Define the model and the rate convention
The examples use European calls and puts in the Black-Scholes-Merton stock model with constant continuous dividend yield. Let S be spot, K strike, T remaining years, σ annual volatility, r the continuously compounded annual interest rate, and q the continuous annual dividend yield. Rates and volatility are decimals in formulas: 5% becomes 0.05. N is the standard normal cumulative distribution function.
- d1 = [ln(S/K) + (r - q + σ²/2)T] / (σ√T), and d2 = d1 - σ√T. Call C = S × exp(-qT) × N(d1) - K × exp(-rT) × N(d2). Put P = K × exp(-rT) × N(-d2) - S × exp(-qT) × N(-d1).
- Rho per percentage point is 0.01 × ∂V/∂r. Call rho = 0.01 × K × T × exp(-rT) × N(d2). Put rho = -0.01 × K × T × exp(-rT) × N(-d2). A library returning the raw derivative will report a number 100 times larger.
- For a small rate change, ΔV ≈ rho × Δr_pp, where Δr_pp is measured in percentage points. Equivalently, use the raw derivative times the decimal rate change. Do not multiply a per-point rho by 0.01 again for a one-point move.
| Rate change from 5% | Change in points | Rho multiplier |
|---|---|---|
| To 6% | +1.00 | +1.00 |
| To 5.25% (25 basis points) | +0.25 | +0.25 |
| To 5.05% (1% relative increase) | +0.05 | +0.05 |
| To 4.75% | -0.25 | -0.25 |