# Rho: It’s Back!

I haven’t had to think about this one for a while. But after 14 years, it’s back in play: the option Greek rho.

###### The Option Greek Rho

Let’s just dig right in, shall we? Rho is the option Greek that measures an option position’s sensitivity to a change in interest rates. It works like some of the other Greeks (theta and vega) measuring the option-value sensitivity in dollars and cents of option premium. For example, if an option has a rho value of 0.35 and interest rates change by 1%, the option value changes by 35 cents.

Calls have positive rho, when you’re long them anyway, and long puts have negative rho. That means when interest rates go up, call values go up and put values go down.

###### How Do Interest Rates Affect Option Prices?

There’s a lot to unpack here. First of all, let me correct an intentional error. Rho—in fact all the Greeks—affects option values, as opposed to prices. Prices are set by the market and are centered around value. That’s fodder for a future post. But I figured I’d throw it out there just to be accurate.

The rationale for why options values are affected by interest rates is rooted in the concepts of hedging, put-call parity and synthetics. But for simplicity, consider this concept: If you’re bullish, you can buy the stock and tie up a lot of cash, or you can buy a call using a fraction of it. Let’s put some numbers on that.

Imagine you’re considering either buying 100 shares of a \$150 stock or buying a 3-month, 150-strike call at \$4.75. The 100 shares would cost you \$15,000. The call will cost you \$475. If you bought the call, you could actually take that remaining \$14,525 and put it into a money market account and earn interest on it. (This is an argument we couldn’t make up until recently.)

It’s very quantifiable. And when you start thinking about it, it probably becomes intuitive that the higher the interest rate, the more money you make squireling away that cash. Consequently, as interest rates rise, calls get more valuable.

A similar argument can be made for puts. If you were to short stock (and are a professional trader or a big-enough retail trader with some clout with your brokerage firm) you would earn interest on the cash you receive when you sell those borrowed shares. This is called a “short stock rebate.” But if you buy the put, you don’t get all that cash, and thus don’t get the interest. (In fact, you have to pay a comparatively small amount of cash.) So as interest rates go higher, the put becomes less valuable, compared with shorting the stock.

###### Rules of Thumb for Rho
• As interest rates rise, calls get more expensive and puts get cheaper.
• As interest rates fall, calls get cheaper and puts get more expensive.
• A one-percent change in the interest rate changes the option by the value of its rho.
• Rho affects longer-term options more than shorter-term options.
###### Put-Call Parity and Rho

Let’s not get all mathematical here. But I do want to point out an interesting phenomenon. You may have read that when options are right at the money, the call and put values are equal. But with equity options, it’s not really like that. The call value is actually higher than the put value. The higher the interest rate the greater the disparity.

###### Interest Rates and Rho

BTW, what interest rates are we talking about? Well, here’s another little textbook fallacy. If you have already read up on interest rates and rho, you probably read something along the lines of “…when the risk-free rate, the rate of U.S. Treasury notes, changes, options change by the value of rho, blah, blah, blah…”

Well, I’m here to tell you that’s a bunch of crap. What a trader really needs to use is the actual interest rates associated with his or her account. If a professional option trader is borrowing money (which they may have to do to carry the long stock they need to hedge their deltas, for example), they’d use the actual interest rate they are paying. Or if they are receiving interest on extra cash they have in their account (maybe they are short a lot of stock), they’d use that short stock rebate rate.

Options educators / writers just say “the risk-free rate” for simplicity, apparently. But it’s real money. Traders, especially big traders need to use the right numbers to get accurate values to estimate interest-rate risk. When I was making markets, I can recall a day when I made (or lost, I forget) around \$10,000 when the Fed announced a surprise rate hike.

And, so there it is too: The connection between the “risk-free rate,” in this case the discount rate, and “interest rates,” the ones you’d actually use, the ones in your account. Clearly, they are related and when one goes up probably the other ones do too. But it’s not an exact correlation. (Anyone who has shopped for a mortgage over the past three months has noticed that. Mortgage rates went up earlier and more than the interest rate the Fed raised.)

###### How Does Rho Affect Your Options Positions?

If you were thinking about it while reading the Rules of Thumb section of this post you may have already figured this out. But unless you have longer-term options, this whole rho thing isn’t likely to be a really big deal. And bigger dollar amount underlyings are intuitively affected more, as there is more interest on a \$2,000 stock than a \$20 stock. And, of course, bigger positions are affected more too. But hey.

You’ll definitely begin making or losing at least some money as a result of interest rates changing. You might as well understand why and how maybe you can preemptively think about it in advance and maybe change random P&L events to ones you can take advantage of. In trading knowledge is power. Put it to good use, my friend.

Dan Passarelli
Founder and President
Market Taker Mentoring, Inc.