Lesson 17 of 20 · 6 min

Rho, and why it rarely matters until rates move

Rho measures rate sensitivity, and it is nearly invisible on short contracts and material on long ones.

Rho is the change in an option's price for a one-point change in the risk-free rate. It is last on every list of the Greeks for a good reason and it is not zero, which is the more interesting half of the story.

In the base case rho is $0.041 per percentage point. Move the rate from 4 to 5 percent and the call goes from $3.02 to $3.06, a four-cent change that is smaller than most bid-ask spreads. For a one-month contract, rho is genuinely negligible.

The reason it grows is discounting. Rho comes from the present value of the strike, and discounting compounds with time. A two-year at-the-money call has rho of $0.960 — twenty-three times the one-month figure. Move that contract's rate from 4 to 5 percent and it goes from $17.68 to $18.65, nearly a dollar. On a long-dated contract, rate exposure is real.

Signs are worth getting right. Rho is positive for calls and negative for puts: a higher rate reduces the present value of the strike, which helps the party who will pay it and hurts the party who will receive it. On the same two-year contract, moving the rate from 4 to 5 percent takes the put from $9.99 to $9.13.

One connection back to Track 1. Someone borrowing against a portfolio at index plus spread already has a large, direct exposure to rates through the loan. Rho on any short-dated option position they hold is a rounding error next to that. Sizing the two exposures against each other is a more useful exercise than tracking rho carefully.

Key terms

Rho: The change in an option's price per one-point change in the risk-free rate. Positive for calls, negative for puts.
Discounting effect: The channel through which rates enter the price: they set the present value of the strike, and the effect compounds with time.
Materiality: Whether a sensitivity is large enough to matter against transaction costs and other risks. Rho usually is not, on short contracts.

Worked example

Rho, one month against two years

Underlying 100, strike 100, volatility 25%. Rate moves from 4% to 5%.

30 days: rho: $0.041 per point
30 days: call at 4% → 5%: $3.02 → $3.06
2 years: rho: $0.960 per point
2 years: call at 4% → 5%: $17.68 → $18.65
2 years: put at 4% → 5%: $9.99 → $9.13
Ratio of rho, 2 years to 30 days: 23×

The same one-point rate move is invisible on a monthly contract and worth a dollar on a two-year one.

Check yourself

Rates rise one point. What happens to a 30-day at-the-money call, and to a two-year one?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.