Interest rates change option prices too. It is easy to forget because the effect on a two-week stock option is tiny, and for years rates barely moved. But a call is a way to control a stock without paying for it today, and when cash earns more, that deferral is worth more. Rho puts a number on it, and knowing when that number is small enough to ignore is the practical skill here.
Rate sensitivity
Rho is the change in an option's price when the interest rate rises by one percentage point, for example from 4% to 5%. Rho is positive for calls and negative for puts.
The reason is carry. Buy a call instead of the shares and you keep the cash in the bank until you exercise. Higher rates mean that cash earns more, so the call is worth more. Buy a put instead of shorting the stock and you give up the interest a short seller earns on the sale proceeds, so higher rates make the put worth less.
Sizing it
A call's rho is roughly the strike, times the time to expiration in years, times the chance the option is exercised, divided by 100. Three examples on a $200 strike with about a 60% chance of finishing in the money:
| Time left | Rough rho per share | Per contract, per 1 point |
|---|---|---|
| 5 days | $0.02 | $2 |
| 30 days | $0.10 | $10 |
| 1 year | $1.20 | $120 |
Rates do not move a full point in a day. A central bank meeting shifts them by a quarter point, which on the 30-day option above is about $2.50 a contract. Set that against a theta of $15 a day or a vega of $30 a point and it is clear why short-dated equity traders hardly look at rho. The chain does not even carry a rho column; the Greeks Heatmap does.
Call rho per 1% rate against days left, out to two years. In the money and long dated is where it stops being a rounding error. Switch to puts to see it turn negative.
When it matters
- Long-dated options. Rho scales with time. A one-year call has about twelve times the rho of a one-month call at the same strike, a two-year call twice that again, and rates can move a full point over a year.
- Deep in-the-money calls. The chance of exercise is close to certain, so rho is close to the strike times the years to expiration. A deep in-the-money call is stock bought on credit, and rho is the interest bill on that credit.
- High rates. The same one-point change matters more to a position priced off a 5% rate than one priced off 0.5%, because the whole carry term is bigger.
Put those together and a long-dated, deep in-the-money call in a high-rate world is where rho stops being a footnote. Traders who run those positions hedge the rate exposure separately, and anyone holding long-dated calls as a stock substitute should at least know the number.
The maths: where rho comes from
A call's rho is K × T × e^(-rT) × N(d2) per unit of rate, divided by 100 per percentage point. N(d2) is the model's chance of exercise, so this is the formula behind the sizing rule above.
Put-call parity gives the intuition in one line: a call minus a put at the same strike equals the stock minus the present value of the strike, S - K e^(-rT). Raise the rate and the present value of the strike falls, so the call gains against the put by about K × T for each unit of rate. Split that between the two by their chances of exercise and you have each one's rho.
Your turn
Open the AAPL chain and pick a far-out expiration. Read how many days it has to run from the strip above the strikes, then read the delta of a call about 5% in the money. Those two numbers, with the strike, are most of what sets that option's rho.