What Is Rho? Definition, How It's Used, Calculation, and Example

What Is Rho?

Rho is the rate at which the price of a derivative changes relative to a change in the risk-free rate of interest. Rho measures the sensitivity of an option or options portfolio to a change in interest rate. Rho may also refer to the aggregated risk exposure to interest rate changes that exist for a book of several options positions.

For example, if an option or options portfolio has a rho of 1.0, then for every 1 percentage-point increase in interest rates, the value of the option (or portfolio) increases 1 percent. Options that are most sensitive to changes in interest rates are those that are at-the-money and with the longest time to expiration.

In mathematical finance, quantities that measure the price sensitivity of a derivative to a change in an underlying parameter are known as the "Greeks." The Greeks are important tools in risk management because they allow a manager, trader, or investor to measure the change in value of an investment or portfolio to a small change in a parameter. More important, this measurement allows the risk to be isolated, thus allowing a manager, trader, or investor to rebalance the portfolio to achieve a desired level of risk relative to that parameter. The most common Greeks are delta, gamma, vega, theta, and rho.

Rho Calculation and Rho in Practice

The exact formula for rho is complicated. But it is calculated as the first derivative of the option's value with respect to the risk-free rate. Rho measures the expected change in an option's price for a 1 percent change in a U.S. Treasury bill's risk-free rate.

For example, assume that a call option is priced at $4 and has a rho of 0.25. If the risk-free rate rises 1 percent, say from 3 percent to 4 percent, the value of the call option would rise from $4 to $4.25.

Call options generally rise in price as interest rates increase and put options generally decrease in price as interest rates increase. Thus, call options have positive rho, while put options have negative rho.

Assume that put option is priced at $9 and has a rho of -0.35. If interest rates were to decrease from 5 percent to 4 percent, then the price of this put option would increase from $9 to $9.35. In this same scenario, assuming the call option mentioned above, its price would decrease from $4 to $3.75.

Rho is larger for options that are in-the-money and decrease steadily as the option changes to become out-of-the-money. Also, rho increases as the time to expiration increases. Long-term equity anticipation securities (LEAPs), which are options that generally have expiration dates that are greater than one year away, are far more sensitive to changes in the risk-free rate and, thus, have larger rho than shorter-term options.

Though rho is a primary input in the Black–Scholes options-pricing model, a change in interest rates generally has a minor overall impact on the pricing of options. Because of this, rho is usually considered to be the least important of all the option Greeks.