Quantitative Finance & Derivatives
Corrado-Miller Closed-Form Implied Volatility for Non-ATM Options
Extract implied volatility away from the money without Solver using the Corrado-Miller (1996) quadratic moneyness correction and spot why ATM Brenner-Subrahmanyam overstates ITM vol.
See why ATM shortcuts explode on in-the-money options
The classic Brenner-Subrahmanyam shortcut SQRT(2*PI()/T) * C / S assumes spot S equals discounted strike X = K*EXP(-r*T). If spot S = 105 (B2), strike K = 100 (B3), risk-free rate r = 0.04 (B4), time T = 0.5 (B5), and observed call price C = 9.50 (B6), discounted strike X = 98.0199 so intrinsic forward moneyness S - X = 6.9801 is already embedded in C. Feeding C = 9.50 into the ATM shortcut hallucinates 32.07% volatility when true Black-Scholes volatility is ~18.42%!
Corrado and Miller (1996) expand N(d1) and N(d2) around (S + X)/2 and solve the resulting quadratic explicitly: sig_CM = (SQRT(2*PI()/T) / (S + X)) * (C - (S - X)/2 + SQRT((C - (S - X)/2)^2 - (S - X)^2 / PI())). On our inputs, C - (S - X)/2 = 6.0099, radicand = 20.6105, and sig_CM = 0.1842 (18.42%).
| Estimator | Formula Logic | Implied Vol on S=105, K=100, C=9.50 |
|---|---|---|
| Naive ATM Shortcut (Bug) | SQRT(2 * PI() / T) * C / S | 0.3207 (Overstated by +13.65 pts!) |
| Corrado-Miller (1996) | Quadratic moneyness root around (S + X) / 2 | 0.1842 (Matches Black-Scholes ~18.4%) |
| Exact ATM Limit (S = X) | Reduces identically to SQRT(2*PI()/T) * C / S | Exact continuity at S = K*EXP(-r*T) |
Corrado-Miller Quadratic Implied Volatility Formula
Guard no-arbitrage bounds (C < MAX(0, S - X)) and the deep-ITM quadratic discriminant rad_cm < 0 before taking SQRT.
=LET(s_px, B2, k_str, B3, r_rf, B4, t_yr, B5, c_obs, B6, x_pv, k_str * EXP(-r_rf * t_yr), m_gap, s_px - x_pv, adj_c, c_obs - 0.5 * m_gap, rad_cm, adj_c^2 - (m_gap^2) / PI(), IF(OR(t_yr <= 0, c_obs < MAX(0, m_gap)), "Arbitrage Violation", IF(rad_cm < 0, "Deep-ITM Radicand Trap", ROUND((SQRT(2 * PI() / t_yr) / (s_px + x_pv)) * (adj_c + SQRT(rad_cm)), 4))))Naive ATM Mispricing Comparison Formula
Place the naive ATM formula alongside Corrado-Miller and verify convergence at the money: if you set S = 98.0199 (or K = 105*EXP(0.02)) with C = 7.40, m_gap -> 0 and both formulas converge.
=ROUND(SQRT(2 * PI() / B5) * (B6 / B2), 4)Try the example
Use the original values or change them to test your understanding.