Rho: Small, Until the Contract Is Long-Dated
Move the rate slider: 30-day options barely notice, LEAPS reprice noticeably.
Worth reading first: Delta: Three Questions, One Number
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Rho measures how an option's price reacts to interest rates. It is the greek people are usually told to ignore, and for 30-day contracts that advice is about right. This lesson is here to show you where it stops being right.
Why rates touch an option at all
The link is not obvious, so it is worth spelling out. An option's strike is money that changes hands in the future, and future money is worth less than money today. The model discounts the strike back to what it is worth now, using the risk-free rate.
Present value of the strike = strike × e−rTThe discounted strike is where rates enter the pricing model.
Raise the rate and that discounted strike shrinks. For a call, the strike is what you would pay, so a smaller present-value cost makes the option more valuable, and calls have positive rho. For a put, the strike is what you would receive, so a smaller present value makes it less valuable, and puts have negative rho.
There is an easier way to picture it. A call lets you control shares without paying for them yet, which leaves your cash earning interest in the meantime. The higher rates go, the more that delay is worth.
Everything hinges on time
Look at the T in the exponent. Rho scales with the time remaining, which is why the table in the widget fans out so sharply. A 30-day contract is discounted over a twelfth of a year. A two-year LEAPS is discounted over two full years.
The practical dividing line is around six months. Below it, rho is smaller than the bid-ask spread you pay getting in and out, so there is no point modelling it. Above it, and especially past a year, it becomes a real part of the price.
What it means for a premium seller
For the strategies at the centre of this course, 30 to 45 day cash-secured puts and covered calls, rho does not enter into any decision. It will not change which strike you pick or when you close. Read the number and move on.
There are two places it does show up.
Long-dated positions. A poor man's covered call is built on a LEAPS call, which carries real positive rho. Falling rates work gently against that leg.
The collateral itself. This is not rho at all, and it matters far more to your returns. A cash-secured put ties up thousands of dollars, and when rates are high that cash earns something in a money market or treasury fund. Your real return is the premium plus what the collateral earns. Compare a covered call with a cash-secured put without counting that, and you will favour the wrong one.
What can go wrong
Trying to trade rho. At retail size, every other greek swamps it. There is no rho strategy here.
Ignoring it on LEAPS. Over two years, a couple of points of rate movement is not noise.
Forgetting the collateral yield. Not technically rho, and far more likely to affect your annual result than rho ever will.
Key takeaways
- Rho measures sensitivity to interest rates, and enters the model through the discounted present value of the strike.
- Calls have positive rho and puts negative. Higher rates make deferring a purchase more valuable.
- Rho scales with time to expiration. Below about six months it is smaller than your transaction costs.
- It matters for LEAPS and poor man's covered calls, and almost nowhere else in premium selling.
- The interest your cash collateral earns is not rho, but it affects your real return far more.
Check your understanding
1. Interest rates rise. What happens to call and put prices, all else equal?
2. For which position does rho genuinely matter?
3. You are running cash-secured puts with rates at 5%. What is most likely to be overlooked?
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