Put Option Rho

Options
Updated Apr 2026 • Has calculator

The change in a put option's price for a 1% increase in the risk-free interest rate.

What is Rho (Put)?

Rho (ρ) for a put option measures how much the put price changes when the risk-free rate rises by 1 percentage point. Put rho is always negative: higher rates increase the opportunity cost of holding a put and reduce the present value of the potential payoff, making puts less valuable. The effect is most significant for deep in-the-money puts and long-dated options.

Formula

ρ_put = −K · T · e^(−rT) · N(−d₂) / 100

Worked Example

Worked example — Apple Inc. (AAPL) — ATM put, representative Q1 2024

Representative Q1 2024 market conditions

Step 1  Stock price (S): $185, Strike (K): $185, r = 5.25%, T = 0.25 yrs, σ = 28%
Step 2  d₂ = 0.083, N(−0.083) ≈ 0.467
Step 3  ρ_put = −185 × 0.25 × e^(−0.013) × 0.467 / 100
Step 4  ρ_put = −185 × 0.25 × 0.987 × 0.467 / 100 ≈ −$0.213 per share
Step 5  → If the Fed raises rates by 1%, this put loses $0.21 per share ($21 per contract)

Source: Hull, J.C. — Options, Futures, and Other Derivatives, 11th ed., Ch. 19 (2024-01-15)

Calculate Rho (Put)

Current market price of the underlying stock

Option strike price

Annual risk-free rate

Time to expiration in years

Annualised implied volatility

Rho (per 1% rate)

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Not investment advice.

How to Interpret Rho (Put)

< -1
Very high negative rho — multi-year deep ITM put
-1 – -0.5
High negative rho — long-dated or deep ITM put
-0.5 – -0.1
Moderate rho — intermediate-term put option
> -0.1
Low rho — short-dated put, rate-insensitive
Sources
  1. Hull, J.C. — Options, Futures, and Other Derivatives, 11th ed.
  2. Investopedia — Rho