Option Rho

Option rho is the estimated change in an option's price, per share, for a one-percentage-point change in interest rates, with the stock price, time to expiration, implied volatility and other inputs held constant. FINRA defines it as "the amount the theoretical price of an options contract is expected to change based on a 1 percentage-point change in interest rates, holding all else constant," and the Options Industry Council calls it "the measure of an option's sensitivity to interest rate changes." It is one of the options Greeks, and for short-dated contracts usually a small one.

Why interest rates affect an option's price. The council lists the interest rate among the pricing inputs behind every option's Greeks, alongside the stock price, strike, time to expiration, implied volatility and anticipated dividends, and notes that the rate assumption may not change at all over a contract's life. A model includes a rate because holding a position, or hedging one, has a financing cost over the option's life. The council explains that rates "are used in pricing models to take into consideration an options price based on its 'hedged value'" and that the effect "has to do with the cost of carrying the position over time." FINRA describes the same idea from the investor's side: a change in rates "can lead to a greater 'cost of carry,' or a greater opportunity cost associated with making the trade versus pursuing another investment."

The sign. The council states that "Rho is positive for purchased calls as higher interest rates increase call premiums" and "negative for purchased puts as higher interest rates decrease put premiums." Sold options reverse it. The council's summary groups a short call with a long put as positions that "would have a negative reaction to increasing interest rates," and says a position hedged by selling stock short, which is how a long call or a short put is hedged, "would be negatively impacted by a decrease in rates."

When it matters. FINRA notes that rho "typically matters most for longer-term options," and the council adds that "the higher the price of the stock and the longer time until expiration generally means a greater sensitivity to changes in interest rates." A two-year option on a high-priced stock can have a rho large enough to notice; a one-month option on the same stock usually has a rho of a few cents. For short-dated positions the stock's price, time passing and implied volatility usually matter far more.

The rate is not one number. The council cautions that "interest rates are not the same industry wide" and "vary from clearing firm to clearing firm, brokerage house to brokerage house and even account to account." The risk-free rate a pricing model uses is an assumption, which is one reason two platforms can show slightly different prices and Greeks for the same contract. Rho should not be read as a bond-style measure of interest rate risk: it describes an option's sensitivity to the rate in its pricing model, not the price behavior of a fixed-income holding.

Rho is quoted per share, for a one-percentage-point change in interest rates. Multiply it by the change in rates, in points, and by the number of shares the contracts cover.

Consider an example with illustrative figures. A stock trades at $160. A call with a $160 strike and two years to expiration is quoted at $29.95 with a rho of 1.50, while the put at the same strike and expiration has a rho of about −1.46.

• If the rate in the pricing model rises from 4% to 5% and nothing else changes, the call should gain about $1.50 a share, to about $31.45, or $150 on one 100-share contract.
• The put should lose about $1.46 a share, or $146 a contract. - A writer of the two-year call has a rho of −1.50 on it, so the same rise in rates costs the writer about $150 a contract.
• A call at the same strike with 30 days to run might have a rho of about 0.07, so the same one-point rise would add only about $7 a contract.

In practice rates rarely move a full point while everything else stands still, and a change in rates usually arrives alongside moves in the stock and in implied volatility that matter more.

Pros

• Shows how much a long-dated option position depends on the level of interest rates.
• Explains why calls and puts on the same stock respond in opposite directions to a change in rates.
• Is small and easy to ignore for short-dated options, which simplifies reading their Greeks.

Cons

• It depends on the interest rate the pricing model assumes, and firms use different rates.
• It assumes rates change while the stock, time and implied volatility stay fixed.
• Rates rarely move a full point quickly, and over the weeks or months a move takes, the stock, time and implied volatility all change too.