The red-headed step child, the forgotten Brady – yes, we are talking about rho. Rho is the one Greek that does not get mentioned with the others, which is not entirely its fault. Rho is the measure of the effect of interest rates on an option's price.
Unfortunately for rho, interest rates do not change that often. When you trade short term options, the effect of interest rates is felt even less. However, to get a complete understanding of the Greeks and how they effect your option’s price, you need to understand rho.
The whole reason rho matters is because of what cost of carry means for options. We will discuss what cost of carry is and why we should know it. We will also talk about why rho is different for calls versus puts, in-the-money versus out-of-the-money, and longer dated options versus short term options.
What Is The Risk-Free Interest Rate
When we talk about interest rates, we mean the risk-free interest rate. The risk-free interest rate is the minimum return you can expect to receive while keeping your risk at zero. The only way you are going to find an investment with zero risk is if you look to the government. Treasury Bonds and Treasury Bills are risk-free because you can assume the government will not default and you will be paid on your bond or bill.
We refer to risk-free investments as minimum return investments because you as an investor would expect a higher return anytime you have money at risk. No one would risk their money if they could make more money risk-free.
Where To Find An Option’s Rho
Before we can even begin to define or talk about all the interesting attributes of rho, you need to know where to find rho. An option’s rho, along with the other Greeks, come out of the option pricing model.
Due to the fact that Greeks are a byproduct of a calculation, means they have a model risk. A model risk means; the outputs are only as good as the inputs. If you plug in the incorrect implied volatility, you can’t expect to receive proper Greeks.
Model risk in terms of option pricing and Greeks is not a serious risk. Out of the 7 factors that affect an option’s price, only one of them is unknown, implied volatility. It would be difficult to mess up the model such that it throws off your Greeks.
It is more important to understand how Greeks are derived, and that you can’t always trust the numbers for how they are.
The best place to find the rho of your option is through an option chain. An option chain displays all the calls and puts for a given expiration and underlying. You can usually customize your option chain to display the various Greeks in which you are interested. Most traders will use their option brokerage, but you can also use free tools such as Nasdaq.com to view an option chain.
What Is Option Rho
The change in the risk-free interest rate has a positive effect on calls and a negative effect on puts. That is to say, when interest rates increase, a call’s premium will also increase, however a put’s premium will decrease. Rho is positive for long calls and short puts and negative for long puts and short calls.
Here is an example of the positive effect rho has on a call option.
You own a long call on The Option Prophet (sym: TOP) which is currently trading at $2.50, with a rho of 0.05, while the risk-free interest rate is at 2%.
The interest rate begins to rise and settles at 5%. Your long call is now worth $2.65.
(2.50(option price) + 0.05(rho) x 3(change in interest rates)) = $2.65
Here is an example of the negative effect rho has on a put option.
You own a long put on TOP, which is currently trading at $4.00, with a rho of 0.06, while the risk-free interest rate is at 4%. If the interest rate increases to 6%, our option is now worth $3.88.
(4.00(option price) - 0.06(rho) x 2(change in interest rates)) = $3.88
As you can see, your put option decreased in value, while your call option increased in value when the interest rate increased.
What Is An Option's Cost Of Carry
Cost of carry specifically refers to the cost incurred from an investment. Typically, we are referring to interest, such as the interest cost on bonds, interest expense due to margin, or the interest on a loan needed to purchase a security.
To drive this concept home, let’s look at the decision making process of trying to invest in TOP while it is trading at $50.
We can buy 100 shares of the stock outright, which would cost us $5,000, or
Instead of buying the stock outright, we can long an at the money call for $5.00. Our total cost here would be $500. Our initial outlay of cash would be smaller, and this would leave us $4,500 remaining. Plus, we will have the same reward potential for half the risk. Now we can take that leftover cash and invest it elsewhere such as Treasury Bills. This would generate a guaranteed return on top of our investment in TOP.
The higher the interest rate, the more attractive the second option becomes; thus, when interest rates go up, calls are a better investment, so their price also increases.
On the flip side of that coin, if we look at a long put versus a long call, we can see a disadvantage. We have two options when we want to play an underlying to the downside.
You can short 100 shares of the stock, which would generate cash into the brokerage and allow us to earn interest on that cash.
You can go long a put option, which will cost you less money overall, but not put extra cash into your brokerage that generates interest income.
The higher the interest rate the more attractive the first option becomes; thus, when interest rates rise the value of put options drops.
How Rho Changes Based On Stock Price
As your stock price increases so does the value of rho, for both puts and calls. As a stock becomes more expensive it will cost more to finance. You can think about this in terms of our cost of carry examples above.
How Rho Changes Based On Time To Expiration
Near term options are going to have a smaller rho value compared to long term options. Short term options are not heavily affected by the changing of interest rates. Interest rates don't typically move by substantial amounts in relatively short periods of time. Longer dated options such as Long-Term Equity Anticipation Securities (LEAPS) which have expiration dates greater than one-year will have greater rho values.
How Rho Changes Based On Volatility
Volatility has an indirect effect on rho. Even though volatility doesn’t directly affect rho it does change the value of our options which causes rho to increase or decrease.
Thinking back to our Complete Guide On Option Delta, we know that your deltas change differently depending if your option is out-of-the-money, in-the-money, or at-the-money. As volatility increases your deltas will move closer to 50, at-the-money.
Your out-of-the-money option will gain in value as the volatility increases, driving your delta higher, thus, your rho increases.
Your at-the-money option will stay flat in value as the volatility increases, keeping your delta the same, thus, no increase to rho.
Your in-the-money option will decrease in value as volatility increases, lowering your delta, thus, decreasing the value of rho.
Even though rho is the least talked about Greek it is still important to understand how it affects your options. More importantly, we learn about the cost of carry and how it drives demand for call and put options and is a fundamental factor in option pricing.
When interest rates are increased or decreased it is usually by a fraction of a percent, such as 0.25%. These incremental changes in interest rates don't have a large effect on short term options but can affect options with expiration dates greater than one year such as LEAPS.
Now that you know Rho, learn about the other Greeks: