Read interest-rate sensitivity without losing the units.
Learn Options and GreeksAdvanced4 min
European options; other inputs fixed
Plain English
The idea
Rho measures how option value changes when the model interest-rate input changes, with other inputs held still.
The effect of a rate change depends on the maturity and other contract inputs. To compare it with price, volatility or time effects, first specify a shock for each and convert them into value contributions.
The raw Black–Scholes derivative is measured per 1.00 of decimal rate. One percentage point is 0.01, so a per-point quotation is the raw sensitivity divided by 100. The derivative unit is not a proposed large rate shock.
Rho measures sensitivity to the risk-free rate input of the pricing model. It does not measure the complete effect of a central-bank announcement, which can also change the underlying price, volatility and expectations about several future rates.
The derivative here uses a continuously compounded annual rate expressed as a decimal. A platform may quote the result per percentage point instead, so a bare rho number is incomplete without its units and the direction of the position.
Worked Example
Reading the rate unit
Suppose the raw rho of a call is 20.00. A one percentage-point rate rise is 0.01 in decimal terms.
The first-order effect is therefore about 0.20, before other effects are considered. A platform may show the same sensitivity as 0.20 per rate point.
That does not mean rates will rise. It only translates the option value sensitivity to the model rate input.
Take a hypothetical raw call rho of £20 per option unit per 1.00 change in the decimal rate. A small rate-input increase from 3% to 3.1% is 0.001, so the local value contribution is about £0.02. A one-percentage-point change from 3% to 4% is 0.01, giving a first-order estimate of £0.20.
These are sensitivity calculations with other inputs fixed, not predictions about interest rates or option prices. The larger change may have greater approximation error because rho changes as the rate input changes.
Raw call rho
20 pounds per 1.00 decimal rate
Small rate change
3% to 3.1%, or +0.001
Small-shock value estimate
About +0.02 pounds per unit
Per-point call rho
0.20 pounds per percentage point
Reading the result
Why calls and puts differ
In the non-dividend European model, the strike is paid or received at expiry if exercise occurs. Discounting that future amount helps explain why a higher rate increases the long call's model value and decreases the long put's model value, with other inputs fixed.
The signs concern the long option. A short position reverses its exposure, and a portfolio combines the signed sensitivities of its positions. Raw rho should not be compared directly with daily theta or per-point vega: convert each into a value contribution using a specified shock and consistent position size first.
Limits and assumptions
One rate input simplifies a richer problem
A single constant rate stands in for the model's financing assumption. Actual discount curves can vary by maturity, and real markets may involve dividends, financing spreads and different contract terms. The simple expression also assumes positive underlying price, strike, volatility and time, European exercise and no dividends; time is in years.
A sensitivity per 1.00 of decimal rate does not invite a linear forecast for a 100-percentage-point rate jump. It is the unit of the derivative. A central-bank move and the relevant maturity's market rate can also differ, so inserting a policy announcement into the model without explanation can misstate the intended scenario.
Common Mistake
Comparing raw and point units
A common mistake is to compare one source quoting raw rho with another source quoting per percentage point. The numbers will differ by a factor of 100.
Another mistake is to treat rho as always unimportant. It may be modest for some short-dated options, but rate sensitivity becomes more relevant as expiry lengthens.
The unit conversion should be written before the answer. Multiplying a raw rho by “1” when intending one percentage point makes the result one hundred times too large, even though the arithmetic itself looks simple.
Formula Summary
Black-Scholes rho and rate-point quoting
European exercise, no dividends, positive S, K, sigma and T; time T is in years, sigma is an annualised decimal and r is a continuously compounded annual rate. Rates and volatility are constant model inputs; jumps and trading frictions are excluded. Do not apply these expressions directly at expiry or zero volatility.
N is the cumulative standard normal distribution; n is its density. The formulas describe long-option model values; signed positions change the exposure.
Call Rho
Call rho equals strike times time to expiry times discount factor times cumulative normal d two.
Strike price.
Time to expiry in years.
Continuously compounded risk-free rate in this convention.
Cumulative standard normal distribution at d2.
Put Rho
Put rho equals negative strike times time to expiry times discount factor times cumulative normal negative d two.
Unit Conversion
Rho per one percentage-point rate move equals raw rho divided by one hundred.
A one percentage-point move is 0.01 in decimal rate units.
Optional derivationExplore the derivation
The compact derivation starts from the call price, uses the same density identity, then gets put rho from put-call parity.
01
Differentiate the call price with respect to r
The rate appears inside d1 and d2, and also inside the discounted strike term.
Rate Derivative
Derivative of call price with respect to r equals S times normal density d one times derivative of d one with respect to r minus strike times discount factor times normal density d two times derivative of d two with respect to r plus strike times time to expiry times discount factor times cumulative normal d two.
02
Use the shared rate derivative
Both d1 and d2 move by the same amount when r changes because d2 equals d1 minus a term that does not contain r.
Shared Derivative
Derivative of d one with respect to r equals derivative of d two with respect to r equals square root of T divided by sigma.
03
Use the density identity
The scaled density terms cancel, leaving the term from differentiating the discount factor.
Cancellation Identity
S times normal density d one equals strike times discount factor times normal density d two.
04
Read the result and convert units
Put rho follows from put-call parity. The raw result is per 1.00 rate move, so per-point quoting divides the raw number by 100.
Rho Result
Call rho equals strike times time to expiry times discount factor times cumulative normal d two. Put rho equals negative strike times time to expiry times discount factor times cumulative normal negative d two. Rho per point equals raw rho divided by one hundred.
Self-check
Check your understanding
What is the decimal-rate change from 3% to 3.1%?
0.001, or one tenth of a percentage point. At the example’s raw rho of £20, the first-order contribution is about +£0.02 per option unit.
Why does a one-percentage-point shock multiply raw rho by 0.01?
Rates are expressed as decimals in the derivative. One percentage point is 0.01, so raw rho 20 corresponds to a per-point sensitivity of 0.20.
Does positive call rho mean an interest-rate announcement must raise the call’s market price?
No. The announcement may also affect the underlying, volatility and other inputs. Rho isolates one model input while holding the rest fixed.
General learning context
Where this helps a public investor
Rho helps readers understand why rate assumptions matter more for some option expiries than others.
Disclaimer
Educational Use Only
This article is for informational and educational purposes only. Options involve risk and are not suitable for every investor. Nothing here is a recommendation to buy, sell, write, or trade an option.
