Rho
The Greek that tracks interest rates: small for most options, large for long-dated ones.
What is Rho?
An option's price depends on five inputs: the share price, the strike, the time to expiration, the volatility, and the interest rate. Each Greek measures how the price responds when one input moves. Rho \(\rho\) is the Greek for the interest rate: how much the option's price changes when the risk-free rate rises by one percentage point.
In the Black–Scholes model, for a European option on a stock that pays no dividend:
\(\rho_{\mathrm{call}} = K\,T\,e^{-rT} N(d_2)\)
\(\rho_{\mathrm{put}} = -K\,T\,e^{-rT} N(-d_2)\)
Here \(K\) is the strike, \(T\) the time to expiration in years, \(r\) the interest rate, and \(N(d_2)\) the model's risk-neutral probability that a call finishes in the money. Divide by 100 to get the change per percentage point.
Calls have positive rho and puts negative rho, and the reason is the time value of money. A call lets you pay the strike later, and the higher rates are, the more that delay is worth. A put pays you the strike later, and the higher rates are, the less that later payment is worth today.
In the Black–Scholes model, for a European option on a stock that pays no dividend:
\(\rho_{\mathrm{call}} = K\,T\,e^{-rT} N(d_2)\)
\(\rho_{\mathrm{put}} = -K\,T\,e^{-rT} N(-d_2)\)
Here \(K\) is the strike, \(T\) the time to expiration in years, \(r\) the interest rate, and \(N(d_2)\) the model's risk-neutral probability that a call finishes in the money. Divide by 100 to get the change per percentage point.
Calls have positive rho and puts negative rho, and the reason is the time value of money. A call lets you pay the strike later, and the higher rates are, the more that delay is worth. A put pays you the strike later, and the higher rates are, the less that later payment is worth today.
Remember
Four things to keep in mind about rho.
-
Calls have positive rho and puts negative rho: rising rates help calls and hurt puts.
-
Rho grows with time to expiration. It is tiny for a one-month option and large for a two-year option.
-
Rho grows as an option moves into the money, because the strike is more likely to be paid or received.
-
Rates usually move slowly, so rho is the Greek traders watch least, until rates move fast or the options are long-dated.
Why Traders Overlook Rho
For most options, rho is small. Take an at-the-money call on a $100 stock, with 25% volatility and rates at 4%. With one month to expiration it costs about $3.04, and a full percentage-point rise in rates adds only about 4 cents. Delta, gamma, theta and vega move that option far more on an ordinary day.
Time changes that. Rho grows roughly in proportion to time to expiration. The same call with two years to run has a rho of about $0.96. A two-point rise in rates, from 4% to 6%, lifts its price from $17.68 to $19.64, while the matching two-year put falls from $9.99 to $8.33.
So does the size of the move. When central banks raise or cut rates by several points within a year or two, as they have more than once, rho stops being a rounding error for anyone holding long-dated options or long-dated hedges.
Rho is built into put–call parity. A call minus a put with the same strike and expiration equals the share price minus the present value of the strike:
\(C - P = S - K\,e^{-rT}\)
As rates rise, that present value falls, so the call gains on the put.
Time changes that. Rho grows roughly in proportion to time to expiration. The same call with two years to run has a rho of about $0.96. A two-point rise in rates, from 4% to 6%, lifts its price from $17.68 to $19.64, while the matching two-year put falls from $9.99 to $8.33.
So does the size of the move. When central banks raise or cut rates by several points within a year or two, as they have more than once, rho stops being a rounding error for anyone holding long-dated options or long-dated hedges.
Rho is built into put–call parity. A call minus a put with the same strike and expiration equals the share price minus the present value of the strike:
\(C - P = S - K\,e^{-rT}\)
As rates rise, that present value falls, so the call gains on the put.
FAQ
-
What is rho in options?
Rho is the Greek that measures how much an option's price changes when the risk-free interest rate rises by one percentage point. A call with a rho of 0.25 gains about 25 cents if rates rise from 4% to 5%, all else equal.
-
Why does a call option have positive rho?
A call lets you control the stock while keeping your cash until you exercise, and pay the strike later. Higher rates make that cash earn more in the meantime and make the later strike payment worth less today, so the call is worth more. A put is the mirror image: it pays you the strike later, and higher rates shrink what that later payment is worth now.
-
Which options have the most rho?
Long-dated options that are in the money. Rho grows roughly in proportion to time to expiration, and with the chance that the option will be exercised. A deep in-the-money two-year call can carry dozens of times the rho of a one-month option.
-
How does rho compare with the other Greeks?
Delta, gamma, theta and vega respond to the share price, the passage of time and volatility, which move every day. Rho responds to interest rates, which usually change in small steps weeks apart. That makes rho the smallest Greek for most short-dated options, and the one most often left out.
-
Do dividends affect rho?
Not directly. Rho measures the response to the interest rate only. Expected dividends lower call prices and raise put prices through a separate sensitivity, sometimes called dividend rho or phi. Rising rates and rising dividends push option prices in opposite directions.
Learn Rho, the Other Greeks, and the Strategies Built on Them
Our Stock Options course covers the Greeks, rho included, along with implied volatility and volatility surfaces, covered calls, spreads, straddles and strangles, in video lessons with quizzes.
