Option Rho

This is part of option Greeks tutorials. See also delta, gamma, theta, vega.

What Is Rho

Rho measures an option's sensitivity to changes in interest rates – how much option premium will change if the risk-free interest rate increases by one percentage point.

Example

Consider a call option on a stock with 3 months left to expiration, which is currently trading at \$2.35 (option premium). Its rho is 0.15 and the 3-month risk-free interest rate is 3%. The option's rho indicates that if the interest rate increases by one percentage point to 4%, the option premium should rise by \$0.15 to \$2.50. Conversely, if the interest rate declines by one percentage point to 2%, the option premium should decrease by \$0.15 to \$2.20. All this assumes that the other factors (the underlying stock's price, implied volatility, and time to expiration) remain the same.

Rho Units

Rho is the ratio of option price change (in dollars) to interest rate change (in percentage points). Therefore, its units are dollars per percentage point, although in practice units are rarely mentioned, as with the other Greeks.

Rho Values

There is no theoretical limit on the values rho can reach. It can be positive or negative, but it is usually a very small number.

In most cases, interest rates have far smaller effect on option prices than underlying price or volatility. Interest rate changes tend to be smaller and take longer time (a one percentage point change in the short-term risk-free interest rate is always a huge event in the markets, while the same change in implied volatility often happens in minutes or seconds on many underlyings).

As a result, rho often receives less attention than the other main Greeks – delta, gamma, theta, and vega – and it is less understood by the typical option trader, also because it is harder to make universal rules about effects of different factors (underlying price, time or volatility) on rho values. These effects depend on option type, underlying type, and the settlement procedures of both. For example, rho of stock options behaves differently from futures options rho or FX options rho (currency options are in fact affected by two interest rates – domestic and foreign – and have two rhos).

Call Option Rho

The main benefit of holding a call option is the optionality, right but not obligation to buy the underlying. If the stock goes up, you make money, but if it goes down, you don't need to exercise the option and you are protected. You have a choice.

That said, a call option has another benefit that is sometimes forgotten: it allows you to control the underlying (at least its upside) without paying for it. It delays payment and improves cash flow.

For example, consider a stock trading at \$50 per share. You can buy 100 shares in the stock market and pay \$5,000 immediately. Alternatively, you can buy a 3-month, \$50 strike call option for \$2 and pay only \$200 (for one contract of 100 shares). In both cases, you control 100 shares of the stock and make money if the stock goes up. However, in the first case you pay \$5,000 now, while in the second case you only pay \$200 now (the option premium) and \$5,000 (the option's strike price) later, if you exercise. The option effectively delays your payment for 3 months (from now to the option's expiration). It is like a 3-month loan of \$5,000. Therefore, the option's time value must reflect not only the optionality, but also the value of the loan.

Call options are more valuable with higher interest rates and have positive rho.

Put Option Rho

It is opposite with put options. Let's say you already own 100 shares of the same stock. You are worried the stock will fall. You can sell the stock in the stock market and get \$5,000 in cash. Alternatively, you can buy a 3-month, \$50 strike put option. If the stock does fall, you can exercise the put later and get \$5,000. Without the put, you receive \$5,000 now (and you can put it in a bank for 3 months and earn interest); with the put you get it in 3 months. The higher the interest rate, the more attractive it is to sell the stock and get the money earlier, rather than buy the put and get the money later.

Put options are less valuable when interest rates are higher. They have negative rho.

Rho and Time to Expiration

From the above examples it should be obvious that the effect of interest rates on options is greater with longer time to expiration. In general, rho approaches zero as an option gets closer to expiration.

Rho of Futures Options

The reasoning in the above examples assumes that the underlying security settles like a stock – you must pay the full price in cash when buying, and you receive the full price in cash when selling. Not all underlyings settle like that. When you buy (go long) a futures contract, you don't pay anything (you only deposit a margin) and only your profits or losses are marked to market and settled daily. Therefore, the above logic of calls having positive rho and puts having negative rho is valid for stock options or currency options, but not for futures options or currency futures options.

It is hard to make any general statements about futures option rho, as there are too many variables, such as the way the options themselves are settled, the marking to market process, or the option's strike. In general, futures options tend to be less sensitive to interest rates than stock or currency options, as the cash flow advantage of calls and disadvantage of puts explained above does not apply.

Rho of Currency Options

With foreign currency options, two interest rates are involved. For example, consider an option on euros, traded on a US exchange in dollars. This option's price will be affected by both the US (domestic) interest rate and the Eurozone (foreign) interest rate.

Rho measures the effect of the domestic rate, which is similar to stock options. Generally, higher domestic interest rate makes foreign currency calls more valuable (positive rho) and puts less valuable (negative rho), because the domestic rate is the cost of financing.

The effect of the foreign interest rate is measured by rho2 (sometimes called phi). It is like the effect dividend yield has on stock options. When you hold the underlying (a stock / euros), you earn it. When you hold a call option instead, you don't. As a result, the higher the EUR interest rate, the less attractive call options are as an alternative to holding the euros directly. It is the opposite with puts. An increase in foreign interest rate makes call options on that currency less valuable and put options more valuable.

Effect of Interest Rates on Underlying

It is important to understand that rho (and rho2) only measures the effect of interest rate on option price if the underlying price and all the other factors remain the same.

In reality, interest rate changes often move the underlying price as well – particularly for currency options (interest rates are the main driver of exchange rates), options on fixed income instruments (bond prices are closely related to yields), but also some stock options (e.g. bank stocks may strongly react on interest rate moves).

Therefore, interest rate changes can affect option prices via two channels: directly and indirectly via moving underlying price. Rho only measures the direct effect, but not the indirect one. The effect of underlying price changes on option premium is measured by delta. For some underlyings, such as bonds or currencies, the indirect effect is often much stronger than the direct one.

How to Calculate Rho

Mathematically, rho is the derivative of option price with respect to interest rate.

If you are interested in the exact formulas, you can find them in Black-Scholes Greeks Formulas and Option Greeks Excel Formulas.

Summary

• Rho measures how option premium will change if the risk-free interest rate increases by one percentage point.
• Call options on most underlyings have positive rho; put options have negative rho.
• Rho is generally greater (in absolute terms) with more time to expiration.
• For many underlyings like currencies or bonds, interest rates may also affect underlying price, and thereby option prices. This indirect effect, though often greater than the direct effect, is not measured by rho.

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