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FRM Exam Part I · Option Sensitivity Measures: The "Greeks"

Rho of an Option and Interest Rate Sensitivity

Updated 11 October 2026 · Fact-checked

Rho is the change in an option's price for a small change in the risk-free rate. For a European call, rho = K × T × e^(−rT) × N(d2), which is positive. For a put, rho = −K × T × e^(−rT) × N(−d2), which is negative. Quote conventions matter: per 1.00 or per 1%.

Understand Rho and Interest Rate Sensitivity

Rho measures how much an option's value changes when the risk-free interest rate changes by a small amount, with everything else held fixed. In maths it is the partial derivative of the option price with respect to r.

Why does the rate matter? A call lets you buy the asset later at strike K. Paying K later is cheaper in present-value terms when rates are higher, so the call gains value. A put lets you sell for K later. Receiving K later is worth less in present-value terms when rates are higher, so the put loses value. This is why call rho is positive and put rho is negative.

Rho depends on maturity and moneyness. The present value of the strike effect grows with time, so rho is larger in size for longer-dated options. Rho is also larger in size for in-the-money options, because they are likely to be exercised and the strike payment (or receipt) really matters. Deep out-of-the-money options have rho close to zero. At very short maturity, rho is close to zero for all options.

Compare rho with the other Greeks. Rho is usually the least important Greek for short-dated equity options, because rates move slowly. It matters more for long-dated options, LEAPS and interest-rate-sensitive books. Vega is the sensitivity to volatility, not to rates. Do not mix the two: vega is always positive for long calls and puts, while rho has opposite signs for calls and puts.

You also need the link to put-call parity. Differentiating c − p = S − K × e^(−rT) with respect to r (no dividends) gives rho_call − rho_put = K × T × e^(−rT). This is a fast consistency check on any answer.

Key formulas to remember

Rho of a European call (no dividends, Black-Scholes-Merton)
ρ_call = K × T × e^(−rT) × N(d2)
Positive. d2 = d1 − σ√T. Units: change in price per 1.00 (100%) change in r.
Rho of a European put (no dividends)
ρ_put = −K × T × e^(−rT) × N(−d2)
Negative. N(−d2) = 1 − N(d2).
Rho difference from put-call parity
ρ_call − ρ_put = K × T × e^(−rT)
Holds for European options on a non-dividend-paying asset. Useful to check or to get one rho from the other.
Price change from a rate move
Δ(option price) ≈ ρ × Δr
Use the same unit for ρ and Δr. If ρ is per 1.00, a 1% move is Δr = 0.01. If ρ is quoted per 1%, use Δr = 1.
d2
d2 = [ln(S/K) + (r − σ²/2) T] ÷ (σ√T)
Needed for N(d2). With continuous compounding r.

How to solve Rho and Interest Rate Sensitivity questions

Use this routine for any rho question, whether it asks for a sign, a size or a price change.

  1. 1Identify the option type (call or put), style (European) and whether the underlying pays income. The formulas above assume no income.
  2. 2Fix the sign first. Call rho is positive, put rho is negative. Reject any answer with the wrong sign.
  3. 3Check the units asked for: per 1.00 change in r, or per 1 percentage point. Note it before you calculate.
  4. 4Compute the discounted strike: K × e^(−rT), then multiply by T to get K × T × e^(−rT).
  5. 5Multiply by N(d2) for a call or by −N(−d2) for a put. If N(d2) is given, use it directly.
  6. 6For a price change, multiply rho by the rate change, in matching units.
  7. 7Sanity check with maturity and moneyness: longer maturity and deeper in the money means larger absolute rho. Check the call-minus-put identity if both are available.

Quickest way: Sign, scale and parity shortcut

When to use it: Use when options offer four numeric answers and you have little time, or when the question gives one rho and asks for the other.

  1. Call rho > 0, put rho < 0. This often removes two options at once.
  2. If you know the call rho, get the put rho as call rho − K × T × e^(−rT).
  3. Estimate magnitude: |rho| cannot exceed K × T × e^(−rT) for either option, because N(·) is at most 1.
  4. For a rate shift of 1 basis point or 1%, scale rho by 0.0001 or 0.01 before comparing with answer choices.

Common mistakes in Rho and Interest Rate Sensitivity

  • Giving a put a positive rho

    Students remember that higher rates lift call prices and assume all options behave the same way.

    Fix: Think of the strike received later. Higher rates cut its present value, so the put falls. Put rho is negative.

  • Confusing rho with vega

    Both are less common Greeks and both are set out in the same chapter, so the names blur.

    Fix: Rho is sensitivity to the risk-free rate. Vega is sensitivity to volatility. Vega is positive for both long calls and puts. Rho has opposite signs.

  • Applying a 1% change as 1 instead of 0.01

    The formula gives rho per 1.00 change in r, but rate changes are quoted in percent.

    Fix: Convert first. A 1% rise is Δr = 0.01. Multiply rho by 0.01. Only skip this if rho is already quoted per 1%.

  • Using N(d2) for the put rho

    Students copy the call formula and just change the sign.

    Fix: The put uses N(−d2). Put rho = −K × T × e^(−rT) × N(−d2).

  • Believing rho is always small

    For short-dated equity options rho is indeed small, and this gets over-generalised.

    Fix: Rho grows with maturity and with strike. For long-dated, in-the-money options it can be significant. Say it is usually less important, not always.

  • Forgetting the discount factor e^(−rT)

    Students remember K × T × N(d2) and drop the present value term.

    Fix: Write the full expression K × T × e^(−rT) × N(d2) each time and compute e^(−rT) explicitly.

Worked examples

Example 1

A 1-year European call has strike K = 100 and a risk-free rate of 5% (continuously compounded). N(d2) = 0.5. The underlying pays no income. Estimate the rho per 1.00 change in r, and the approximate price change if the rate rises by 1%. Use e^(−0.05) = 0.9512.

Show the solution
  1. Formula: ρ_call = K × T × e^(−rT) × N(d2).
  2. Discounted strike: 100 × 0.9512 = 95.12.
  3. Multiply by T = 1: 95.12.
  4. Multiply by N(d2) = 0.5: 95.12 × 0.5 = 47.56.
  5. Price change: Δr = 0.01, so 47.56 × 0.01 = 0.4756.

Answer: Rho ≈ 47.56 per 1.00 change in r. A 1% rise in the rate raises the call price by about 0.476.

Example 2

For the same option and assumptions, a European put has the same strike K = 100, T = 1 and r = 5%. What is the rho of the put per 1.00 change in r? Use the call rho of 47.56 and e^(−0.05) = 0.9512.

Show the solution
  1. Parity relationship: ρ_call − ρ_put = K × T × e^(−rT).
  2. Right-hand side: 100 × 1 × 0.9512 = 95.12.
  3. Rearrange: ρ_put = ρ_call − 95.12 = 47.56 − 95.12.
  4. ρ_put = −47.56.
  5. Check with the direct formula: −95.12 × N(−d2) = −95.12 × 0.5 = −47.56. Matches.

Answer: The put rho is −47.56 per 1.00 change in r, or about −0.476 per 1% rise in rates.

Exam tips

  • Sign questions are common. Memorise: call rho positive, put rho negative, and say why in terms of the present value of the strike.
  • Watch the unit trap. Read whether the answer is per 1.00 or per 1% before choosing among close numeric options.
  • Use put-call parity on rho to move between call and put values. It saves computing N(d2) twice.
  • For maturity and moneyness questions, remember rho rises in size with maturity and is near zero for deep out-of-the-money or very short-dated options.
  • Distinguish rho from vega, theta and delta in conceptual questions by naming the input each Greek responds to.

Practice questions from Option Sensitivity Measures: The "Greeks"

Rho and Interest Rate Sensitivity: frequently asked questions

What is rho of an option in simple terms?

Rho tells you how much an option's price changes when the risk-free interest rate changes by a small amount. A positive rho means the option gains when rates rise. A negative rho means it loses.

How do interest rates affect call and put prices?

Higher rates raise the value of a call and lower the value of a put, other things equal. This happens because the present value of the strike falls as rates rise. The effect grows with maturity.

What is the difference between rho and vega?

Rho measures sensitivity to the risk-free rate, while vega measures sensitivity to volatility. Vega is positive for long calls and long puts. Rho is positive for calls and negative for puts.

Is rho larger for in-the-money or out-of-the-money options?

In absolute terms, rho is larger for in-the-money options and close to zero for deep out-of-the-money ones. It is also larger for longer maturities. At very short maturities it is small for all options.