Rho (sensitivity to interest rates)
Rho is the option greek that measures how much an option's value changes for a one-percentage-point change in the risk-free interest rate. It is positive for calls and negative for puts on shares or an index, grows with time to expiry, and is usually the least important greek for short-dated index options.
Senzoukria · Glossary · Updated September 2026
At a glance
- Call (Black-Scholes)
- ρ = K·τ·e^(−rτ)·N(d2) per 1.00 of rate
- Put (Black-Scholes)
- ρ = −K·τ·e^(−rτ)·N(−d2)
- Usual quote
- Per 1 percentage point of rate (divide by 100)
Why rates move option prices
In a no-arbitrage model the holder of a call defers paying the strike until exercise, and the holder of a put defers receiving it. A higher rate raises the forward price of the underlying and lowers the present value of the strike, which increases call values and decreases put values. Rho measures that effect: the change in value for a one-point change in the continuously compounded risk-free rate, holding spot, volatility and time fixed.
Worked example
An at-the-money six-month call with spot and strike 100, implied volatility 20% and a 4% rate is worth about 6.63 in Black-Scholes. Its rho is 0.259 per rate point, so a rate of 5% instead of 4% adds roughly 0.26. The matching put has a rho of −0.231. The same call with 30 days left has a rho of only 0.042: over a month, a full point of rate changes the price by about four cents on a 100 underlying, much less than a single volatility point would.
| Contract | Days to expiry | Rho per rate point |
|---|---|---|
| Call | 182 | +0.259 |
| Put | 182 | −0.231 |
| Call | 30 | +0.042 |
Options on futures behave differently
For an option on a futures contract priced with Black-76, the futures price is the input and the rate only discounts the premium. Rho then equals −τ times the option value for both calls and puts. For a one-week at-the-money E-mini option worth about 49.7 index points, that is roughly −0.0095 points per rate point, a negligible amount. The rate still matters to futures traders through the basis between the future and the cash index, which is where a change in financing cost shows up.
Where rho matters and where it is assumed
- Long-dated options (LEAPS) and deep in-the-money options carry the largest rho.
- Put-call parity, implied forwards and implied volatility solvers all need a rate input; a wrong rate shifts the implied volatility of calls and puts in opposite directions.
- Senzoukria does not display rho. Its own greek computations, such as the vanna and charm exposures and the locally computed Black-Scholes greeks on the Databento path, use fixed assumed rate constants rather than a market rate, and the vanna and charm snapshot records the assumed rate and dividend yield so they cannot be mistaken for measurements.
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Frequently asked questions
- Why is rho usually ignored for 0DTE and weekly options?
- Because rho scales with time to expiry. With a few days left, a full percentage point of rate changes the value by a fraction of a cent to a few cents, far below the bid-ask spread, while gamma and theta change it by much more every hour.
- Does rho explain why call and put implied volatilities differ?
- It can contribute. If the rate or dividend assumed by a calculator differs from the one embedded in market prices, parity is violated in the model and the solver assigns different implied volatilities to the call and the put of the same strike. Checking the implied forward is the usual way to catch it.