First-order Greek · ∂V/∂r

Rho: an option's sensitivity to interest rates

Rho is the change in an option's value for a one-percentage-point change in the risk-free rate. It's the quietest of the first-order Greeks, until you hold long-dated options.

The formula

d₁ = [ ln(S/K) + (r + σ²/2)·T ] / (σ√T)
d₂ = d₁ − σ√T

ρ_call = K·T·e^−rT·N(d₂)
ρ_put = −K·T·e^−rT·N(−d₂)

Black-Scholes, European options, no dividends. S spot, K strike, T years to expiry, r the risk-free rate, σ implied volatility, N(·) the standard normal CDF and φ(·) its density.

Per 1.00 change in rates; divide by 100 for a one-point move.

Why calls like higher rates

A call lets you buy later instead of now, so you keep the cash and earn interest on it meanwhile; higher rates make that worth more. A put is the mirror image: it pays the strike later, and that payment is worth less in today's money when rates are higher.

When it matters

Rho scales with T. For a same-day or one-week option it rounds to nothing; for LEAPS and long-dated index options it can be a real risk. That's why this site, built for 0DTE and short-dated index flow, gives it a short page.

NDX implied close distribution from the option chain, with the numbers behind it, in Zanshin Flow — Where the option chain prices the NDX close, and how that moved.

Questions

Do interest rates affect 0DTE options?

Barely. Rho is proportional to time to expiry, so for options expiring today it is close to zero.

Why is put rho negative?

A put's payoff includes receiving the strike at expiry. Higher rates reduce the present value of that payment, so the put is worth less.

How is rho quoted?

Usually per one percentage point change in the risk-free rate, which is the Black-Scholes value divided by 100.

Not financial advice. This article explains how options and dealer hedging work. It is not a recommendation to buy or sell anything.