Quantitative Finance
Bachelier Normal ATM Option Price & Straddle Breakeven Formula
Price at-the-money options on interest rates, spreads, and commodities that can trade near or below zero using Bachelier’s (1900) arithmetic normal formula C_ATM = DF * sigma_N * SQRT(T / (2 * PI())).
Price rate swaptions, yield-curve spreads, and negative-capable futures cleanly!
Black-76 lognormal option pricing breaks down whenever underlying yields, swap spreads, or energy futures approach or cross zero! Bachelier’s (1900) arithmetic Brownian motion model quotes basis-point or dollar normal volatility sigma_N directly in price units per root-year, remaining 100% well-defined across positive, zero, and negative strikes!
At the forward strike K = F_0 (with discount factor DF = EXP(-r * T)), the exact closed-form Bachelier ATM Call (and Put) premium simplifies to C_ATM = P_ATM = DF * sigma_N * SQRT(T / (2 * PI())), while the undiscounted ATM Straddle Breakeven Move equals 2 * sigma_N * SQRT(T / (2 * PI())) ≈ 0.7979 * sigma_N * SQRT(T)!
| Expiry T (sigma_N = 12.00, r = 0%) | Undiscounted ATM Call / Put | ATM Straddle Breakeven (+/-) | Ratio to 1-SD Move |
|---|---|---|---|
| 3 Months (T = 0.25 yr) | 2.3937 | +/- 4.7873 | 79.79% of 6.00 SD |
| 1 Year (T = 1.00 yr) | 4.7873 | +/- 9.5746 | 79.79% of 12.00 SD |
| Discounted 1Y (r = 4.00%) | 4.5996 | +/- 9.5746 (terminal) | DF = 0.9608 |
Discounted Bachelier ATM Call / Put Premium
Computes exact EXP(-r * T) * sigma_N * SQRT(T / (2 * PI())) from normal volatility B3, expiry T (B4), and risk-free rate B5 (works for any positive, zero, or negative forward B2).
=IF(OR(B3 <= 0, B4 <= 0), "Invalid Vol or Expiry", ROUND(EXP(-B5 * B4) * B3 * SQRT(B4 / (2 * PI())), 4))Terminal ATM Straddle Breakeven Distance (+/- Price Units)
Calculates the terminal forward move required at expiry to break even on an undiscounted ATM straddle (Call + Put).
=IF(OR(B3 <= 0, B4 <= 0), "Invalid Vol or Expiry", ROUND(2 * B3 * SQRT(B4 / (2 * PI())), 4))Try the example
Use the original values or change them to test your understanding.