Rho Calculator (Options)
This calculator computes the Rho of a European option under the Black‑Scholes framework. Rho is the partial derivative of the option price with respect to the continuously compounded risk‑free interest rate and expresses how the option price changes when rates move.
The tool returns the Black‑Scholes call price and both call and put Rho values expressed per unit change in the annual rate (i.e. per 1.0 = 100 percentage points) and per one percentage point change (multiply per unit by 0.01). Use inputs for underlying price, strike, days to expiry, annual volatility, risk‑free rate and dividend yield.
Inputs
Results
Black‑Scholes call price
—
Call Rho (change in price per 1.0 = 100% point change in r)
—
Call Rho (per 1 percentage point change in r)
—
Put Rho (change in price per 1.0 = 100% point change in r)
—
Put Rho (per 1 percentage point change in r)
—
| Output | Value | Unit |
|---|---|---|
| Black‑Scholes call price | — | currency |
| Call Rho (change in price per 1.0 = 100% point change in r) | — | currency |
| Call Rho (per 1 percentage point change in r) | — | currency |
| Put Rho (change in price per 1.0 = 100% point change in r) | — | currency |
| Put Rho (per 1 percentage point change in r) | — | currency |
Visualization
Methodology
Calculations use the Black‑Scholes closed‑form expressions for European options. Key assumptions: lognormal asset returns, constant volatility and interest rates, continuous dividend yield, and European exercise (no early exercise).
Rho is calculated analytically from the Black‑Scholes formula: for a call C, rho = ∂C/∂r = K * T * exp(-rT) * N(d2). For a put P, rho = ∂P/∂r = -K * T * exp(-rT) * N(-d2). The calculator exposes both the per‑unit and per‑one‑percentage‑point versions.
Expert Q&A
Is this Rho valid for American options or options on futures?
No. The formula implemented is the Black‑Scholes closed form for European options on underlying assets with continuous dividend yield. American options and certain futures options require different models (binomial, finite difference, or Black 76). Use models appropriate to exercise style and underlying.
What are the main limitations of the Black‑Scholes Rho?
It assumes constant volatility and interest rates and no early exercise; Rho reflects sensitivity for small instantaneous changes in the risk‑free rate under those assumptions. Large rate moves, stochastic rates, discrete dividends, or volatility skew will make results approximate.
How should I interpret Rho per percentage point versus per unit?
Rho per unit is the derivative ∂price/∂r when r is expressed as a decimal (e.g., change per 1.0 = 100 percentage points). Rho per 1 percentage point multiplies that derivative by 0.01 and shows the expected price change for a 1 percentage‑point (100 basis points) change in the annual rate.
How can I improve accuracy for traded instruments?
Calibrate implied volatility from market option prices for matching strike and tenor, use market yields for the risk‑free curve and account for discrete dividends if material. Consider Monte Carlo or local volatility models when path‑dependent or early exercise features are present.
Are results guaranteed to be exact?
No. Results are mathematically correct under Black‑Scholes assumptions but subject to input quality, numerical precision, and model limitations. See accuracy and compliance notes below.
Sources & citations
- NIST — Publications and standards — https://www.nist.gov/publications
- ISO — Risk management (example standard) — https://www.iso.org/iso-31000-risk-management.html
- IEEE Standards Association — https://standards.ieee.org
- OSHA — Occupational Safety and Health Administration — https://www.osha.gov
- Black‑Scholes framework (academic reference overview) — https://www.math.uah.edu/stat/BlackScholes.html