Rho

Formula

ρ_call = K T e^{−rT} N(d2) / 100
ρ_put  = −K T e^{−rT} N(−d2) / 100

d2 = [ln(S/K) + (r − q − σ²/2) T] / (σ √T)

Per 1 percentage point of r. The raw derivative ∂V/∂r is 100 times larger.
q enters d2. It does not appear as its own discount in the rho formula;
∂V/∂q is a different Greek.

Sign

• Call rho is greater than or equal to zero in this model. Put rho is less than or equal to zero. A book that is long calls and short puts of the same strike picks up both.
• The rate inside Black-Scholes is the continuous risk-free rate used to discount the strike, not a credit spread and not the dividend. Changing q instead is not rho.
• The Greeks calculator reports rho per 1 percent with the formulas above. Put-call parity is where that discount shows up as K e^{−rT}.

Pitfalls

• One point of rate, not one point of volatility. The /100 here is the same kind of unit choice as vega, applied to r rather than to σ.
• Short-dated options have small rho because of the factor T. A one-week option barely changes when the rate moves by a point. A five-year option moves much more, in the model.
• The rate is flat and constant. A real curve has many rates. One rho cannot describe a shift of a single tenor, a slope change, and a parallel move as three different risks.

Python reference implementation

Requires scipy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download rho.py · All snippets

from math import exp, log, sqrt
from scipy.stats import norm

def bs_rho(S, K, T, r, sigma, call=True, q=0.0) -> float:
    """Black-Scholes-Merton rho per 1 percentage point of the risk-free rate.

    call=False returns the put rho, which is negative or zero.
    """
    if S <= 0 or K <= 0 or T <= 0 or sigma <= 0:
        raise ValueError("S, K, T and sigma must be positive")
    d2 = (log(S / K) + (r - q - 0.5 * sigma * sigma) * T) / (sigma * sqrt(T))
    if call:
        return float(K * T * exp(-r * T) * norm.cdf(d2) / 100.0)
    return float(-K * T * exp(-r * T) * norm.cdf(-d2) / 100.0)

FAQ

Why is put rho negative?

A higher risk-free rate reduces the present value of the strike the put might receive, and it raises the forward. Both hurt the model value of a European put, so the derivative is negative. A European call is helped by the same two facts.

Does rho include dividend sensitivity?

No. q is held fixed. Rho moves only r. The dividend yield changes d2, so a different q produces a different rho, but the derivative itself is with respect to the interest rate.