Put Rho
Sensitivity of put price to a 1.00 (i.e. 100 percentage point) change in the risk-free rate. Negative for puts — higher rates decrease put values.
When to use: Use to size interest-rate exposure on put positions. Like call rho, generally a minor Greek for short-dated trades but material for long-dated puts.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| Rho | Rho | Change in option price per 1.00 (i.e. 100%) change in the risk-free rate | $ |
| S | Stock Price | Current price of the underlying | $ |
| Strike | Strike Price | Exercise price of the option contract | $ |
| Rf | Risk-Free Rate | Continuously compounded annual risk-free rate as a decimal | % |
| Sigma | Volatility | Annualized volatility of the underlying as a decimal (e.g. 0.25 for 25%) | % |
| T | Time to Expiration | Time to expiration in years (e.g. 0.25 for 3 months) | years |
Real-Life Examples
Example 1: ATM Put Rho, 30% Vol, 90 Days
Stock at $100, $100-strike 90-day put, r = 4%, σ = 30%.
Given
Step-by-Step
A 1.00 (100 percentage point) rise in r decreases the put price by ~$12.46. Per-1%-rate convention: ~−$0.125. Note ρ_call − ρ_put = K·T·e^(−rT) — the rho version of put-call parity.
Frequently Asked Questions
Higher rates lower the present value of the strike (which the put holder receives at exercise), so the put is worth less. Equivalently, higher rates raise the forward price of the stock, pushing more puts OTM under the risk-neutral measure.
Differentiate put-call parity (C − P = S − K·e^(−rT)) with respect to r: ∂(K·e^(−rT))/∂r = −K·T·e^(−rT), so ∂C/∂r − ∂P/∂r = K·T·e^(−rT).