |
Brenner-Subrahmanyam Closed-Form ATM Implied Volatility Formula
Guard T <= 0 or S <= 0, and multiply SQRT(2 * PI() / B4) by ATM call-to-spot ratio B3 / B2.
=IF(OR(B2 <= 0, B4 <= 0), "INVALID_INPUT", ROUND(SQRT((2 * PI()) / B4) * (B3 / B2), 4))Daily 1-SD Expected Move & Straddle Percentage Breakeven
Convert annualized IV to daily standard deviation via SQRT(252) trading days, and compute 2 * C_ATM / S for the zero-carry ATM straddle percentage move.
=ROUND(B2 * (E2 / SQRT(252)), 2)Try the example
Use the original values or change them to test your understanding.