Option Rho Calculator
Rho is the option Greek that measures sensitivity to interest rates: the change in an option's theoretical value for a 1 percentage point change in the risk-free rate. This calculator uses the standard Black-Scholes closed form for a European call or put. Enter the spot price, strike, time to expiry, volatility and the risk-free rate to see the option's rho. Rho tends to be small for short-dated options and meaningful for long-dated ones.
Black-Scholes rho formula
d2 = (ln(S / K) + (r - 0.5 * sigma^2) * T) / (sigma * sqrt(T))
Call rho = K * T * e^(-rT) * N(d2) / 100
Put rho = -K * T * e^(-rT) * N(-d2) / 100
N(x) is the cumulative standard normal distribution. Dividing by 100 expresses rho per 1 percentage point change in r, the market convention. The rate r and volatility sigma are entered as percentages and converted to decimals internally.
Worked example
For S = 100, K = 100, T = 1 year, sigma = 20%, r = 5%: d2 = (ln(1) + (0.05 - 0.02) * 1) / (0.20 * 1) = 0.03 / 0.20 = 0.15. N(0.15) is about 0.5596. Call rho = 100 * 1 * e^(-0.05) * 0.5596 / 100 = 0.9512 * 0.5596 / 1 = 0.5323 per 1% rate change. A 1% rise in rates lifts this one-year at-the-money call by about $0.53.
Option rho: frequently asked questions
What is option rho?
Rho measures how much an option's theoretical price changes when the risk-free interest rate changes. By convention it is quoted per 1 percentage point (1%) move in the rate. Calls have positive rho, puts have negative rho, because a higher rate raises the present-value benefit of deferring the purchase price under a call and lowers it under a put.
How is rho calculated in Black-Scholes?
For a European call, rho = K * T * e^(-rT) * N(d2). For a put, rho = -K * T * e^(-rT) * N(-d2). Here K is the strike, T is time to expiry in years, r is the continuously compounded risk-free rate, N is the standard normal cumulative distribution, and d2 = (ln(S/K) + (r - 0.5 * sigma^2) * T) / (sigma * sqrt(T)). The result is scaled by 1/100 to express it per 1% rate change.
Why is rho usually the least watched Greek?
For short-dated options, time to expiry T is small, so rho is small: interest rate moves barely affect the price. Rho matters most for long-dated options (LEAPS) and in environments where rates move sharply. Delta, gamma, theta and vega typically dominate day-to-day option pricing.
Sources and method
- U.S. Securities and Exchange Commission investor education on options: Investor.gov: Options.
- Rho is the analytical derivative of the Black and Scholes (1973) European option price with respect to the interest rate; a standard, public mathematical result.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.